Examples involving exponential function from Albert Rich testsuite.
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-- Integrands of the form x^m F^(c (a+b x)^n)
-- n>0
integrate(f^(c*(a+b*x)^2)*x^m, x)
x 2 2 2
++ m (%A b + 2 %A a b + a )c
(1) | %A f d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0.06 (OT) = 0.07 sec
integrate(f^(c*(a+b*x)^2)*x^3, x)
(2)
+------------+
| 2
3 2 2 +---+ (b x + a)\|- b c log(f)
(- 2 a b c log(f) + 3 a b c log(f))\|%pi erf(------------------------)
b
+
2 2 2
((2 b c x - 2 a b c x + 2 a c)log(f) - 2)
*
2 2 2 +------------+
(b c x + 2 a b c x + a c)log(f) | 2
%e \|- b c log(f)
/
+------------+
4 2 2 | 2
4 b c log(f) \|- b c log(f)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^2)*x^2, x)
(3)
+------------+
| 2
2 +---+ (b x + a)\|- b c log(f)
(2 a b c log(f) - b)\|%pi erf(------------------------)
b
+
2 2 2 +------------+
(b c x + 2 a b c x + a c)log(f) | 2
(2 b x - 2 a)%e \|- b c log(f)
/
+------------+
3 | 2
4 b c log(f)\|- b c log(f)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^2)*x, x)
(4)
+------------+
| 2
+---+ (b x + a)\|- b c log(f)
- a b c log(f)\|%pi erf(------------------------)
b
+
2 2 2 +------------+
(b c x + 2 a b c x + a c)log(f) | 2
%e \|- b c log(f)
/
+------------+
2 | 2
2 b c log(f)\|- b c log(f)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^2), x)
+------------+
| 2
+---+ (b x + a)\|- b c log(f)
\|%pi erf(------------------------)
b
(5) -----------------------------------
+------------+
| 2
2 \|- b c log(f)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^2)/x, x)
2 2 2
x (%A b + 2 %A a b + a )c
++ f
(6) | ------------------------- d%A
++ %A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^2)/x^2, x)
2 2 2
x (%A b + 2 %A a b + a )c
++ f
(7) | ------------------------- d%A
++ 2
%A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^2)/x^3, x)
2 2 2
x (%A b + 2 %A a b + a )c
++ f
(8) | ------------------------- d%A
++ 3
%A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^3)*x^m, x)
x 3 3 2 2 2 3
++ m (%A b + 3 %A a b + 3 %A a b + a )c
(9) | %A f d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^3)*x^2, x)
(10)
3 3 2 2 2 3 +------------+2
(b c x + 3 a b c x + 3 a b c x + a c)log(f) 3| 3
%e \|- b c log(f)
+
-
2 1 3 3 2 2 2 3
a b c log(f)Gamma(-,(- b c x - 3 a b c x - 3 a b c x - a c)log(f))
3
*
+------------+
3| 3
\|- b c log(f)
+
2 2 3 3 2 2 2 3
2 a b c log(f)Gamma(-,(- b c x - 3 a b c x - 3 a b c x - a c)log(f))
3
/
+------------+2
3 3| 3
3 b c log(f)\|- b c log(f)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^3)*x, x)
(11)
+------------+
1 3 3 2 2 2 3 3| 3
a Gamma(-,(- b c x - 3 a b c x - 3 a b c x - a c)log(f))\|- b c log(f)
3
+
2 3 3 2 2 2 3
- b Gamma(-,(- b c x - 3 a b c x - 3 a b c x - a c)log(f))
3
/
+------------+2
3| 3
3 b \|- b c log(f)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^3), x)
1 3 3 2 2 2 3
Gamma(-,(- b c x - 3 a b c x - 3 a b c x - a c)log(f))
3
(12) - --------------------------------------------------------
+------------+
3| 3
3 \|- b c log(f)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^3)/x, x)
3 3 2 2 2 3
x (%A b + 3 %A a b + 3 %A a b + a )c
++ f
(13) | ------------------------------------- d%A
++ %A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^3)/x^2, x)
3 3 2 2 2 3
x (%A b + 3 %A a b + 3 %A a b + a )c
++ f
(14) | ------------------------------------- d%A
++ 2
%A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^3)/x^3, x)
3 3 2 2 2 3
x (%A b + 3 %A a b + 3 %A a b + a )c
++ f
(15) | ------------------------------------- d%A
++ 3
%A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)*x^m, x)
x 3 3 2 2 2 3
++ %A b + 3 %A a b + 3 %A a b + a m
(16) | %e %A d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)*x^4, x)
(17)
3 3 2 2 2 3 +----+2
2 2 2 b x + 3 a b x + 3 a b x + a 3| 3
(3 b x - 6 a b x + 9 a )%e \|- b
+
+----+
4 1 3 3 2 2 2 3 3| 3
(- 3 a - 4 a)b Gamma(-,- b x - 3 a b x - 3 a b x - a )\|- b
3
+
3 2 2 3 3 2 2 2 3
(12 a + 2)b Gamma(-,- b x - 3 a b x - 3 a b x - a )
3
/
+----+2
5 3| 3
9 b \|- b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)*x^3, x)
(18)
3 3 2 2 2 3 +----+2
b x + 3 a b x + 3 a b x + a 3| 3
(3 b x - 6 a)%e \|- b
+
+----+
3 1 3 3 2 2 2 3 3| 3
(3 a + 1)b Gamma(-,- b x - 3 a b x - 3 a b x - a )\|- b
3
+
2 2 2 3 3 2 2 2 3
- 9 a b Gamma(-,- b x - 3 a b x - 3 a b x - a )
3
/
+----+2
4 3| 3
9 b \|- b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)*x^2, x)
(19)
3 3 2 2 2 3 +----+2
b x + 3 a b x + 3 a b x + a 3| 3
%e \|- b
+
+----+
2 1 3 3 2 2 2 3 3| 3
- a b Gamma(-,- b x - 3 a b x - 3 a b x - a )\|- b
3
+
2 2 3 3 2 2 2 3
2 a b Gamma(-,- b x - 3 a b x - 3 a b x - a )
3
/
+----+2
3 3| 3
3 b \|- b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)*x, x)
(20)
+----+
1 3 3 2 2 2 3 3| 3
a Gamma(-,- b x - 3 a b x - 3 a b x - a )\|- b
3
+
2 3 3 2 2 2 3
- b Gamma(-,- b x - 3 a b x - 3 a b x - a )
3
/
+----+2
3| 3
3 b \|- b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3), x)
1 3 3 2 2 2 3
Gamma(-,- b x - 3 a b x - 3 a b x - a )
3
(21) - -----------------------------------------
+----+
3| 3
3 \|- b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)/x, x)
3 3 2 2 2 3
x %A b + 3 %A a b + 3 %A a b + a
++ %e
(22) | ----------------------------------- d%A
++ %A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^sqrt(5+3*x), x)
+-------+
+-------+ \|3 x + 5
(2 \|3 x + 5 - 2)%e
(23) ------------------------------
3
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
-- n<0
integrate(f^(c/(a+b*x))*x^m, x)
c
x --------
++ m %A b + a
(24) | %A f d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x))*x^4, x)
(25)
4 4 4 2 3 2 3 2 3 3
(b c x + a c )log(f) + (b c x - 18 a b c x - 19 a c )log(f)
+
3 2 3 2 2 2 2 2 3 2 2
(2 b c x - 14 a b c x + 86 a b c x + 102 a c )log(f)
+
4 4 3 3 2 2 2 3 4
(6 b c x - 16 a b c x + 36 a b c x - 96 a b c x - 154 a c)log(f)
+
5 5 5
24 b x + 24 a
*
c log(f)
--------
b x + a
%e
+
5 5 4 4 2 3 3 3 2 2
- c log(f) + 20 a c log(f) - 120 a c log(f) + 240 a c log(f)
+
4
- 120 a c log(f)
*
c log(f)
Ei(--------)
b x + a
/
5
120 b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x))*x^3, x)
(26)
3 3 3 2 2 2 2 2 2 2
(b c x + a c )log(f) + (b c x - 10 a b c x - 11 a c )log(f)
+
3 3 2 2 2 3 4 4 4
(2 b c x - 6 a b c x + 18 a b c x + 26 a c)log(f) + 6 b x - 6 a
*
c log(f)
--------
b x + a
%e
+
4 4 3 3 2 2 2 3 c log(f)
(- c log(f) + 12 a c log(f) - 36 a c log(f) + 24 a c log(f))Ei(--------)
b x + a
/
4
24 b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x))*x^2, x)
(27)
2 2 2 2 2 2 3 3
(b c x + a c )log(f) + (b c x - 4 a b c x - 5 a c)log(f) + 2 b x
+
3
2 a
*
c log(f)
--------
b x + a
%e
+
3 3 2 2 2 c log(f)
(- c log(f) + 6 a c log(f) - 6 a c log(f))Ei(--------)
b x + a
/
3
6 b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x))*x, x)
(28)
c log(f)
--------
2 2 2 b x + a
((b c x + a c)log(f) + b x - a )%e
+
2 2 c log(f)
(- c log(f) + 2 a c log(f))Ei(--------)
b x + a
/
2
2 b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)), x)
c log(f)
--------
b x + a c log(f)
(b x + a)%e - c log(f)Ei(--------)
b x + a
(29) ------------------------------------------
b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x))/x, x)
c log(f)
--------
b c x log(f) a c log(f)
(30) Ei(- ------------)%e - Ei(--------)
2 b x + a
a b x + a
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x))/x^2, x)
c log(f) c log(f)
-------- --------
b c x log(f) a 2 b x + a
- b c x log(f)Ei(- ------------)%e + (- a b x - a )%e
2
a b x + a
(31) ---------------------------------------------------------------------
2
a x
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x))/x^3, x)
(32)
c log(f)
--------
2 2 2 2 2 2 b c x log(f) a
(b c x log(f) + 2 a b c x log(f))Ei(- ------------)%e
2
a b x + a
+
c log(f)
--------
2 2 2 2 2 2 4 b x + a
((a b c x + a b c x)log(f) + a b x - a )%e
/
4 2
2 a x
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^2)*x^m, x)
c
---------------------
x 2 2 2
++ m %A b + 2 %A a b + a
(33) | %A f d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^2)*x^4, x)
(34)
+----------+
| c log(f)
b |- --------
| 2
3 3 2 2 2 4 +---+ \| b
(- 4 c log(f) - 60 a c log(f) - 15 a c log(f))\|%pi erf(--------------)
b x + a
+
2 2 2 2
(4 b c x + 4 a b c )log(f)
+
4 3 3 2 2 2 3 6 5
(2 b c x - 9 a b c x + 36 a b c x + 47 a b c)log(f) + 3 b x
+
5
3 a b
*
c log(f)
-------------------
2 2 2
b x + 2 a b x + a
%e
+
2 2 3 c log(f)
(15 a b c log(f) + 30 a b c log(f))Ei(-------------------)
2 2 2
b x + 2 a b x + a
*
+----------+
| c log(f)
|- --------
| 2
\| b
/
+----------+
6 | c log(f)
15 b |- --------
| 2
\| b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^2)*x^3, x)
(35)
+----------+
| c log(f)
b |- --------
| 2
2 2 3 +---+ \| b
(8 a c log(f) + 4 a c log(f))\|%pi erf(--------------)
b x + a
+
3 2 2 2 5 4 4
((b c x - 6 a b c x - 7 a b c)log(f) + b x - a b)
*
c log(f)
-------------------
2 2 2
b x + 2 a b x + a
%e
+
2 2 2 c log(f)
(- b c log(f) - 6 a b c log(f))Ei(-------------------)
2 2 2
b x + 2 a b x + a
*
+----------+
| c log(f)
|- --------
| 2
\| b
/
+----------+
5 | c log(f)
4 b |- --------
| 2
\| b
Type: Union(Expression(Integer),...)
fricas
Time: 0.02 sec
integrate(f^(c/(a+b*x)^2)*x^2, x)
(36)
+----------+
| c log(f)
b |- --------
| 2
2 2 2 +---+ \| b
(- 2 c log(f) - 3 a c log(f))\|%pi erf(--------------)
b x + a
+
c log(f)
-------------------
2 2 2
2 4 3 3 b x + 2 a b x + a
((2 b c x + 2 a b c)log(f) + b x + a b)%e
+
c log(f)
3 a b c log(f)Ei(-------------------)
2 2 2
b x + 2 a b x + a
*
+----------+
| c log(f)
|- --------
| 2
\| b
/
+----------+
4 | c log(f)
3 b |- --------
| 2
\| b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^2)*x, x)
(37)
+----------+
| c log(f)
b |- --------
| 2
+---+ \| b
2 a c log(f)\|%pi erf(--------------)
b x + a
+
c log(f)
-------------------
2 2 2
3 2 2 b x + 2 a b x + a c log(f)
((b x - a b)%e - b c log(f)Ei(-------------------))
2 2 2
b x + 2 a b x + a
*
+----------+
| c log(f)
|- --------
| 2
\| b
/
+----------+
3 | c log(f)
2 b |- --------
| 2
\| b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^2), x)
(38)
+----------+
| c log(f)
b |- --------
| 2
+---+ \| b
- c log(f)\|%pi erf(--------------)
b x + a
+
c log(f)
-------------------
2 2 2 +----------+
2 b x + 2 a b x + a | c log(f)
(b x + a b)%e |- --------
| 2
\| b
/
+----------+
2 | c log(f)
b |- --------
| 2
\| b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^2)/x, x)
c
---------------------
2 2 2
x %A b + 2 %A a b + a
++ f
(39) | ---------------------- d%A
++ %A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^2)/x^2, x)
c
---------------------
2 2 2
x %A b + 2 %A a b + a
++ f
(40) | ---------------------- d%A
++ 2
%A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^2)/x^3, x)
c
---------------------
2 2 2
x %A b + 2 %A a b + a
++ f
(41) | ---------------------- d%A
++ 3
%A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^3)*x^m, x)
c
---------------------------------
x 3 3 2 2 2 3
++ m %A b + 3 %A a b + 3 %A a b + a
(42) | %A f d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^3)*x^4, x)
(43)
4 2 3 2 2 7 5 5 2
((3 b c x - 24 a b c x - 27 a b c)log(f) + 2 b x + 2 a b )
*
c log(f)
------------------------------
3 3 2 2 2 3
b x + 3 a b x + 3 a b x + a
%e
+
2 2 c log(f)
- 20 a b c log(f)Ei(------------------------------)
3 3 2 2 2 3
b x + 3 a b x + 3 a b x + a
*
+----------+2
| c log(f)
|- --------
3| 3
\| b
+
2 2 3
(3 b c log(f) - 20 a b c log(f))
*
+----------+
1 c log(f) | c log(f)
Gamma(-,- ------------------------------) |- --------
3 3 3 2 2 2 3 3| 3
b x + 3 a b x + 3 a b x + a \| b
+
2 2 4 2 c log(f)
(- 30 a c log(f) + 10 a c log(f))Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
b x + 3 a b x + 3 a b x + a
/
+----------+2
7 | c log(f)
10 b |- --------
3| 3
\| b
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(f^(c/(a+b*x)^3)*x^3, x)
(44)
3 2 6 4 4 2
((3 b c x + 3 a b c)log(f) + b x - a b )
*
c log(f)
------------------------------
3 3 2 2 2 3
b x + 3 a b x + 3 a b x + a
%e
+
2 c log(f)
4 a b c log(f)Ei(------------------------------)
3 3 2 2 2 3
b x + 3 a b x + 3 a b x + a
*
+----------+2
| c log(f)
|- --------
3| 3
\| b
+
+----------+
2 1 c log(f) | c log(f)
6 a b c log(f)Gamma(-,- ------------------------------) |- --------
3 3 3 2 2 2 3 3| 3
b x + 3 a b x + 3 a b x + a \| b
+
2 2 3 2 c log(f)
(3 c log(f) - 4 a c log(f))Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
b x + 3 a b x + 3 a b x + a
/
+----------+2
6 | c log(f)
4 b |- --------
3| 3
\| b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^3)*x^2, x)
(45)
c log(f)
------------------------------
3 3 2 2 2 3
5 3 3 2 b x + 3 a b x + 3 a b x + a
(b x + a b )%e
+
2 c log(f)
- b c log(f)Ei(------------------------------)
3 3 2 2 2 3
b x + 3 a b x + 3 a b x + a
*
+----------+2
| c log(f)
|- --------
3| 3
\| b
+
+----------+
1 c log(f) | c log(f)
- 3 a b c log(f)Gamma(-,- ------------------------------) |- --------
3 3 3 2 2 2 3 3| 3
b x + 3 a b x + 3 a b x + a \| b
+
2 2 c log(f)
3 a c log(f)Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
b x + 3 a b x + 3 a b x + a
/
+----------+2
5 | c log(f)
3 b |- --------
3| 3
\| b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^3)*x, x)
(46)
c log(f)
------------------------------
3 3 2 2 2 3 +----------+2
4 2 2 2 b x + 3 a b x + 3 a b x + a | c log(f)
(b x - a b )%e |- --------
3| 3
\| b
+
+----------+
1 c log(f) | c log(f)
b c log(f)Gamma(-,- ------------------------------) |- --------
3 3 3 2 2 2 3 3| 3
b x + 3 a b x + 3 a b x + a \| b
+
2 c log(f)
- 2 a c log(f)Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
b x + 3 a b x + 3 a b x + a
/
+----------+2
4 | c log(f)
2 b |- --------
3| 3
\| b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^3), x)
(47)
c log(f)
------------------------------
3 3 2 2 2 3 +----------+2
3 2 b x + 3 a b x + 3 a b x + a | c log(f)
(b x + a b )%e |- --------
3| 3
\| b
+
2 c log(f)
c log(f)Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
b x + 3 a b x + 3 a b x + a
/
+----------+2
3 | c log(f)
b |- --------
3| 3
\| b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^3)/x, x)
c
---------------------------------
3 3 2 2 2 3
x %A b + 3 %A a b + 3 %A a b + a
++ f
(48) | ---------------------------------- d%A
++ %A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^3)/x^2, x)
c
---------------------------------
3 3 2 2 2 3
x %A b + 3 %A a b + 3 %A a b + a
++ f
(49) | ---------------------------------- d%A
++ 2
%A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c/(a+b*x)^3)/x^3, x)
c
---------------------------------
3 3 2 2 2 3
x %A b + 3 %A a b + 3 %A a b + a
++ f
(50) | ---------------------------------- d%A
++ 3
%A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
-- n symbolic
integrate(f^(c*(a+b*x)^n)*x^m, x)
x n
++ m c (%A b + a)
(51) | %A f d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^n)*x^3, x)
x n
++ 3 c (%A b + a)
(52) | %A f d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^n)*x^2, x)
x n
++ 2 c (%A b + a)
(53) | %A f d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^n)*x, x)
x n
++ c (%A b + a)
(54) | %A f d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^n), x)
x n
++ c (%A b + a)
(55) | f d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^n)/x, x)
n
x c (%A b + a)
++ f
(56) | -------------- d%A
++ %A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^n)/x^2, x)
n
x c (%A b + a)
++ f
(57) | -------------- d%A
++ 2
%A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(c*(a+b*x)^n)/x^3, x)
n
x c (%A b + a)
++ f
(58) | -------------- d%A
++ 3
%A
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
-- Integrands of the form (c+d x)^m F^(a+b (c+d x)^n)
-- n>0
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^m, x)
(59)
m + 1 2 2 2
Gamma(-----,(- b d x - 2 b c d x - b c )log(F))
2
*
(- m + 1)log(- b log(F)) + 2 a log(F)
-------------------------------------
2
%e
/
2 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^11, x)
(60)
5 10 10 5 9 9 5 2 8 8 5 3 7 7 5 4 6 6
b d x + 10 b c d x + 45 b c d x + 120 b c d x + 210 b c d x
+
5 5 5 5 5 6 4 4 5 7 3 3 5 8 2 2
252 b c d x + 210 b c d x + 120 b c d x + 45 b c d x
+
5 9 5 10
10 b c d x + b c
*
5
log(F)
+
4 8 8 4 7 7 4 2 6 6 4 3 5 5
- 5 b d x - 40 b c d x - 140 b c d x - 280 b c d x
+
4 4 4 4 4 5 3 3 4 6 2 2 4 7 4 8
- 350 b c d x - 280 b c d x - 140 b c d x - 40 b c d x - 5 b c
*
4
log(F)
+
3 6 6 3 5 5 3 2 4 4 3 3 3 3
20 b d x + 120 b c d x + 300 b c d x + 400 b c d x
+
3 4 2 2 3 5 3 6
300 b c d x + 120 b c d x + 20 b c
*
3
log(F)
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4
(- 60 b d x - 240 b c d x - 360 b c d x - 240 b c d x - 60 b c )
*
2
log(F)
+
2 2 2
(120 b d x + 240 b c d x + 120 b c )log(F) - 120
*
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
%e
/
6 6
2 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^9, x)
(61)
4 8 8 4 7 7 4 2 6 6 4 3 5 5 4 4 4 4
b d x + 8 b c d x + 28 b c d x + 56 b c d x + 70 b c d x
+
4 5 3 3 4 6 2 2 4 7 4 8
56 b c d x + 28 b c d x + 8 b c d x + b c
*
4
log(F)
+
3 6 6 3 5 5 3 2 4 4 3 3 3 3 3 4 2 2
- 4 b d x - 24 b c d x - 60 b c d x - 80 b c d x - 60 b c d x
+
3 5 3 6
- 24 b c d x - 4 b c
*
3
log(F)
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(12 b d x + 48 b c d x + 72 b c d x + 48 b c d x + 12 b c )log(F)
+
2 2 2
(- 24 b d x - 48 b c d x - 24 b c )log(F) + 24
*
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
%e
/
5 5
2 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^7, x)
(62)
3 6 6 3 5 5 3 2 4 4 3 3 3 3 3 4 2 2
b d x + 6 b c d x + 15 b c d x + 20 b c d x + 15 b c d x
+
3 5 3 6
6 b c d x + b c
*
3
log(F)
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(- 3 b d x - 12 b c d x - 18 b c d x - 12 b c d x - 3 b c )log(F)
+
2 2 2
(6 b d x + 12 b c d x + 6 b c )log(F) - 6
*
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
%e
/
4 4
2 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^5, x)
(63)
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(b d x + 4 b c d x + 6 b c d x + 4 b c d x + b c )log(F)
+
2 2 2
(- 2 b d x - 4 b c d x - 2 b c )log(F) + 2
*
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
%e
/
3 3
2 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^3, x)
(64)
2 2 2
((b d x + 2 b c d x + b c )log(F) - 1)
*
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
%e
/
2 2
2 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x), x)
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
%e
(65) ---------------------------------------
2 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x), x)
2 2 2 a log(F)
Ei((b d x + 2 b c d x + b c )log(F))%e
(66) -----------------------------------------------
2 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^3, x)
(67)
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
- %e
+
2 2 2 2 2 2
(b d x + 2 b c d x + b c )log(F)Ei((b d x + 2 b c d x + b c )log(F))
*
a log(F)
%e
/
3 2 2 2
2 d x + 4 c d x + 2 c d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^5, x)
(68)
2 2 2
((- b d x - 2 b c d x - b c )log(F) - 1)
*
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
%e
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(b d x + 4 b c d x + 6 b c d x + 4 b c d x + b c )log(F)
*
2 2 2 a log(F)
Ei((b d x + 2 b c d x + b c )log(F))%e
/
5 4 4 3 2 3 2 3 2 4
4 d x + 16 c d x + 24 c d x + 16 c d x + 4 c d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^7, x)
(69)
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(- b d x - 4 b c d x - 6 b c d x - 4 b c d x - b c )log(F)
+
2 2 2
(- b d x - 2 b c d x - b c )log(F) - 2
*
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
%e
+
3 6 6 3 5 5 3 2 4 4 3 3 3 3 3 4 2 2
b d x + 6 b c d x + 15 b c d x + 20 b c d x + 15 b c d x
+
3 5 3 6
6 b c d x + b c
*
3 2 2 2 a log(F)
log(F) Ei((b d x + 2 b c d x + b c )log(F))%e
/
7 6 6 5 2 5 4 3 4 3 4 3 2 5 2
12 d x + 72 c d x + 180 c d x + 240 c d x + 180 c d x + 72 c d x
+
6
12 c d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^9, x)
(70)
3 6 6 3 5 5 3 2 4 4 3 3 3 3 3 4 2 2
- b d x - 6 b c d x - 15 b c d x - 20 b c d x - 15 b c d x
+
3 5 3 6
- 6 b c d x - b c
*
3
log(F)
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(- b d x - 4 b c d x - 6 b c d x - 4 b c d x - b c )log(F)
+
2 2 2
(- 2 b d x - 4 b c d x - 2 b c )log(F) - 6
*
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
%e
+
4 8 8 4 7 7 4 2 6 6 4 3 5 5 4 4 4 4
b d x + 8 b c d x + 28 b c d x + 56 b c d x + 70 b c d x
+
4 5 3 3 4 6 2 2 4 7 4 8
56 b c d x + 28 b c d x + 8 b c d x + b c
*
4 2 2 2 a log(F)
log(F) Ei((b d x + 2 b c d x + b c )log(F))%e
/
9 8 8 7 2 7 6 3 6 5 4 5 4
48 d x + 384 c d x + 1344 c d x + 2688 c d x + 3360 c d x
+
5 4 3 6 3 2 7 2 8
2688 c d x + 1344 c d x + 384 c d x + 48 c d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^11, x)
(71)
4 8 8 4 7 7 4 2 6 6 4 3 5 5 4 4 4 4
- b d x - 8 b c d x - 28 b c d x - 56 b c d x - 70 b c d x
+
4 5 3 3 4 6 2 2 4 7 4 8
- 56 b c d x - 28 b c d x - 8 b c d x - b c
*
4
log(F)
+
3 6 6 3 5 5 3 2 4 4 3 3 3 3 3 4 2 2
- b d x - 6 b c d x - 15 b c d x - 20 b c d x - 15 b c d x
+
3 5 3 6
- 6 b c d x - b c
*
3
log(F)
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(- 2 b d x - 8 b c d x - 12 b c d x - 8 b c d x - 2 b c )log(F)
+
2 2 2
(- 6 b d x - 12 b c d x - 6 b c )log(F) - 24
*
2 2 2
(b d x + 2 b c d x + b c + a)log(F)
%e
+
5 10 10 5 9 9 5 2 8 8 5 3 7 7 5 4 6 6
b d x + 10 b c d x + 45 b c d x + 120 b c d x + 210 b c d x
+
5 5 5 5 5 6 4 4 5 7 3 3 5 8 2 2 5 9
252 b c d x + 210 b c d x + 120 b c d x + 45 b c d x + 10 b c d x
+
5 10
b c
*
5 2 2 2 a log(F)
log(F) Ei((b d x + 2 b c d x + b c )log(F))%e
/
11 10 10 9 2 9 8 3 8 7 4 7 6
240 d x + 2400 c d x + 10800 c d x + 28800 c d x + 50400 c d x
+
5 6 5 6 5 4 7 4 3 8 3 2 9 2
60480 c d x + 50400 c d x + 28800 c d x + 10800 c d x + 2400 c d x
+
10
240 c d
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^12, x)
(72)
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
10395 d\|%pi %e erf(------------------------)
d
+
5 11 11 5 10 10 5 2 9 9 5 3 8 8
64 b d x + 704 b c d x + 3520 b c d x + 10560 b c d x
+
5 4 7 7 5 5 6 6 5 6 5 5 5 7 4 4
21120 b c d x + 29568 b c d x + 29568 b c d x + 21120 b c d x
+
5 8 3 3 5 9 2 2 5 10 5 11
10560 b c d x + 3520 b c d x + 704 b c d x + 64 b c
*
5
log(F)
+
4 9 9 4 8 8 4 2 7 7 4 3 6 6
- 352 b d x - 3168 b c d x - 12672 b c d x - 29568 b c d x
+
4 4 5 5 4 5 4 4 4 6 3 3
- 44352 b c d x - 44352 b c d x - 29568 b c d x
+
4 7 2 2 4 8 4 9
- 12672 b c d x - 3168 b c d x - 352 b c
*
4
log(F)
+
3 7 7 3 6 6 3 2 5 5 3 3 4 4
1584 b d x + 11088 b c d x + 33264 b c d x + 55440 b c d x
+
3 4 3 3 3 5 2 2 3 6 3 7
55440 b c d x + 33264 b c d x + 11088 b c d x + 1584 b c
*
3
log(F)
+
2 5 5 2 4 4 2 2 3 3 2 3 2 2
- 5544 b d x - 27720 b c d x - 55440 b c d x - 55440 b c d x
+
2 4 2 5
- 27720 b c d x - 5544 b c
*
2
log(F)
+
3 3 2 2 2 3
(13860 b d x + 41580 b c d x + 41580 b c d x + 13860 b c )log(F)
+
- 20790 d x - 20790 c
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
+------------+
6 6 | 2
128 b d log(F) \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^10, x)
(73)
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
- 945 d\|%pi %e erf(------------------------)
d
+
4 9 9 4 8 8 4 2 7 7 4 3 6 6
32 b d x + 288 b c d x + 1152 b c d x + 2688 b c d x
+
4 4 5 5 4 5 4 4 4 6 3 3 4 7 2 2
4032 b c d x + 4032 b c d x + 2688 b c d x + 1152 b c d x
+
4 8 4 9
288 b c d x + 32 b c
*
4
log(F)
+
3 7 7 3 6 6 3 2 5 5 3 3 4 4
- 144 b d x - 1008 b c d x - 3024 b c d x - 5040 b c d x
+
3 4 3 3 3 5 2 2 3 6 3 7
- 5040 b c d x - 3024 b c d x - 1008 b c d x - 144 b c
*
3
log(F)
+
2 5 5 2 4 4 2 2 3 3 2 3 2 2
504 b d x + 2520 b c d x + 5040 b c d x + 5040 b c d x
+
2 4 2 5
2520 b c d x + 504 b c
*
2
log(F)
+
3 3 2 2 2 3
(- 1260 b d x - 3780 b c d x - 3780 b c d x - 1260 b c )log(F)
+
1890 d x + 1890 c
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
+------------+
5 5 | 2
64 b d log(F) \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^8, x)
(74)
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
105 d\|%pi %e erf(------------------------)
d
+
3 7 7 3 6 6 3 2 5 5 3 3 4 4
16 b d x + 112 b c d x + 336 b c d x + 560 b c d x
+
3 4 3 3 3 5 2 2 3 6 3 7
560 b c d x + 336 b c d x + 112 b c d x + 16 b c
*
3
log(F)
+
2 5 5 2 4 4 2 2 3 3 2 3 2 2
- 56 b d x - 280 b c d x - 560 b c d x - 560 b c d x
+
2 4 2 5
- 280 b c d x - 56 b c
*
2
log(F)
+
3 3 2 2 2 3
(140 b d x + 420 b c d x + 420 b c d x + 140 b c )log(F) - 210 d x
+
- 210 c
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
+------------+
4 4 | 2
32 b d log(F) \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^6, x)
(75)
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
- 15 d\|%pi %e erf(------------------------)
d
+
2 5 5 2 4 4 2 2 3 3 2 3 2 2 2 4
8 b d x + 40 b c d x + 80 b c d x + 80 b c d x + 40 b c d x
+
2 5
8 b c
*
2
log(F)
+
3 3 2 2 2 3
(- 20 b d x - 60 b c d x - 60 b c d x - 20 b c )log(F) + 30 d x
+
30 c
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
+------------+
3 3 | 2
16 b d log(F) \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^4, x)
(76)
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
3 d\|%pi %e erf(------------------------)
d
+
3 3 2 2 2 3
((4 b d x + 12 b c d x + 12 b c d x + 4 b c )log(F) - 6 d x - 6 c)
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
+------------+
2 2 | 2
8 b d log(F) \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(c+d*x)^2, x)
(77)
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
- d\|%pi %e erf(------------------------)
d
+
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
(2 d x + 2 c)%e \|- b d log(F)
/
+------------+
| 2
4 b d log(F)\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2), x)
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
\|%pi %e erf(------------------------)
d
(78) ---------------------------------------------
+------------+
| 2
2 \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^2, x)
(79)
+------------+
| 2
2 +---+ a log(F) (d x + c)\|- b d log(F)
(b d x + b c d)log(F)\|%pi %e erf(------------------------)
d
+
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
- %e \|- b d log(F)
/
+------------+
2 | 2
(d x + c d)\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^4, x)
(80)
2 4 3 2 3 2 2 2 2 2 3 2 +---+ a log(F)
(2 b d x + 6 b c d x + 6 b c d x + 2 b c d)log(F) \|%pi %e
*
+------------+
| 2
(d x + c)\|- b d log(F)
erf(------------------------)
d
+
2 2 2
((- 2 b d x - 4 b c d x - 2 b c )log(F) - 1)
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
+------------+
4 3 3 2 2 2 3 | 2
(3 d x + 9 c d x + 9 c d x + 3 c d)\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^6, x)
(81)
3 6 5 3 5 4 3 2 4 3 3 3 3 2 3 4 2
4 b d x + 20 b c d x + 40 b c d x + 40 b c d x + 20 b c d x
+
3 5
4 b c d
*
+------------+
| 2
3 +---+ a log(F) (d x + c)\|- b d log(F)
log(F) \|%pi %e erf(------------------------)
d
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(- 4 b d x - 16 b c d x - 24 b c d x - 16 b c d x - 4 b c )log(F)
+
2 2 2
(- 2 b d x - 4 b c d x - 2 b c )log(F) - 3
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
6 5 5 4 2 4 3 3 3 2 4 2 5
(15 d x + 75 c d x + 150 c d x + 150 c d x + 75 c d x + 15 c d)
*
+------------+
| 2
\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^8, x)
(82)
4 8 7 4 7 6 4 2 6 5 4 3 5 4 4 4 4 3
8 b d x + 56 b c d x + 168 b c d x + 280 b c d x + 280 b c d x
+
4 5 3 2 4 6 2 4 7
168 b c d x + 56 b c d x + 8 b c d
*
+------------+
| 2
4 +---+ a log(F) (d x + c)\|- b d log(F)
log(F) \|%pi %e erf(------------------------)
d
+
3 6 6 3 5 5 3 2 4 4 3 3 3 3
- 8 b d x - 48 b c d x - 120 b c d x - 160 b c d x
+
3 4 2 2 3 5 3 6
- 120 b c d x - 48 b c d x - 8 b c
*
3
log(F)
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(- 4 b d x - 16 b c d x - 24 b c d x - 16 b c d x - 4 b c )log(F)
+
2 2 2
(- 6 b d x - 12 b c d x - 6 b c )log(F) - 15
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
8 7 7 6 2 6 5 3 5 4 4 4 3
105 d x + 735 c d x + 2205 c d x + 3675 c d x + 3675 c d x
+
5 3 2 6 2 7
2205 c d x + 735 c d x + 105 c d
*
+------------+
| 2
\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^10, x)
(83)
5 10 9 5 9 8 5 2 8 7 5 3 7 6
16 b d x + 144 b c d x + 576 b c d x + 1344 b c d x
+
5 4 6 5 5 5 5 4 5 6 4 3 5 7 3 2
2016 b c d x + 2016 b c d x + 1344 b c d x + 576 b c d x
+
5 8 2 5 9
144 b c d x + 16 b c d
*
+------------+
| 2
5 +---+ a log(F) (d x + c)\|- b d log(F)
log(F) \|%pi %e erf(------------------------)
d
+
4 8 8 4 7 7 4 2 6 6 4 3 5 5
- 16 b d x - 128 b c d x - 448 b c d x - 896 b c d x
+
4 4 4 4 4 5 3 3 4 6 2 2 4 7
- 1120 b c d x - 896 b c d x - 448 b c d x - 128 b c d x
+
4 8
- 16 b c
*
4
log(F)
+
3 6 6 3 5 5 3 2 4 4 3 3 3 3
- 8 b d x - 48 b c d x - 120 b c d x - 160 b c d x
+
3 4 2 2 3 5 3 6
- 120 b c d x - 48 b c d x - 8 b c
*
3
log(F)
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4
(- 12 b d x - 48 b c d x - 72 b c d x - 48 b c d x - 12 b c )
*
2
log(F)
+
2 2 2
(- 30 b d x - 60 b c d x - 30 b c )log(F) - 105
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
10 9 9 8 2 8 7 3 7 6 4 6 5
945 d x + 8505 c d x + 34020 c d x + 79380 c d x + 119070 c d x
+
5 5 4 6 4 3 7 3 2 8 2 9
119070 c d x + 79380 c d x + 34020 c d x + 8505 c d x + 945 c d
*
+------------+
| 2
\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(F^(a+b*(c+d*x)^2)/(c+d*x)^12, x)
(84)
6 12 11 6 11 10 6 2 10 9 6 3 9 8
32 b d x + 352 b c d x + 1760 b c d x + 5280 b c d x
+
6 4 8 7 6 5 7 6 6 6 6 5 6 7 5 4
10560 b c d x + 14784 b c d x + 14784 b c d x + 10560 b c d x
+
6 8 4 3 6 9 3 2 6 10 2 6 11
5280 b c d x + 1760 b c d x + 352 b c d x + 32 b c d
*
+------------+
| 2
6 +---+ a log(F) (d x + c)\|- b d log(F)
log(F) \|%pi %e erf(------------------------)
d
+
5 10 10 5 9 9 5 2 8 8 5 3 7 7
- 32 b d x - 320 b c d x - 1440 b c d x - 3840 b c d x
+
5 4 6 6 5 5 5 5 5 6 4 4 5 7 3 3
- 6720 b c d x - 8064 b c d x - 6720 b c d x - 3840 b c d x
+
5 8 2 2 5 9 5 10
- 1440 b c d x - 320 b c d x - 32 b c
*
5
log(F)
+
4 8 8 4 7 7 4 2 6 6 4 3 5 5
- 16 b d x - 128 b c d x - 448 b c d x - 896 b c d x
+
4 4 4 4 4 5 3 3 4 6 2 2 4 7
- 1120 b c d x - 896 b c d x - 448 b c d x - 128 b c d x
+
4 8
- 16 b c
*
4
log(F)
+
3 6 6 3 5 5 3 2 4 4 3 3 3 3
- 24 b d x - 144 b c d x - 360 b c d x - 480 b c d x
+
3 4 2 2 3 5 3 6
- 360 b c d x - 144 b c d x - 24 b c
*
3
log(F)
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4
(- 60 b d x - 240 b c d x - 360 b c d x - 240 b c d x - 60 b c )
*
2
log(F)
+
2 2 2
(- 210 b d x - 420 b c d x - 210 b c )log(F) - 945
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
12 11 11 10 2 10 9 3 9 8
10395 d x + 114345 c d x + 571725 c d x + 1715175 c d x
+
4 8 7 5 7 6 6 6 5 7 5 4
3430350 c d x + 4802490 c d x + 4802490 c d x + 3430350 c d x
+
8 4 3 9 3 2 10 2 11
1715175 c d x + 571725 c d x + 114345 c d x + 10395 c d
*
+------------+
| 2
\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.02 (EV) = 0.02 sec
integrate(F^(a+b*(c+d*x)^3)*(c+d*x)^m, x)
(85)
m + 1 3 3 2 2 2 3
Gamma(-----,(- b d x - 3 b c d x - 3 b c d x - b c )log(F))
3
*
(- m + 2)log(- b log(F)) + 3 a log(F)
-------------------------------------
3
%e
/
3 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)*(c+d*x)^17, x)
(86)
5 15 15 5 14 14 5 2 13 13 5 3 12 12
b d x + 15 b c d x + 105 b c d x + 455 b c d x
+
5 4 11 11 5 5 10 10 5 6 9 9 5 7 8 8
1365 b c d x + 3003 b c d x + 5005 b c d x + 6435 b c d x
+
5 8 7 7 5 9 6 6 5 10 5 5 5 11 4 4
6435 b c d x + 5005 b c d x + 3003 b c d x + 1365 b c d x
+
5 12 3 3 5 13 2 2 5 14 5 15
455 b c d x + 105 b c d x + 15 b c d x + b c
*
5
log(F)
+
4 12 12 4 11 11 4 2 10 10 4 3 9 9
- 5 b d x - 60 b c d x - 330 b c d x - 1100 b c d x
+
4 4 8 8 4 5 7 7 4 6 6 6 4 7 5 5
- 2475 b c d x - 3960 b c d x - 4620 b c d x - 3960 b c d x
+
4 8 4 4 4 9 3 3 4 10 2 2 4 11
- 2475 b c d x - 1100 b c d x - 330 b c d x - 60 b c d x
+
4 12
- 5 b c
*
4
log(F)
+
3 9 9 3 8 8 3 2 7 7 3 3 6 6
20 b d x + 180 b c d x + 720 b c d x + 1680 b c d x
+
3 4 5 5 3 5 4 4 3 6 3 3 3 7 2 2
2520 b c d x + 2520 b c d x + 1680 b c d x + 720 b c d x
+
3 8 3 9
180 b c d x + 20 b c
*
3
log(F)
+
2 6 6 2 5 5 2 2 4 4 2 3 3 3
- 60 b d x - 360 b c d x - 900 b c d x - 1200 b c d x
+
2 4 2 2 2 5 2 6
- 900 b c d x - 360 b c d x - 60 b c
*
2
log(F)
+
3 3 2 2 2 3
(120 b d x + 360 b c d x + 360 b c d x + 120 b c )log(F) - 120
*
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
%e
/
6 6
3 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(F^(a+b*(c+d*x)^3)*(c+d*x)^14, x)
(87)
4 12 12 4 11 11 4 2 10 10 4 3 9 9
b d x + 12 b c d x + 66 b c d x + 220 b c d x
+
4 4 8 8 4 5 7 7 4 6 6 6 4 7 5 5
495 b c d x + 792 b c d x + 924 b c d x + 792 b c d x
+
4 8 4 4 4 9 3 3 4 10 2 2 4 11 4 12
495 b c d x + 220 b c d x + 66 b c d x + 12 b c d x + b c
*
4
log(F)
+
3 9 9 3 8 8 3 2 7 7 3 3 6 6
- 4 b d x - 36 b c d x - 144 b c d x - 336 b c d x
+
3 4 5 5 3 5 4 4 3 6 3 3 3 7 2 2
- 504 b c d x - 504 b c d x - 336 b c d x - 144 b c d x
+
3 8 3 9
- 36 b c d x - 4 b c
*
3
log(F)
+
2 6 6 2 5 5 2 2 4 4 2 3 3 3
12 b d x + 72 b c d x + 180 b c d x + 240 b c d x
+
2 4 2 2 2 5 2 6
180 b c d x + 72 b c d x + 12 b c
*
2
log(F)
+
3 3 2 2 2 3
(- 24 b d x - 72 b c d x - 72 b c d x - 24 b c )log(F) + 24
*
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
%e
/
5 5
3 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)*(c+d*x)^11, x)
(88)
3 9 9 3 8 8 3 2 7 7 3 3 6 6 3 4 5 5
b d x + 9 b c d x + 36 b c d x + 84 b c d x + 126 b c d x
+
3 5 4 4 3 6 3 3 3 7 2 2 3 8 3 9
126 b c d x + 84 b c d x + 36 b c d x + 9 b c d x + b c
*
3
log(F)
+
2 6 6 2 5 5 2 2 4 4 2 3 3 3 2 4 2 2
- 3 b d x - 18 b c d x - 45 b c d x - 60 b c d x - 45 b c d x
+
2 5 2 6
- 18 b c d x - 3 b c
*
2
log(F)
+
3 3 2 2 2 3
(6 b d x + 18 b c d x + 18 b c d x + 6 b c )log(F) - 6
*
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
%e
/
4 4
3 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)*(c+d*x)^8, x)
(89)
2 6 6 2 5 5 2 2 4 4 2 3 3 3 2 4 2 2
b d x + 6 b c d x + 15 b c d x + 20 b c d x + 15 b c d x
+
2 5 2 6
6 b c d x + b c
*
2
log(F)
+
3 3 2 2 2 3
(- 2 b d x - 6 b c d x - 6 b c d x - 2 b c )log(F) + 2
*
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
%e
/
3 3
3 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)*(c+d*x)^5, x)
(90)
3 3 2 2 2 3
((b d x + 3 b c d x + 3 b c d x + b c )log(F) - 1)
*
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
%e
/
2 2
3 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)*(c+d*x)^2, x)
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
%e
(91) ----------------------------------------------------
3 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)/(c+d*x), x)
3 3 2 2 2 3 a log(F)
Ei((b d x + 3 b c d x + 3 b c d x + b c )log(F))%e
(92) ------------------------------------------------------------
3 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)/(c+d*x)^4, x)
(93)
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
- %e
+
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c )log(F)
*
3 3 2 2 2 3 a log(F)
Ei((b d x + 3 b c d x + 3 b c d x + b c )log(F))%e
/
4 3 3 2 2 2 3
3 d x + 9 c d x + 9 c d x + 3 c d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)/(c+d*x)^7, x)
(94)
3 3 2 2 2 3
((- b d x - 3 b c d x - 3 b c d x - b c )log(F) - 1)
*
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
%e
+
2 6 6 2 5 5 2 2 4 4 2 3 3 3 2 4 2 2
b d x + 6 b c d x + 15 b c d x + 20 b c d x + 15 b c d x
+
2 5 2 6
6 b c d x + b c
*
2 3 3 2 2 2 3 a log(F)
log(F) Ei((b d x + 3 b c d x + 3 b c d x + b c )log(F))%e
/
7 6 6 5 2 5 4 3 4 3 4 3 2 5 2 6
6 d x + 36 c d x + 90 c d x + 120 c d x + 90 c d x + 36 c d x + 6 c d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)/(c+d*x)^10, x)
(95)
2 6 6 2 5 5 2 2 4 4 2 3 3 3 2 4 2 2
- b d x - 6 b c d x - 15 b c d x - 20 b c d x - 15 b c d x
+
2 5 2 6
- 6 b c d x - b c
*
2
log(F)
+
3 3 2 2 2 3
(- b d x - 3 b c d x - 3 b c d x - b c )log(F) - 2
*
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
%e
+
3 9 9 3 8 8 3 2 7 7 3 3 6 6 3 4 5 5
b d x + 9 b c d x + 36 b c d x + 84 b c d x + 126 b c d x
+
3 5 4 4 3 6 3 3 3 7 2 2 3 8 3 9
126 b c d x + 84 b c d x + 36 b c d x + 9 b c d x + b c
*
3 3 3 2 2 2 3 a log(F)
log(F) Ei((b d x + 3 b c d x + 3 b c d x + b c )log(F))%e
/
10 9 9 8 2 8 7 3 7 6 4 6 5
18 d x + 162 c d x + 648 c d x + 1512 c d x + 2268 c d x
+
5 5 4 6 4 3 7 3 2 8 2 9
2268 c d x + 1512 c d x + 648 c d x + 162 c d x + 18 c d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)/(c+d*x)^13, x)
(96)
3 9 9 3 8 8 3 2 7 7 3 3 6 6 3 4 5 5
- b d x - 9 b c d x - 36 b c d x - 84 b c d x - 126 b c d x
+
3 5 4 4 3 6 3 3 3 7 2 2 3 8 3 9
- 126 b c d x - 84 b c d x - 36 b c d x - 9 b c d x - b c
*
3
log(F)
+
2 6 6 2 5 5 2 2 4 4 2 3 3 3 2 4 2 2
- b d x - 6 b c d x - 15 b c d x - 20 b c d x - 15 b c d x
+
2 5 2 6
- 6 b c d x - b c
*
2
log(F)
+
3 3 2 2 2 3
(- 2 b d x - 6 b c d x - 6 b c d x - 2 b c )log(F) - 6
*
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
%e
+
4 12 12 4 11 11 4 2 10 10 4 3 9 9
b d x + 12 b c d x + 66 b c d x + 220 b c d x
+
4 4 8 8 4 5 7 7 4 6 6 6 4 7 5 5
495 b c d x + 792 b c d x + 924 b c d x + 792 b c d x
+
4 8 4 4 4 9 3 3 4 10 2 2 4 11 4 12
495 b c d x + 220 b c d x + 66 b c d x + 12 b c d x + b c
*
4 3 3 2 2 2 3 a log(F)
log(F) Ei((b d x + 3 b c d x + 3 b c d x + b c )log(F))%e
/
13 12 12 11 2 11 10 3 10 9 4 9 8
72 d x + 864 c d x + 4752 c d x + 15840 c d x + 35640 c d x
+
5 8 7 6 7 6 7 6 5 8 5 4 9 4 3
57024 c d x + 66528 c d x + 57024 c d x + 35640 c d x + 15840 c d x
+
10 3 2 11 2 12
4752 c d x + 864 c d x + 72 c d
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(F^(a+b*(c+d*x)^3)/(c+d*x)^16, x)
(97)
4 12 12 4 11 11 4 2 10 10 4 3 9 9
- b d x - 12 b c d x - 66 b c d x - 220 b c d x
+
4 4 8 8 4 5 7 7 4 6 6 6 4 7 5 5
- 495 b c d x - 792 b c d x - 924 b c d x - 792 b c d x
+
4 8 4 4 4 9 3 3 4 10 2 2 4 11 4 12
- 495 b c d x - 220 b c d x - 66 b c d x - 12 b c d x - b c
*
4
log(F)
+
3 9 9 3 8 8 3 2 7 7 3 3 6 6 3 4 5 5
- b d x - 9 b c d x - 36 b c d x - 84 b c d x - 126 b c d x
+
3 5 4 4 3 6 3 3 3 7 2 2 3 8 3 9
- 126 b c d x - 84 b c d x - 36 b c d x - 9 b c d x - b c
*
3
log(F)
+
2 6 6 2 5 5 2 2 4 4 2 3 3 3
- 2 b d x - 12 b c d x - 30 b c d x - 40 b c d x
+
2 4 2 2 2 5 2 6
- 30 b c d x - 12 b c d x - 2 b c
*
2
log(F)
+
3 3 2 2 2 3
(- 6 b d x - 18 b c d x - 18 b c d x - 6 b c )log(F) - 24
*
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
%e
+
5 15 15 5 14 14 5 2 13 13 5 3 12 12
b d x + 15 b c d x + 105 b c d x + 455 b c d x
+
5 4 11 11 5 5 10 10 5 6 9 9 5 7 8 8
1365 b c d x + 3003 b c d x + 5005 b c d x + 6435 b c d x
+
5 8 7 7 5 9 6 6 5 10 5 5 5 11 4 4
6435 b c d x + 5005 b c d x + 3003 b c d x + 1365 b c d x
+
5 12 3 3 5 13 2 2 5 14 5 15
455 b c d x + 105 b c d x + 15 b c d x + b c
*
5 3 3 2 2 2 3 a log(F)
log(F) Ei((b d x + 3 b c d x + 3 b c d x + b c )log(F))%e
/
16 15 15 14 2 14 13 3 13 12
360 d x + 5400 c d x + 37800 c d x + 163800 c d x
+
4 12 11 5 11 10 6 10 9 7 9 8
491400 c d x + 1081080 c d x + 1801800 c d x + 2316600 c d x
+
8 8 7 9 7 6 10 6 5 11 5 4
2316600 c d x + 1801800 c d x + 1081080 c d x + 491400 c d x
+
12 4 3 13 3 2 14 2 15
163800 c d x + 37800 c d x + 5400 c d x + 360 c d
Type: Union(Expression(Integer),...)
fricas
Time: 0.02 (EV) = 0.02 sec
integrate(F^(a+b*(c+d*x)^3)*(c+d*x)^3, x)
(98)
3 3 2 2 2 3
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F)
(3 d x + 3 c)%e
*
+------------+
3| 3
\|- b d log(F)
+
a log(F) 1 3 3 2 2 2 3
d %e Gamma(-,(- b d x - 3 b c d x - 3 b c d x - b c )log(F))
3
/
+------------+
3| 3
9 b d log(F)\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)*(c+d*x), x)
(99)
a log(F) 2 3 3 2 2 2 3
d %e Gamma(-,(- b d x - 3 b c d x - 3 b c d x - b c )log(F))
3
- ---------------------------------------------------------------------
+------------+2
3| 3
3 \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3), x)
(100)
a log(F) 1 3 3 2 2 2 3
%e Gamma(-,(- b d x - 3 b c d x - 3 b c d x - b c )log(F))
3
- -------------------------------------------------------------------
+------------+
3| 3
3 \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)/(c+d*x)^2, x)
(101)
3 3 2 2 2 3 +------------+2
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F) 3| 3
- %e \|- b d log(F)
+
3 2 a log(F)
(- b d x - b c d )log(F)%e
*
2 3 3 2 2 2 3
Gamma(-,(- b d x - 3 b c d x - 3 b c d x - b c )log(F))
3
/
+------------+2
2 3| 3
(d x + c d)\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)/(c+d*x)^3, x)
(102)
3 3 2 2 2 3 +------------+
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F) 3| 3
- %e \|- b d log(F)
+
3 2 2 2 a log(F)
(- b d x - 2 b c d x - b c d)log(F)%e
*
1 3 3 2 2 2 3
Gamma(-,(- b d x - 3 b c d x - 3 b c d x - b c )log(F))
3
/
+------------+
3 2 2 2 3| 3
(2 d x + 4 c d x + 2 c d)\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^3)/(c+d*x)^5, x)
(103)
3 3 2 2 2 3
((- 3 b d x - 9 b c d x - 9 b c d x - 3 b c )log(F) - 1)
*
3 3 2 2 2 3 +------------+2
(b d x + 3 b c d x + 3 b c d x + b c + a)log(F) 3| 3
%e \|- b d log(F)
+
2 6 4 2 5 3 2 2 4 2 2 3 3 2 4 2 2
(- 3 b d x - 12 b c d x - 18 b c d x - 12 b c d x - 3 b c d )log(F)
*
a log(F) 2 3 3 2 2 2 3
%e Gamma(-,(- b d x - 3 b c d x - 3 b c d x - b c )log(F))
3
/
+------------+2
5 4 4 3 2 3 2 3 2 4 3| 3
(4 d x + 16 c d x + 24 c d x + 16 c d x + 4 c d)\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(a+b*(c+d*x)^(1/2)), x)
+-------+
+-------+ b log(f)\|d x + c + a log(f)
(2 b log(f)\|d x + c - 2)%e
(104) ---------------------------------------------------------
2 2
b d log(f)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(f^(a+b*(c+d*x)^(1/3)), x)
(105)
2 2 3+-------+2 3+-------+
(3 b log(f) \|d x + c - 6 b log(f)\|d x + c + 6)
*
3+-------+
b log(f)\|d x + c + a log(f)
%e
/
3 3
b d log(f)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
-- n<0
integrate(F^(a+b/(c+d*x))*(c+d*x)^m, x)
%A a d + a c + b
x ----------------
++ %A d + c m
(106) | F (%A d + c) d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))*(c+d*x)^4, x)
(107)
4 4 4 3 2 2 3 3 2 3
(b d x + b c)log(F) + (b d x + 2 b c d x + b c )log(F)
+
2 3 3 2 2 2 2 2 2 3 2
(2 b d x + 6 b c d x + 6 b c d x + 2 b c )log(F)
+
4 4 3 3 2 2 2 3 4
(6 b d x + 24 b c d x + 36 b c d x + 24 b c d x + 6 b c )log(F)
+
5 5 4 4 2 3 3 3 2 2 4 5
24 d x + 120 c d x + 240 c d x + 240 c d x + 120 c d x + 24 c
*
(a d x + a c + b)log(F)
-----------------------
d x + c
%e
+
5 5 b log(F) a log(F)
- b log(F) Ei(--------)%e
d x + c
/
120 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))*(c+d*x)^3, x)
(108)
3 3 3 2 2 2 2 2 2 2
(b d x + b c)log(F) + (b d x + 2 b c d x + b c )log(F)
+
3 3 2 2 2 3 4 4
(2 b d x + 6 b c d x + 6 b c d x + 2 b c )log(F) + 6 d x
+
3 3 2 2 2 3 4
24 c d x + 36 c d x + 24 c d x + 6 c
*
(a d x + a c + b)log(F)
-----------------------
d x + c
%e
+
4 4 b log(F) a log(F)
- b log(F) Ei(--------)%e
d x + c
/
24 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))*(c+d*x)^2, x)
(109)
2 2 2 2 2 2 3 3
(b d x + b c)log(F) + (b d x + 2 b c d x + b c )log(F) + 2 d x
+
2 2 2 3
6 c d x + 6 c d x + 2 c
*
(a d x + a c + b)log(F)
-----------------------
d x + c
%e
+
3 3 b log(F) a log(F)
- b log(F) Ei(--------)%e
d x + c
/
6 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))*(c+d*x), x)
(110)
(a d x + a c + b)log(F)
-----------------------
2 2 2 d x + c
((b d x + b c)log(F) + d x + 2 c d x + c )%e
+
2 2 b log(F) a log(F)
- b log(F) Ei(--------)%e
d x + c
/
2 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)), x)
(a d x + a c + b)log(F)
-----------------------
d x + c b log(F) a log(F)
(d x + c)%e - b log(F)Ei(--------)%e
d x + c
(111) -------------------------------------------------------------------
d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))/(c+d*x), x)
b log(F) a log(F)
Ei(--------)%e
d x + c
(112) - ----------------------
d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))/(c+d*x)^2, x)
(a d x + a c + b)log(F)
-----------------------
d x + c
%e
(113) - -------------------------
b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))/(c+d*x)^3, x)
(a d x + a c + b)log(F)
-----------------------
d x + c
(- b log(F) + d x + c)%e
(114) -----------------------------------------------
2 2 2 2
(b d x + b c d)log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))/(c+d*x)^4, x)
(115)
2 2 2 2 2
(- b log(F) + (2 b d x + 2 b c)log(F) - 2 d x - 4 c d x - 2 c )
*
(a d x + a c + b)log(F)
-----------------------
d x + c
%e
/
3 3 2 3 2 3 2 3
(b d x + 2 b c d x + b c d)log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))/(c+d*x)^5, x)
(116)
3 3 2 2 2
- b log(F) + (3 b d x + 3 b c)log(F)
+
2 2 2 3 3 2 2
(- 6 b d x - 12 b c d x - 6 b c )log(F) + 6 d x + 18 c d x
+
2 3
18 c d x + 6 c
*
(a d x + a c + b)log(F)
-----------------------
d x + c
%e
/
4 4 3 4 3 2 4 2 2 4 3 4
(b d x + 3 b c d x + 3 b c d x + b c d)log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))/(c+d*x)^6, x)
(117)
4 4 3 3 3
- b log(F) + (4 b d x + 4 b c)log(F)
+
2 2 2 2 2 2 2
(- 12 b d x - 24 b c d x - 12 b c )log(F)
+
3 3 2 2 2 3 4 4
(24 b d x + 72 b c d x + 72 b c d x + 24 b c )log(F) - 24 d x
+
3 3 2 2 2 3 4
- 96 c d x - 144 c d x - 96 c d x - 24 c
*
(a d x + a c + b)log(F)
-----------------------
d x + c
%e
/
5 5 4 5 4 3 5 2 3 2 5 3 2 5 4 5
(b d x + 4 b c d x + 6 b c d x + 4 b c d x + b c d)log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))/(c+d*x)^7, x)
(118)
5 5 4 4 4
- b log(F) + (5 b d x + 5 b c)log(F)
+
3 2 2 3 3 2 3
(- 20 b d x - 40 b c d x - 20 b c )log(F)
+
2 3 3 2 2 2 2 2 2 3 2
(60 b d x + 180 b c d x + 180 b c d x + 60 b c )log(F)
+
4 4 3 3 2 2 2 3 4
(- 120 b d x - 480 b c d x - 720 b c d x - 480 b c d x - 120 b c )
*
log(F)
+
5 5 4 4 2 3 3 3 2 2 4 5
120 d x + 600 c d x + 1200 c d x + 1200 c d x + 600 c d x + 120 c
*
(a d x + a c + b)log(F)
-----------------------
d x + c
%e
/
6 6 5 6 5 4 6 2 4 3 6 3 3 2 6 4 2 6 5
(b d x + 5 b c d x + 10 b c d x + 10 b c d x + 5 b c d x + b c d)
*
6
log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x)^m, x)
2 2 2
%A a d + 2 %A a c d + a c + b
-------------------------------
x 2 2 2
++ %A d + 2 %A c d + c m
(119) | F (%A d + c) d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x)^9, x)
(120)
4 2 2 4 4 2 4
(b d x + 2 b c d x + b c )log(F)
+
3 4 4 3 3 3 3 2 2 2 3 3 3 4 3
(b d x + 4 b c d x + 6 b c d x + 4 b c d x + b c )log(F)
+
2 6 6 2 5 5 2 2 4 4 2 3 3 3 2 4 2 2
2 b d x + 12 b c d x + 30 b c d x + 40 b c d x + 30 b c d x
+
2 5 2 6
12 b c d x + 2 b c
*
2
log(F)
+
8 8 7 7 2 6 6 3 5 5
6 b d x + 48 b c d x + 168 b c d x + 336 b c d x
+
4 4 4 5 3 3 6 2 2 7 8
420 b c d x + 336 b c d x + 168 b c d x + 48 b c d x + 6 b c
*
log(F)
+
10 10 9 9 2 8 8 3 7 7 4 6 6
24 d x + 240 c d x + 1080 c d x + 2880 c d x + 5040 c d x
+
5 5 5 6 4 4 7 3 3 8 2 2 9
6048 c d x + 5040 c d x + 2880 c d x + 1080 c d x + 240 c d x
+
10
24 c
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
d x + 2 c d x + c
%e
+
5 5 b log(F) a log(F)
- b log(F) Ei(-------------------)%e
2 2 2
d x + 2 c d x + c
/
240 d
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x)^7, x)
(121)
3 2 2 3 3 2 3
(b d x + 2 b c d x + b c )log(F)
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(b d x + 4 b c d x + 6 b c d x + 4 b c d x + b c )log(F)
+
6 6 5 5 2 4 4 3 3 3 4 2 2
2 b d x + 12 b c d x + 30 b c d x + 40 b c d x + 30 b c d x
+
5 6
12 b c d x + 2 b c
*
log(F)
+
8 8 7 7 2 6 6 3 5 5 4 4 4
6 d x + 48 c d x + 168 c d x + 336 c d x + 420 c d x
+
5 3 3 6 2 2 7 8
336 c d x + 168 c d x + 48 c d x + 6 c
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
d x + 2 c d x + c
%e
+
4 4 b log(F) a log(F)
- b log(F) Ei(-------------------)%e
2 2 2
d x + 2 c d x + c
/
48 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x)^5, x)
(122)
2 2 2 2 2 2 2
(b d x + 2 b c d x + b c )log(F)
+
4 4 3 3 2 2 2 3 4 6 6
(b d x + 4 b c d x + 6 b c d x + 4 b c d x + b c )log(F) + 2 d x
+
5 5 2 4 4 3 3 3 4 2 2 5 6
12 c d x + 30 c d x + 40 c d x + 30 c d x + 12 c d x + 2 c
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
d x + 2 c d x + c
%e
+
3 3 b log(F) a log(F)
- b log(F) Ei(-------------------)%e
2 2 2
d x + 2 c d x + c
/
12 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x)^3, x)
(123)
2 2 2 4 4 3 3 2 2 2
(b d x + 2 b c d x + b c )log(F) + d x + 4 c d x + 6 c d x
+
3 4
4 c d x + c
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
d x + 2 c d x + c
%e
+
2 2 b log(F) a log(F)
- b log(F) Ei(-------------------)%e
2 2 2
d x + 2 c d x + c
/
4 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x), x)
(124)
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
2 2 2 d x + 2 c d x + c
(d x + 2 c d x + c )%e
+
b log(F) a log(F)
- b log(F)Ei(-------------------)%e
2 2 2
d x + 2 c d x + c
/
2 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x), x)
b log(F) a log(F)
Ei(-------------------)%e
2 2 2
d x + 2 c d x + c
(125) - ---------------------------------
2 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^3, x)
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
d x + 2 c d x + c
%e
(126) - ---------------------------------------
2 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^5, x)
(127)
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
2 2 2 d x + 2 c d x + c
(- b log(F) + d x + 2 c d x + c )%e
-------------------------------------------------------------------------
2 3 2 2 2 2 2 2
(2 b d x + 4 b c d x + 2 b c d)log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^7, x)
(128)
2 2 2 2 2 4 4 3 3
- b log(F) + (2 b d x + 4 b c d x + 2 b c )log(F) - 2 d x - 8 c d x
+
2 2 2 3 4
- 12 c d x - 8 c d x - 2 c
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
d x + 2 c d x + c
%e
/
3 5 4 3 4 3 3 2 3 2 3 3 2 3 4 3
(2 b d x + 8 b c d x + 12 b c d x + 8 b c d x + 2 b c d)log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^9, x)
(129)
3 3 2 2 2 2 2 2 2
- b log(F) + (3 b d x + 6 b c d x + 3 b c )log(F)
+
4 4 3 3 2 2 2 3 4
(- 6 b d x - 24 b c d x - 36 b c d x - 24 b c d x - 6 b c )log(F)
+
6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
6 d x + 36 c d x + 90 c d x + 120 c d x + 90 c d x + 36 c d x + 6 c
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
d x + 2 c d x + c
%e
/
4 7 6 4 6 5 4 2 5 4 4 3 4 3 4 4 3 2
2 b d x + 12 b c d x + 30 b c d x + 40 b c d x + 30 b c d x
+
4 5 2 4 6
12 b c d x + 2 b c d
*
4
log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 (EV) = 0.01 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^11, x)
(130)
4 4 3 2 2 3 3 2 3
- b log(F) + (4 b d x + 8 b c d x + 4 b c )log(F)
+
2 4 4 2 3 3 2 2 2 2 2 3 2 4 2
(- 12 b d x - 48 b c d x - 72 b c d x - 48 b c d x - 12 b c )log(F)
+
6 6 5 5 2 4 4 3 3 3
24 b d x + 144 b c d x + 360 b c d x + 480 b c d x
+
4 2 2 5 6
360 b c d x + 144 b c d x + 24 b c
*
log(F)
+
8 8 7 7 2 6 6 3 5 5 4 4 4
- 24 d x - 192 c d x - 672 c d x - 1344 c d x - 1680 c d x
+
5 3 3 6 2 2 7 8
- 1344 c d x - 672 c d x - 192 c d x - 24 c
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
d x + 2 c d x + c
%e
/
5 9 8 5 8 7 5 2 7 6 5 3 6 5 5 4 5 4
2 b d x + 16 b c d x + 56 b c d x + 112 b c d x + 140 b c d x
+
5 5 4 3 5 6 3 2 5 7 2 5 8
112 b c d x + 56 b c d x + 16 b c d x + 2 b c d
*
5
log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.02 (EV) = 0.02 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^13, x)
(131)
5 5 4 2 2 4 4 2 4
- b log(F) + (5 b d x + 10 b c d x + 5 b c )log(F)
+
3 4 4 3 3 3 3 2 2 2 3 3 3 4
(- 20 b d x - 80 b c d x - 120 b c d x - 80 b c d x - 20 b c )
*
3
log(F)
+
2 6 6 2 5 5 2 2 4 4 2 3 3 3
60 b d x + 360 b c d x + 900 b c d x + 1200 b c d x
+
2 4 2 2 2 5 2 6
900 b c d x + 360 b c d x + 60 b c
*
2
log(F)
+
8 8 7 7 2 6 6 3 5 5
- 120 b d x - 960 b c d x - 3360 b c d x - 6720 b c d x
+
4 4 4 5 3 3 6 2 2 7
- 8400 b c d x - 6720 b c d x - 3360 b c d x - 960 b c d x
+
8
- 120 b c
*
log(F)
+
10 10 9 9 2 8 8 3 7 7 4 6 6
120 d x + 1200 c d x + 5400 c d x + 14400 c d x + 25200 c d x
+
5 5 5 6 4 4 7 3 3 8 2 2 9
30240 c d x + 25200 c d x + 14400 c d x + 5400 c d x + 1200 c d x
+
10
120 c
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2
d x + 2 c d x + c
%e
/
6 11 10 6 10 9 6 2 9 8 6 3 8 7 6 4 7 6
2 b d x + 20 b c d x + 90 b c d x + 240 b c d x + 420 b c d x
+
6 5 6 5 6 6 5 4 6 7 4 3 6 8 3 2 6 9 2
504 b c d x + 420 b c d x + 240 b c d x + 90 b c d x + 20 b c d x
+
6 10
2 b c d
*
6
log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.03 (EV) = 0.04 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x)^10, x)
(132)
+----------+
| b log(F)
d |- --------
| 2
6 6 +---+ a log(F) \| d
- 32 b log(F) \|%pi %e erf(--------------)
d x + c
+
5 2 5 5
(32 b d x + 32 b c d)log(F)
+
4 4 3 4 3 2 4 2 2 4 3 4
(16 b d x + 48 b c d x + 48 b c d x + 16 b c d)log(F)
+
3 6 5 3 5 4 3 2 4 3 3 3 3 2
24 b d x + 120 b c d x + 240 b c d x + 240 b c d x
+
3 4 2 3 5
120 b c d x + 24 b c d
*
3
log(F)
+
2 8 7 2 7 6 2 2 6 5 2 3 5 4
60 b d x + 420 b c d x + 1260 b c d x + 2100 b c d x
+
2 4 4 3 2 5 3 2 2 6 2 2 7
2100 b c d x + 1260 b c d x + 420 b c d x + 60 b c d
*
2
log(F)
+
10 9 9 8 2 8 7 3 7 6
210 b d x + 1890 b c d x + 7560 b c d x + 17640 b c d x
+
4 6 5 5 5 4 6 4 3 7 3 2
26460 b c d x + 26460 b c d x + 17640 b c d x + 7560 b c d x
+
8 2 9
1890 b c d x + 210 b c d
*
log(F)
+
12 11 11 10 2 10 9 3 9 8
945 d x + 10395 c d x + 51975 c d x + 155925 c d x
+
4 8 7 5 7 6 6 6 5 7 5 4
311850 c d x + 436590 c d x + 436590 c d x + 311850 c d x
+
8 4 3 9 3 2 10 2 11
155925 c d x + 51975 c d x + 10395 c d x + 945 c d
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
%e |- --------
| 2
\| d
/
+----------+
2 | b log(F)
10395 d |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 (EV) = 0.02 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x)^8, x)
(133)
+----------+
| b log(F)
d |- --------
| 2
5 5 +---+ a log(F) \| d
- 16 b log(F) \|%pi %e erf(--------------)
d x + c
+
4 2 4 4
(16 b d x + 16 b c d)log(F)
+
3 4 3 3 3 2 3 2 2 3 3 3
(8 b d x + 24 b c d x + 24 b c d x + 8 b c d)log(F)
+
2 6 5 2 5 4 2 2 4 3 2 3 3 2
12 b d x + 60 b c d x + 120 b c d x + 120 b c d x
+
2 4 2 2 5
60 b c d x + 12 b c d
*
2
log(F)
+
8 7 7 6 2 6 5 3 5 4
30 b d x + 210 b c d x + 630 b c d x + 1050 b c d x
+
4 4 3 5 3 2 6 2 7
1050 b c d x + 630 b c d x + 210 b c d x + 30 b c d
*
log(F)
+
10 9 9 8 2 8 7 3 7 6 4 6 5
105 d x + 945 c d x + 3780 c d x + 8820 c d x + 13230 c d x
+
5 5 4 6 4 3 7 3 2 8 2 9
13230 c d x + 8820 c d x + 3780 c d x + 945 c d x + 105 c d
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
%e |- --------
| 2
\| d
/
+----------+
2 | b log(F)
945 d |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 (EV) = 0.01 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x)^6, x)
(134)
+----------+
| b log(F)
d |- --------
| 2
4 4 +---+ a log(F) \| d
- 8 b log(F) \|%pi %e erf(--------------)
d x + c
+
3 2 3 3
(8 b d x + 8 b c d)log(F)
+
2 4 3 2 3 2 2 2 2 2 3 2
(4 b d x + 12 b c d x + 12 b c d x + 4 b c d)log(F)
+
6 5 5 4 2 4 3 3 3 2 4 2
6 b d x + 30 b c d x + 60 b c d x + 60 b c d x + 30 b c d x
+
5
6 b c d
*
log(F)
+
8 7 7 6 2 6 5 3 5 4 4 4 3
15 d x + 105 c d x + 315 c d x + 525 c d x + 525 c d x
+
5 3 2 6 2 7
315 c d x + 105 c d x + 15 c d
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
%e |- --------
| 2
\| d
/
+----------+
2 | b log(F)
105 d |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x)^4, x)
(135)
+----------+
| b log(F)
d |- --------
| 2
3 3 +---+ a log(F) \| d
- 4 b log(F) \|%pi %e erf(--------------)
d x + c
+
2 2 2 2
(4 b d x + 4 b c d)log(F)
+
4 3 3 2 2 2 3 6 5
(2 b d x + 6 b c d x + 6 b c d x + 2 b c d)log(F) + 3 d x
+
5 4 2 4 3 3 3 2 4 2 5
15 c d x + 30 c d x + 30 c d x + 15 c d x + 3 c d
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
%e |- --------
| 2
\| d
/
+----------+
2 | b log(F)
15 d |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)*(c+d*x)^2, x)
(136)
+----------+
| b log(F)
d |- --------
| 2
2 2 +---+ a log(F) \| d
- 2 b log(F) \|%pi %e erf(--------------)
d x + c
+
2 4 3 3 2 2 2 3
((2 b d x + 2 b c d)log(F) + d x + 3 c d x + 3 c d x + c d)
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
%e |- --------
| 2
\| d
/
+----------+
2 | b log(F)
3 d |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2), x)
(137)
+----------+
| b log(F)
d |- --------
| 2
+---+ a log(F) \| d
- b log(F)\|%pi %e erf(--------------)
d x + c
+
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
2 d x + 2 c d x + c | b log(F)
(d x + c d)%e |- --------
| 2
\| d
/
+----------+
2 | b log(F)
d |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^2, x)
+----------+
| b log(F)
d |- --------
| 2
+---+ a log(F) \| d
\|%pi %e erf(--------------)
d x + c
(138) - -----------------------------------
+----------+
2 | b log(F)
2 d |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^4, x)
(139)
+----------+
| b log(F)
d |- --------
| 2
+---+ a log(F) \| d
(d x + c)\|%pi %e erf(--------------)
d x + c
+
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
- 2 d %e |- --------
| 2
\| d
/
+----------+
3 2 | b log(F)
(4 b d x + 4 b c d )log(F) |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^6, x)
(140)
+----------+
| b log(F)
d |- --------
| 2
3 3 2 2 2 3 +---+ a log(F) \| d
(- 3 d x - 9 c d x - 9 c d x - 3 c )\|%pi %e erf(--------------)
d x + c
+
3 2 2 2
(- 4 b d log(F) + 6 d x + 12 c d x + 6 c d)
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
%e |- --------
| 2
\| d
/
+----------+
2 5 3 2 4 2 2 2 3 2 3 2 2 | b log(F)
(8 b d x + 24 b c d x + 24 b c d x + 8 b c d )log(F) |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^8, x)
(141)
5 5 4 4 2 3 3 3 2 2 4 5
(15 d x + 75 c d x + 150 c d x + 150 c d x + 75 c d x + 15 c )
*
+----------+
| b log(F)
d |- --------
| 2
+---+ a log(F) \| d
\|%pi %e erf(--------------)
d x + c
+
2 2 3 2 2 2 5 4
- 8 b d log(F) + (20 b d x + 40 b c d x + 20 b c d)log(F) - 30 d x
+
4 3 2 3 2 3 2 4
- 120 c d x - 180 c d x - 120 c d x - 30 c d
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
%e |- --------
| 2
\| d
/
3 7 5 3 6 4 3 2 5 3 3 3 4 2 3 4 3
16 b d x + 80 b c d x + 160 b c d x + 160 b c d x + 80 b c d x
+
3 5 2
16 b c d
*
+----------+
3 | b log(F)
log(F) |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 (EV) = 0.02 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^10, x)
(142)
7 7 6 6 2 5 5 3 4 4 4 3 3
- 105 d x - 735 c d x - 2205 c d x - 3675 c d x - 3675 c d x
+
5 2 2 6 7
- 2205 c d x - 735 c d x - 105 c
*
+----------+
| b log(F)
d |- --------
| 2
+---+ a log(F) \| d
\|%pi %e erf(--------------)
d x + c
+
3 3 2 3 2 2 2 2 2 2
- 16 b d log(F) + (56 b d x + 112 b c d x + 56 b c d)log(F)
+
5 4 4 3 2 3 2 3 2
- 140 b d x - 560 b c d x - 840 b c d x - 560 b c d x
+
4
- 140 b c d
*
log(F)
+
7 6 6 5 2 5 4 3 4 3 4 3 2
210 d x + 1260 c d x + 3150 c d x + 4200 c d x + 3150 c d x
+
5 2 6
1260 c d x + 210 c d
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
%e |- --------
| 2
\| d
/
4 9 7 4 8 6 4 2 7 5 4 3 6 4 4 4 5 3
32 b d x + 224 b c d x + 672 b c d x + 1120 b c d x + 1120 b c d x
+
4 5 4 2 4 6 3 4 7 2
672 b c d x + 224 b c d x + 32 b c d
*
+----------+
4 | b log(F)
log(F) |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0.02 (EV) = 0.02 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^12, x)
(143)
9 9 8 8 2 7 7 3 6 6 4 5 5
945 d x + 8505 c d x + 34020 c d x + 79380 c d x + 119070 c d x
+
5 4 4 6 3 3 7 2 2 8 9
119070 c d x + 79380 c d x + 34020 c d x + 8505 c d x + 945 c
*
+----------+
| b log(F)
d |- --------
| 2
+---+ a log(F) \| d
\|%pi %e erf(--------------)
d x + c
+
4 4 3 3 2 3 2 3 2 3
- 32 b d log(F) + (144 b d x + 288 b c d x + 144 b c d)log(F)
+
2 5 4 2 4 3 2 2 3 2 2 3 2
- 504 b d x - 2016 b c d x - 3024 b c d x - 2016 b c d x
+
2 4
- 504 b c d
*
2
log(F)
+
7 6 6 5 2 5 4 3 4 3
1260 b d x + 7560 b c d x + 18900 b c d x + 25200 b c d x
+
4 3 2 5 2 6
18900 b c d x + 7560 b c d x + 1260 b c d
*
log(F)
+
9 8 8 7 2 7 6 3 6 5
- 1890 d x - 15120 c d x - 52920 c d x - 105840 c d x
+
4 5 4 5 4 3 6 3 2 7 2 8
- 132300 c d x - 105840 c d x - 52920 c d x - 15120 c d x - 1890 c d
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
%e |- --------
| 2
\| d
/
5 11 9 5 10 8 5 2 9 7 5 3 8 6
64 b d x + 576 b c d x + 2304 b c d x + 5376 b c d x
+
5 4 7 5 5 5 6 4 5 6 5 3 5 7 4 2
8064 b c d x + 8064 b c d x + 5376 b c d x + 2304 b c d x
+
5 8 3 5 9 2
576 b c d x + 64 b c d
*
+----------+
5 | b log(F)
log(F) |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0.04 (EV) = 0.04 sec
integrate(F^(a+b/(c+d*x)^2)/(c+d*x)^14, x)
(144)
11 11 10 10 2 9 9 3 8 8
- 10395 d x - 114345 c d x - 571725 c d x - 1715175 c d x
+
4 7 7 5 6 6 6 5 5 7 4 4
- 3430350 c d x - 4802490 c d x - 4802490 c d x - 3430350 c d x
+
8 3 3 9 2 2 10 11
- 1715175 c d x - 571725 c d x - 114345 c d x - 10395 c
*
+----------+
| b log(F)
d |- --------
| 2
+---+ a log(F) \| d
\|%pi %e erf(--------------)
d x + c
+
5 5 4 3 2 4 2 4 2 4
- 64 b d log(F) + (352 b d x + 704 b c d x + 352 b c d)log(F)
+
3 5 4 3 4 3 3 2 3 2 3 3 2
- 1584 b d x - 6336 b c d x - 9504 b c d x - 6336 b c d x
+
3 4
- 1584 b c d
*
3
log(F)
+
2 7 6 2 6 5 2 2 5 4 2 3 4 3
5544 b d x + 33264 b c d x + 83160 b c d x + 110880 b c d x
+
2 4 3 2 2 5 2 2 6
83160 b c d x + 33264 b c d x + 5544 b c d
*
2
log(F)
+
9 8 8 7 2 7 6
- 13860 b d x - 110880 b c d x - 388080 b c d x
+
3 6 5 4 5 4 5 4 3
- 776160 b c d x - 970200 b c d x - 776160 b c d x
+
6 3 2 7 2 8
- 388080 b c d x - 110880 b c d x - 13860 b c d
*
log(F)
+
11 10 10 9 2 9 8 3 8 7
20790 d x + 207900 c d x + 935550 c d x + 2494800 c d x
+
4 7 6 5 6 5 6 5 4 7 4 3
4365900 c d x + 5239080 c d x + 4365900 c d x + 2494800 c d x
+
8 3 2 9 2 10
935550 c d x + 207900 c d x + 20790 c d
*
2 2 2
(a d x + 2 a c d x + a c + b)log(F)
-------------------------------------
2 2 2 +----------+
d x + 2 c d x + c | b log(F)
%e |- --------
| 2
\| d
/
6 13 11 6 12 10 6 2 11 9 6 3 10 8
128 b d x + 1408 b c d x + 7040 b c d x + 21120 b c d x
+
6 4 9 7 6 5 8 6 6 6 7 5 6 7 6 4
42240 b c d x + 59136 b c d x + 59136 b c d x + 42240 b c d x
+
6 8 5 3 6 9 4 2 6 10 3 6 11 2
21120 b c d x + 7040 b c d x + 1408 b c d x + 128 b c d
*
+----------+
6 | b log(F)
log(F) |- --------
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0.05 (EV) = 0.05 sec
integrate(F^(a+b/(c+d*x)^3)*(c+d*x)^m, x)
3 3 2 2 2 3
%A a d + 3 %A a c d + 3 %A a c d + a c + b
---------------------------------------------
x 3 3 2 2 2 3
++ %A d + 3 %A c d + 3 %A c d + c m
(145) | F (%A d + c) d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 (EV) = 0.01 sec
integrate(F^(a+b/(c+d*x)^3)*(c+d*x)^14, x)
(146)
4 3 3 4 2 2 4 2 4 3 4
(b d x + 3 b c d x + 3 b c d x + b c )log(F)
+
3 6 6 3 5 5 3 2 4 4 3 3 3 3 3 4 2 2
b d x + 6 b c d x + 15 b c d x + 20 b c d x + 15 b c d x
+
3 5 3 6
6 b c d x + b c
*
3
log(F)
+
2 9 9 2 8 8 2 2 7 7 2 3 6 6
2 b d x + 18 b c d x + 72 b c d x + 168 b c d x
+
2 4 5 5 2 5 4 4 2 6 3 3 2 7 2 2
252 b c d x + 252 b c d x + 168 b c d x + 72 b c d x
+
2 8 2 9
18 b c d x + 2 b c
*
2
log(F)
+
12 12 11 11 2 10 10 3 9 9
6 b d x + 72 b c d x + 396 b c d x + 1320 b c d x
+
4 8 8 5 7 7 6 6 6 7 5 5
2970 b c d x + 4752 b c d x + 5544 b c d x + 4752 b c d x
+
8 4 4 9 3 3 10 2 2 11
2970 b c d x + 1320 b c d x + 396 b c d x + 72 b c d x
+
12
6 b c
*
log(F)
+
15 15 14 14 2 13 13 3 12 12
24 d x + 360 c d x + 2520 c d x + 10920 c d x
+
4 11 11 5 10 10 6 9 9 7 8 8
32760 c d x + 72072 c d x + 120120 c d x + 154440 c d x
+
8 7 7 9 6 6 10 5 5 11 4 4
154440 c d x + 120120 c d x + 72072 c d x + 32760 c d x
+
12 3 3 13 2 2 14 15
10920 c d x + 2520 c d x + 360 c d x + 24 c
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
+
5 5 b log(F) a log(F)
- b log(F) Ei(------------------------------)%e
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
360 d
Type: Union(Expression(Integer),...)
fricas
Time: 0.02 (EV) = 0.02 sec
integrate(F^(a+b/(c+d*x)^3)*(c+d*x)^11, x)
(147)
3 3 3 3 2 2 3 2 3 3 3
(b d x + 3 b c d x + 3 b c d x + b c )log(F)
+
2 6 6 2 5 5 2 2 4 4 2 3 3 3 2 4 2 2
b d x + 6 b c d x + 15 b c d x + 20 b c d x + 15 b c d x
+
2 5 2 6
6 b c d x + b c
*
2
log(F)
+
9 9 8 8 2 7 7 3 6 6
2 b d x + 18 b c d x + 72 b c d x + 168 b c d x
+
4 5 5 5 4 4 6 3 3 7 2 2
252 b c d x + 252 b c d x + 168 b c d x + 72 b c d x
+
8 9
18 b c d x + 2 b c
*
log(F)
+
12 12 11 11 2 10 10 3 9 9 4 8 8
6 d x + 72 c d x + 396 c d x + 1320 c d x + 2970 c d x
+
5 7 7 6 6 6 7 5 5 8 4 4 9 3 3
4752 c d x + 5544 c d x + 4752 c d x + 2970 c d x + 1320 c d x
+
10 2 2 11 12
396 c d x + 72 c d x + 6 c
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
+
4 4 b log(F) a log(F)
- b log(F) Ei(------------------------------)%e
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
72 d
Type: Union(Expression(Integer),...)
fricas
Time: 0.02 (EV) = 0.02 sec
integrate(F^(a+b/(c+d*x)^3)*(c+d*x)^8, x)
(148)
2 3 3 2 2 2 2 2 2 3 2
(b d x + 3 b c d x + 3 b c d x + b c )log(F)
+
6 6 5 5 2 4 4 3 3 3 4 2 2
b d x + 6 b c d x + 15 b c d x + 20 b c d x + 15 b c d x
+
5 6
6 b c d x + b c
*
log(F)
+
9 9 8 8 2 7 7 3 6 6 4 5 5 5 4 4
2 d x + 18 c d x + 72 c d x + 168 c d x + 252 c d x + 252 c d x
+
6 3 3 7 2 2 8 9
168 c d x + 72 c d x + 18 c d x + 2 c
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
+
3 3 b log(F) a log(F)
- b log(F) Ei(------------------------------)%e
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
18 d
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(F^(a+b/(c+d*x)^3)*(c+d*x)^5, x)
(149)
3 3 2 2 2 3 6 6 5 5
(b d x + 3 b c d x + 3 b c d x + b c )log(F) + d x + 6 c d x
+
2 4 4 3 3 3 4 2 2 5 6
15 c d x + 20 c d x + 15 c d x + 6 c d x + c
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
+
2 2 b log(F) a log(F)
- b log(F) Ei(------------------------------)%e
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
6 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^3)*(c+d*x)^2, x)
(150)
3 3 2 2 2 3
(d x + 3 c d x + 3 c d x + c )
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
+
b log(F) a log(F)
- b log(F)Ei(------------------------------)%e
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
3 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^3)/(c+d*x), x)
b log(F) a log(F)
Ei(------------------------------)%e
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
(151) - --------------------------------------------
3 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^3)/(c+d*x)^4, x)
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
(152) - ----------------------------------------------------
3 b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^3)/(c+d*x)^7, x)
(153)
3 3 2 2 2 3
(- b log(F) + d x + 3 c d x + 3 c d x + c )
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
/
2 4 3 2 3 2 2 2 2 2 3 2
(3 b d x + 9 b c d x + 9 b c d x + 3 b c d)log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^3)/(c+d*x)^10, x)
(154)
2 2 3 3 2 2 2 3
- b log(F) + (2 b d x + 6 b c d x + 6 b c d x + 2 b c )log(F)
+
6 6 5 5 2 4 4 3 3 3 4 2 2 5
- 2 d x - 12 c d x - 30 c d x - 40 c d x - 30 c d x - 12 c d x
+
6
- 2 c
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
/
3 7 6 3 6 5 3 2 5 4 3 3 4 3 3 4 3 2
3 b d x + 18 b c d x + 45 b c d x + 60 b c d x + 45 b c d x
+
3 5 2 3 6
18 b c d x + 3 b c d
*
3
log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 (EV) = 0.02 sec
integrate(F^(a+b/(c+d*x)^3)/(c+d*x)^13, x)
(155)
3 3 2 3 3 2 2 2 2 2 2 3 2
- b log(F) + (3 b d x + 9 b c d x + 9 b c d x + 3 b c )log(F)
+
6 6 5 5 2 4 4 3 3 3 4 2 2
- 6 b d x - 36 b c d x - 90 b c d x - 120 b c d x - 90 b c d x
+
5 6
- 36 b c d x - 6 b c
*
log(F)
+
9 9 8 8 2 7 7 3 6 6 4 5 5 5 4 4
6 d x + 54 c d x + 216 c d x + 504 c d x + 756 c d x + 756 c d x
+
6 3 3 7 2 2 8 9
504 c d x + 216 c d x + 54 c d x + 6 c
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
/
4 10 9 4 9 8 4 2 8 7 4 3 7 6 4 4 6 5
3 b d x + 27 b c d x + 108 b c d x + 252 b c d x + 378 b c d x
+
4 5 5 4 4 6 4 3 4 7 3 2 4 8 2 4 9
378 b c d x + 252 b c d x + 108 b c d x + 27 b c d x + 3 b c d
*
4
log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.02 (EV) = 0.03 sec
integrate(F^(a+b/(c+d*x)^3)/(c+d*x)^16, x)
(156)
4 4 3 3 3 3 2 2 3 2 3 3 3
- b log(F) + (4 b d x + 12 b c d x + 12 b c d x + 4 b c )log(F)
+
2 6 6 2 5 5 2 2 4 4 2 3 3 3
- 12 b d x - 72 b c d x - 180 b c d x - 240 b c d x
+
2 4 2 2 2 5 2 6
- 180 b c d x - 72 b c d x - 12 b c
*
2
log(F)
+
9 9 8 8 2 7 7 3 6 6
24 b d x + 216 b c d x + 864 b c d x + 2016 b c d x
+
4 5 5 5 4 4 6 3 3 7 2 2
3024 b c d x + 3024 b c d x + 2016 b c d x + 864 b c d x
+
8 9
216 b c d x + 24 b c
*
log(F)
+
12 12 11 11 2 10 10 3 9 9 4 8 8
- 24 d x - 288 c d x - 1584 c d x - 5280 c d x - 11880 c d x
+
5 7 7 6 6 6 7 5 5 8 4 4
- 19008 c d x - 22176 c d x - 19008 c d x - 11880 c d x
+
9 3 3 10 2 2 11 12
- 5280 c d x - 1584 c d x - 288 c d x - 24 c
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
/
5 13 12 5 12 11 5 2 11 10 5 3 10 9
3 b d x + 36 b c d x + 198 b c d x + 660 b c d x
+
5 4 9 8 5 5 8 7 5 6 7 6 5 7 6 5
1485 b c d x + 2376 b c d x + 2772 b c d x + 2376 b c d x
+
5 8 5 4 5 9 4 3 5 10 3 2 5 11 2 5 12
1485 b c d x + 660 b c d x + 198 b c d x + 36 b c d x + 3 b c d
*
5
log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.06 (EV) = 0.06 sec
integrate(F^(a+b/(c+d*x)^3)/(c+d*x)^19, x)
(157)
5 5 4 3 3 4 2 2 4 2 4 3 4
- b log(F) + (5 b d x + 15 b c d x + 15 b c d x + 5 b c )log(F)
+
3 6 6 3 5 5 3 2 4 4 3 3 3 3
- 20 b d x - 120 b c d x - 300 b c d x - 400 b c d x
+
3 4 2 2 3 5 3 6
- 300 b c d x - 120 b c d x - 20 b c
*
3
log(F)
+
2 9 9 2 8 8 2 2 7 7 2 3 6 6
60 b d x + 540 b c d x + 2160 b c d x + 5040 b c d x
+
2 4 5 5 2 5 4 4 2 6 3 3 2 7 2 2
7560 b c d x + 7560 b c d x + 5040 b c d x + 2160 b c d x
+
2 8 2 9
540 b c d x + 60 b c
*
2
log(F)
+
12 12 11 11 2 10 10 3 9 9
- 120 b d x - 1440 b c d x - 7920 b c d x - 26400 b c d x
+
4 8 8 5 7 7 6 6 6
- 59400 b c d x - 95040 b c d x - 110880 b c d x
+
7 5 5 8 4 4 9 3 3 10 2 2
- 95040 b c d x - 59400 b c d x - 26400 b c d x - 7920 b c d x
+
11 12
- 1440 b c d x - 120 b c
*
log(F)
+
15 15 14 14 2 13 13 3 12 12
120 d x + 1800 c d x + 12600 c d x + 54600 c d x
+
4 11 11 5 10 10 6 9 9 7 8 8
163800 c d x + 360360 c d x + 600600 c d x + 772200 c d x
+
8 7 7 9 6 6 10 5 5 11 4 4
772200 c d x + 600600 c d x + 360360 c d x + 163800 c d x
+
12 3 3 13 2 2 14 15
54600 c d x + 12600 c d x + 1800 c d x + 120 c
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
/
6 16 15 6 15 14 6 2 14 13 6 3 13 12
3 b d x + 45 b c d x + 315 b c d x + 1365 b c d x
+
6 4 12 11 6 5 11 10 6 6 10 9 6 7 9 8
4095 b c d x + 9009 b c d x + 15015 b c d x + 19305 b c d x
+
6 8 8 7 6 9 7 6 6 10 6 5 6 11 5 4
19305 b c d x + 15015 b c d x + 9009 b c d x + 4095 b c d x
+
6 12 4 3 6 13 3 2 6 14 2 6 15
1365 b c d x + 315 b c d x + 45 b c d x + 3 b c d
*
6
log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.11 (EV) = 0.11 sec
integrate(F^(a+b/(c+d*x)^3)*(c+d*x)^3, x)
(158)
3 2 6 4 5 3 2 4 2 3 3
(3 b d x + 3 b c d )log(F) + d x + 4 c d x + 6 c d x + 4 c d x
+
4 2
c d
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3 +----------+2
d x + 3 c d x + 3 c d x + c | b log(F)
%e |- --------
3| 3
\| d
+
2 2 a log(F) 2 b log(F)
3 b log(F) %e Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
+----------+2
3 | b log(F)
4 d |- --------
3| 3
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^3)*(c+d*x), x)
(159)
3 2 2 2
(d x + 2 c d x + c d)
*
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3 +----------+
d x + 3 c d x + 3 c d x + c | b log(F)
%e |- --------
3| 3
\| d
+
a log(F) 1 b log(F)
b log(F)%e Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
+----------+
2 | b log(F)
2 d |- --------
3| 3
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^3), x)
(160)
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3
3 2 d x + 3 c d x + 3 c d x + c
(d x + c d )%e
*
+----------+2
| b log(F)
|- --------
3| 3
\| d
+
a log(F) 2 b log(F)
b log(F)%e Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
+----------+2
3 | b log(F)
d |- --------
3| 3
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^3)/(c+d*x)^2, x)
a log(F) 1 b log(F)
%e Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
(161) ---------------------------------------------------
+----------+
2 | b log(F)
3 d |- --------
3| 3
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^3)/(c+d*x)^3, x)
a log(F) 2 b log(F)
%e Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
(162) ---------------------------------------------------
+----------+2
3 | b log(F)
3 d |- --------
3| 3
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x)^3)/(c+d*x)^5, x)
(163)
3 3 2 2 2 3
(a d x + 3 a c d x + 3 a c d x + a c + b)log(F)
--------------------------------------------------
3 3 2 2 2 3 +----------+
d x + 3 c d x + 3 c d x + c | b log(F)
- 3 d %e |- --------
3| 3
\| d
+
a log(F) 1 b log(F)
(- d x - c)%e Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
+----------+
3 2 | b log(F)
(9 b d x + 9 b c d )log(F) |- --------
3| 3
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
-- n symbolic
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^m, x)
x n
++ m b (%A d + c) + a
(164) | (%A d + c) F d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^3, x)
x n
++ 3 3 2 2 2 3 b (%A d + c) + a
(165) | (%A d + 3 %A c d + 3 %A c d + c )F d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^2, x)
x n
++ 2 2 2 b (%A d + c) + a
(166) | (%A d + 2 %A c d + c )F d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x), x)
x n
++ b (%A d + c) + a
(167) | (%A d + c)F d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n), x)
x n
++ b (%A d + c) + a
(168) | F d%A
++
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)/(c+d*x), x)
a log(F) n log(d x + c)
%e Ei(b log(F)%e )
(169) --------------------------------------
d n
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)/(c+d*x)^2, x)
n
x b (%A d + c) + a
++ F
(170) | --------------------- d%A
++ 2 2 2
%A d + 2 %A c d + c
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)/(c+d*x)^3, x)
n
x b (%A d + c) + a
++ F
(171) | --------------------------------- d%A
++ 3 3 2 2 2 3
%A d + 3 %A c d + 3 %A c d + c
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)/(c+d*x)^4, x)
n
x b (%A d + c) + a
++ F
(172) | --------------------------------------------- d%A
++ 4 4 3 3 2 2 2 3 4
%A d + 4 %A c d + 6 %A c d + 4 %A c d + c
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1+6*n), x)
(173)
5 5 n log(d x + c) 5 4 4 n log(d x + c) 4
b log(F) (%e ) - 5 b log(F) (%e )
+
3 3 n log(d x + c) 3 2 2 n log(d x + c) 2
20 b log(F) (%e ) - 60 b log(F) (%e )
+
n log(d x + c)
120 b log(F)%e - 120
*
n log(d x + c)
b log(F)%e + a log(F)
%e
/
6 6
b d n log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1+5*n), x)
(174)
4 4 n log(d x + c) 4 3 3 n log(d x + c) 3
b log(F) (%e ) - 4 b log(F) (%e )
+
2 2 n log(d x + c) 2 n log(d x + c)
12 b log(F) (%e ) - 24 b log(F)%e + 24
*
n log(d x + c)
b log(F)%e + a log(F)
%e
/
5 5
b d n log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1+4*n), x)
(175)
3 3 n log(d x + c) 3 2 2 n log(d x + c) 2
b log(F) (%e ) - 3 b log(F) (%e )
+
n log(d x + c)
6 b log(F)%e - 6
*
n log(d x + c)
b log(F)%e + a log(F)
%e
/
4 4
b d n log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1+3*n), x)
(176)
2 2 n log(d x + c) 2 n log(d x + c)
(b log(F) (%e ) - 2 b log(F)%e + 2)
*
n log(d x + c)
b log(F)%e + a log(F)
%e
/
3 3
b d n log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1+2*n), x)
n log(d x + c)
n log(d x + c) b log(F)%e + a log(F)
(b log(F)%e - 1)%e
(177) -------------------------------------------------------------------
2 2
b d n log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1+n), x)
n log(d x + c)
b log(F)%e + a log(F)
%e
(178) -------------------------------------
b d n log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)/(c+d*x), x)
a log(F) n log(d x + c)
%e Ei(b log(F)%e )
(179) --------------------------------------
d n
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1-n), x)
(180)
n log(d x + c)
b log(F)%e + a log(F)
- %e
+
a log(F) n log(d x + c) n log(d x + c)
b log(F)%e %e Ei(b log(F)%e )
/
n log(d x + c)
d n %e
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1-2*n), x)
(181)
n log(d x + c)
n log(d x + c) b log(F)%e + a log(F)
(- b log(F)%e - 1)%e
+
2 2 a log(F) n log(d x + c) 2 n log(d x + c)
b log(F) %e (%e ) Ei(b log(F)%e )
/
n log(d x + c) 2
2 d n (%e )
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1-3*n), x)
(182)
2 2 n log(d x + c) 2 n log(d x + c)
(- b log(F) (%e ) - b log(F)%e - 2)
*
n log(d x + c)
b log(F)%e + a log(F)
%e
+
3 3 a log(F) n log(d x + c) 3 n log(d x + c)
b log(F) %e (%e ) Ei(b log(F)%e )
/
n log(d x + c) 3
6 d n (%e )
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1-4*n), x)
(183)
3 3 n log(d x + c) 3 2 2 n log(d x + c) 2
- b log(F) (%e ) - b log(F) (%e )
+
n log(d x + c)
- 2 b log(F)%e - 6
*
n log(d x + c)
b log(F)%e + a log(F)
%e
+
4 4 a log(F) n log(d x + c) 4 n log(d x + c)
b log(F) %e (%e ) Ei(b log(F)%e )
/
n log(d x + c) 4
24 d n (%e )
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^n)*(c+d*x)^(-1-5*n), x)
(184)
4 4 n log(d x + c) 4 3 3 n log(d x + c) 3
- b log(F) (%e ) - b log(F) (%e )
+
2 2 n log(d x + c) 2 n log(d x + c)
- 2 b log(F) (%e ) - 6 b log(F)%e - 24
*
n log(d x + c)
b log(F)%e + a log(F)
%e
+
5 5 a log(F) n log(d x + c) 5 n log(d x + c)
b log(F) %e (%e ) Ei(b log(F)%e )
/
n log(d x + c) 5
120 d n (%e )
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(c*(a+b*x)^n)*(a+b*x)^(-1+1/2*n), x)
(n - 2)log(b x + a)
-------------------
+---+ 2 +----------+
\|%pi erf((b x + a)%e \|- c log(F) )
(185) ------------------------------------------------------
+----------+
b n\|- c log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate((a+b*x)^(-1+1/2*n)/F^(c*(a+b*x)^n), x)
(n - 2)log(b x + a)
-------------------
+---+ 2 +--------+
\|%pi erf((b x + a)%e \|c log(F) )
(186) ----------------------------------------------------
+--------+
b n\|c log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
-- Integrands of the form (e+f x)^m F^(a+b (c+d x)^n)
-- n>0
integrate(F^(a+b*(c+d*x)^2)*(e+f*x)^5, x)
(187)
3 5 5 3 4 2 4 3 3 3 2 3 3 2 4 3 2
- 4 b c d f + 20 b c d e f - 40 b c d e f + 40 b c d e f
+
3 5 4 3 6 5
- 20 b c d e f + 4 b d e
*
3
log(F)
+
2 3 5 2 2 2 4 2 3 2 3 2 4 3 2 2
(20 b c d f - 60 b c d e f + 60 b c d e f - 20 b d e f )log(F)
+
5 2 4
(- 15 b c d f + 15 b d e f )log(F)
*
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
\|%pi %e erf(------------------------)
d
+
2 4 5 4 2 3 5 2 4 4 3
4 b d f x + (- 4 b c d f + 20 b d e f )x
+
2 2 2 5 2 3 4 2 4 2 3 2
(4 b c d f - 20 b c d e f + 40 b d e f )x
+
2 3 5 2 2 2 4 2 3 2 3 2 4 3 2
(- 4 b c d f + 20 b c d e f - 40 b c d e f + 40 b d e f )x
+
2 4 5 2 3 4 2 2 2 2 3 2 3 3 2
4 b c f - 20 b c d e f + 40 b c d e f - 40 b c d e f
+
2 4 4
20 b d e f
*
2
log(F)
+
2 5 2 5 2 4 2 5
- 8 b d f x + (14 b c d f - 30 b d e f )x - 18 b c f
+
4 2 2 3
50 b c d e f - 40 b d e f
*
log(F)
+
5
8 f
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
+------------+
3 6 3 | 2
8 b d log(F) \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(F^(a+b*(c+d*x)^2)*(e+f*x)^4, x)
(188)
2 4 4 2 3 2 3 2 2 3 2 2 2 4 3
4 b c d f - 16 b c d e f + 24 b c d e f - 16 b c d e f
+
2 5 4
4 b d e
*
2
log(F)
+
2 4 2 3 3 2 2 4
(- 12 b c d f + 24 b c d e f - 12 b d e f )log(F) + 3 d f
*
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
\|%pi %e erf(------------------------)
d
+
3 4 3 2 4 3 3 2
4 b d f x + (- 4 b c d f + 16 b d e f )x
+
2 4 2 3 3 2 2 3 4
(4 b c d f - 16 b c d e f + 24 b d e f )x - 4 b c f
+
2 3 2 2 2 3 3
16 b c d e f - 24 b c d e f + 16 b d e f
*
log(F)
+
4 4 3
- 6 d f x + 10 c f - 16 d e f
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
+------------+
2 5 2 | 2
8 b d log(F) \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(e+f*x)^3, x)
(189)
2 3 3 2 2 2 2 2 3 2 2 4 3 2
(- 2 b c d f + 6 b c d e f - 6 b c d e f + 2 b d e )log(F)
+
3 2 2
(3 b c d f - 3 b d e f )log(F)
*
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
\|%pi %e erf(------------------------)
d
+
2 3 2 3 2 2 2 3
2 b d f x + (- 2 b c d f + 6 b d e f )x + 2 b c f
+
2 2 2
- 6 b c d e f + 6 b d e f
*
log(F)
+
3
- 2 f
*
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
%e \|- b d log(F)
/
+------------+
2 4 2 | 2
4 b d log(F) \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(e+f*x)^2, x)
(190)
2 2 2 3 2 2 +---+ a log(F)
((2 b c d f - 4 b c d e f + 2 b d e )log(F) - d f )\|%pi %e
*
+------------+
| 2
(d x + c)\|- b d log(F)
erf(------------------------)
d
+
2 2 2
2 2 (b d x + 2 b c d x + b c + a)log(F)
(2 d f x - 2 c f + 4 d e f)%e
*
+------------+
| 2
\|- b d log(F)
/
+------------+
3 | 2
4 b d log(F)\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)*(e+f*x), x)
(191)
+------------+
| 2
2 +---+ a log(F) (d x + c)\|- b d log(F)
(- b c d f + b d e)log(F)\|%pi %e erf(------------------------)
d
+
2 2 2 +------------+
(b d x + 2 b c d x + b c + a)log(F) | 2
f %e \|- b d log(F)
/
+------------+
2 | 2
2 b d log(F)\|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2), x)
+------------+
| 2
+---+ a log(F) (d x + c)\|- b d log(F)
\|%pi %e erf(------------------------)
d
(192) ---------------------------------------------
+------------+
| 2
2 \|- b d log(F)
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(e+f*x), x)
2 2 2
x %A b d + 2 %A b c d + b c + a
++ F
(193) | -------------------------------- d%A
++ %A f + e
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(e+f*x)^2, x)
2 2 2
x %A b d + 2 %A b c d + b c + a
++ F
(194) | -------------------------------- d%A
++ 2 2 2
%A f + 2 %A e f + e
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b*(c+d*x)^2)/(e+f*x)^3, x)
2 2 2
x %A b d + 2 %A b c d + b c + a
++ F
(195) | --------------------------------- d%A
++ 3 3 2 2 2 3
%A f + 3 %A e f + 3 %A e f + e
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e*(c+d*x)^3)*(a+b*x)^3, x)
(196)
3 3 2 2 2 3
3 2 3 d e x + 3 c d e x + 3 c d e x + c e
(3 b d x + 9 a b d - 6 b c)%e
*
+-----+2
3| 3
\|- d e
+
3 4 2 3 2 2 2 3 3 3
((- 3 a d + 9 a b c d - 9 a b c d + 3 b c d)e + b d)
*
+-----+
1 3 3 2 2 2 3 3| 3
Gamma(-,- d e x - 3 c d e x - 3 c d e x - c e)\|- d e
3
+
2 4 2 3 3 2 2
(- 9 a b d + 18 a b c d - 9 b c d )e
*
2 3 3 2 2 2 3
Gamma(-,- d e x - 3 c d e x - 3 c d e x - c e)
3
/
+-----+2
4 3| 3
9 d e \|- d e
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e*(c+d*x)^3)*(a+b*x)^2, x)
(197)
3 3 2 2 2 3 +-----+2
2 d e x + 3 c d e x + 3 c d e x + c e 3| 3
b %e \|- d e
+
2 3 2 2 2
(- a d + 2 a b c d - b c d)e
*
+-----+
1 3 3 2 2 2 3 3| 3
Gamma(-,- d e x - 3 c d e x - 3 c d e x - c e)\|- d e
3
+
3 2 2 2 3 3 2 2 2 3
(- 2 a b d + 2 b c d )e Gamma(-,- d e x - 3 c d e x - 3 c d e x - c e)
3
/
+-----+2
3 3| 3
3 d e \|- d e
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e*(c+d*x)^3)*(a+b*x), x)
(198)
+-----+
1 3 3 2 2 2 3 3| 3
(- a d + b c)Gamma(-,- d e x - 3 c d e x - 3 c d e x - c e)\|- d e
3
+
2 3 3 2 2 2 3
- b d Gamma(-,- d e x - 3 c d e x - 3 c d e x - c e)
3
/
+-----+2
3| 3
3 d \|- d e
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e*(c+d*x)^3), x)
1 3 3 2 2 2 3
Gamma(-,- d e x - 3 c d e x - 3 c d e x - c e)
3
(199) - ------------------------------------------------
+-----+
3| 3
3 \|- d e
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e*(c+d*x)^3)/(a+b*x), x)
3 3 2 2 2 3
x (%A d + 3 %A c d + 3 %A c d + c )e
++ %e
(200) | -------------------------------------- d%A
++ %A b + a
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e*(c+d*x)^3)/(a+b*x)^2, x)
3 3 2 2 2 3
x (%A d + 3 %A c d + 3 %A c d + c )e
++ %e
(201) | -------------------------------------- d%A
++ 2 2 2
%A b + 2 %A a b + a
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
-- n<0
integrate(F^(a+b/(c+d*x))/(e+f*x), x)
(202)
((a c + b)f - a d e)log(F)
--------------------------
(- b d f x - b d e)log(F) c f - d e
Ei(----------------------------)%e
2 2
(c d f - d e)x + c f - c d e
+
b log(F) a log(F)
- Ei(--------)%e
d x + c
/
f
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))/(e+f*x)^2, x)
(203)
(- b d f x - b d e)log(F)
(- b d f x - b d e)log(F)Ei(----------------------------)
2 2
(c d f - d e)x + c f - c d e
*
((a c + b)f - a d e)log(F)
--------------------------
c f - d e
%e
+
(a d x + a c + b)log(F)
-----------------------
2 2 d x + c
((- c d f + d e)x - c f + c d e)%e
/
2 3 2 2 2 2 2 2 2 3
(c f - 2 c d e f + d e f)x + c e f - 2 c d e f + d e
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(F^(a+b/(c+d*x))/(e+f*x)^3, x)
(204)
2 2 3 2 2 2 2 2 2 2 2
(b d f x + 2 b d e f x + b d e f)log(F)
+
2 3 3 2 2 2 2 3 2
(2 b c d f - 2 b d e f )x + (4 b c d e f - 4 b d e f)x
+
2 2 3 3
2 b c d e f - 2 b d e
*
log(F)
*
((a c + b)f - a d e)log(F)
--------------------------
(- b d f x - b d e)log(F) c f - d e
Ei(----------------------------)%e
2 2
(c d f - d e)x + c f - c d e
+
2 3 3 2 2 2 3 3 2 2 2
(b c d f - b d e f )x + (b c d f - b d e f)x + b c d e f
+
2 2
- b c d e f
*
log(F)
+
2 2 3 3 2 4 2 2 2 2 2 3 2 4 3
(c d f - 2 c d e f + d e f)x + (2 c d e f - 4 c d e f + 2 d e )x
+
4 3 3 2 2 2 2 3 3
- c f + 4 c d e f - 5 c d e f + 2 c d e
*
(a d x + a c + b)log(F)
-----------------------
d x + c
%e
/
4 6 3 5 2 2 2 4 3 3 3 4 4 2 2
(2 c f - 8 c d e f + 12 c d e f - 8 c d e f + 2 d e f )x
+
4 5 3 2 4 2 2 3 3 3 4 2 4 5
(4 c e f - 16 c d e f + 24 c d e f - 16 c d e f + 4 d e f)x
+
4 2 4 3 3 3 2 2 4 2 3 5 4 6
2 c e f - 8 c d e f + 12 c d e f - 8 c d e f + 2 d e
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 (EV) = 0.02 sec
integrate(F^(a+b/(c+d*x))/(e+f*x)^4, x)
(205)
3 3 5 3 3 3 4 2 3 3 2 3 3 3 3 2 3
(- b d f x - 3 b d e f x - 3 b d e f x - b d e f )log(F)
+
2 3 5 2 4 4 3 2 3 4 2 4 2 3 2
(- 6 b c d f + 6 b d e f )x + (- 18 b c d e f + 18 b d e f )x
+
2 3 2 3 2 4 3 2 2 3 3 2 2 4 4
(- 18 b c d e f + 18 b d e f )x - 6 b c d e f + 6 b d e f
*
2
log(F)
+
2 3 5 4 4 5 2 3 3
(- 6 b c d f + 12 b c d e f - 6 b d e f )x
+
2 3 4 4 2 3 5 3 2 2
(- 18 b c d e f + 36 b c d e f - 18 b d e f )x
+
2 3 2 3 4 3 2 5 4 2 3 3 2
(- 18 b c d e f + 36 b c d e f - 18 b d e f)x - 6 b c d e f
+
4 4 5 5
12 b c d e f - 6 b d e
*
log(F)
*
((a c + b)f - a d e)log(F)
--------------------------
(- b d f x - b d e)log(F) c f - d e
Ei(----------------------------)%e
2 2
(c d f - d e)x + c f - c d e
+
2 3 5 2 4 4 3
(- b c d f + b d e f )x
+
2 2 2 5 2 3 4 2 4 2 3 2
(- b c d f - b c d e f + 2 b d e f )x
+
2 2 2 4 2 3 2 3 2 4 3 2 2 2 2 2 3 2 3 3 2
(- 2 b c d e f + b c d e f + b d e f )x - b c d e f + b c d e f
*
2
log(F)
+
2 3 5 4 4 5 2 3 3
(- 5 b c d f + 10 b c d e f - 5 b d e f )x
+
3 2 5 2 3 4 4 2 3 5 3 2 2
(- 4 b c d f - 3 b c d e f + 18 b c d e f - 11 b d e f )x
+
4 5 3 2 4 2 3 2 3 4 3 2
b c d f - 12 b c d e f + 15 b c d e f + 2 b c d e f
+
5 4
- 6 b d e f
*
x
+
4 4 3 2 2 3 2 3 3 2 4 4
b c d e f - 8 b c d e f + 13 b c d e f - 6 b c d e f
*
log(F)
+
3 3 5 2 4 4 5 2 3 6 3 2 3
(- 2 c d f + 6 c d e f - 6 c d e f + 2 d e f )x
+
3 3 4 2 4 2 3 5 3 2 6 4 2
(- 6 c d e f + 18 c d e f - 18 c d e f + 6 d e f)x
+
3 3 2 3 2 4 3 2 5 4 6 5 6 5
(- 6 c d e f + 18 c d e f - 18 c d e f + 6 d e )x - 2 c f
+
5 4 4 2 2 3 3 3 3 2 2 4 4 5 5
12 c d e f - 30 c d e f + 38 c d e f - 24 c d e f + 6 c d e
*
(a d x + a c + b)log(F)
-----------------------
d x + c
%e
/
6 9 5 8 4 2 2 7 3 3 3 6 2 4 4 5
6 c f - 36 c d e f + 90 c d e f - 120 c d e f + 90 c d e f
+
5 5 4 6 6 3
- 36 c d e f + 6 d e f
*
3
x
+
6 8 5 2 7 4 2 3 6 3 3 4 5 2 4 5 4
18 c e f - 108 c d e f + 270 c d e f - 360 c d e f + 270 c d e f
+
5 6 3 6 7 2
- 108 c d e f + 18 d e f
*
2
x
+
6 2 7 5 3 6 4 2 4 5 3 3 5 4 2 4 6 3
18 c e f - 108 c d e f + 270 c d e f - 360 c d e f + 270 c d e f
+
5 7 2 6 8
- 108 c d e f + 18 d e f
*
x
+
6 3 6 5 4 5 4 2 5 4 3 3 6 3 2 4 7 2
6 c e f - 36 c d e f + 90 c d e f - 120 c d e f + 90 c d e f
+
5 8 6 9
- 36 c d e f + 6 d e
Type: Union(Expression(Integer),...)
fricas
Time: 0.04 (EV) = 0.04 sec
integrate(%e^(e/(c+d*x))*(a+b*x)^4, x)
(206)
4 5 5 4 4 3 5 4
24 b d x + (6 b d e + 120 a b d )x
+
4 3 2 3 4 4 3 2 2 5 3
(2 b d e + (40 a b d - 16 b c d )e + 240 a b d )x
+
4 2 3 3 3 4 2 2
b d e + (20 a b d - 14 b c d )e
+
2 2 4 3 3 4 2 2 3 5
(120 a b d - 120 a b c d + 36 b c d )e + 240 a b d
*
2
x
+
4 4 3 2 4 3
b d e + (20 a b d - 18 b c d)e
+
2 2 3 3 2 4 2 2
(120 a b d - 200 a b c d + 86 b c d)e
+
3 4 2 2 3 3 2 2 4 3 4 5
(240 a b d - 480 a b c d + 360 a b c d - 96 b c d)e + 120 a d
*
x
+
4 4 3 4 2 3
b c e + (20 a b c d - 19 b c )e
+
2 2 2 3 2 4 3 2
(120 a b c d - 220 a b c d + 102 b c )e
+
3 3 2 2 2 2 3 3 4 4 4 4
(240 a b c d - 600 a b c d + 520 a b c d - 154 b c )e + 120 a c d
+
3 2 3 2 2 3 2 3 4 4 5
- 240 a b c d + 240 a b c d - 120 a b c d + 24 b c
*
e
-------
d x + c
%e
+
4 5 3 4 4
- b e + (- 20 a b d + 20 b c)e
+
2 2 2 3 4 2 3
(- 120 a b d + 240 a b c d - 120 b c )e
+
3 3 2 2 2 3 2 4 3 2
(- 240 a b d + 720 a b c d - 720 a b c d + 240 b c )e
+
4 4 3 3 2 2 2 2 3 3 4 4
(- 120 a d + 480 a b c d - 720 a b c d + 480 a b c d - 120 b c )e
*
e
Ei(-------)
d x + c
/
5
120 d
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 (EV) = 0.02 sec
integrate(%e^(e/(c+d*x))*(a+b*x)^3, x)
(207)
3 4 4 3 3 2 4 3
6 b d x + (2 b d e + 24 a b d )x
+
3 2 2 2 3 3 2 2 4 2
(b d e + (12 a b d - 6 b c d )e + 36 a b d )x
+
3 3 2 2 3 2
b d e + (12 a b d - 10 b c d)e
+
2 3 2 2 3 2 3 4
(36 a b d - 48 a b c d + 18 b c d)e + 24 a d
*
x
+
3 3 2 3 2 2
b c e + (12 a b c d - 11 b c )e
+
2 2 2 2 3 3 3 3 2 2 2
(36 a b c d - 60 a b c d + 26 b c )e + 24 a c d - 36 a b c d
+
2 3 3 4
24 a b c d - 6 b c
*
e
-------
d x + c
%e
+
3 4 2 3 3
- b e + (- 12 a b d + 12 b c)e
+
2 2 2 3 2 2
(- 36 a b d + 72 a b c d - 36 b c )e
+
3 3 2 2 2 2 3 3
(- 24 a d + 72 a b c d - 72 a b c d + 24 b c )e
*
e
Ei(-------)
d x + c
/
4
24 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x))*(a+b*x)^2, x)
(208)
2 3 3 2 2 3 2
2 b d x + (b d e + 6 a b d )x
+
2 2 2 2 2 3 2 2
(b d e + (6 a b d - 4 b c d)e + 6 a d )x + b c e
+
2 2 2 2 2 2 3
(6 a b c d - 5 b c )e + 6 a c d - 6 a b c d + 2 b c
*
e
-------
d x + c
%e
+
2 3 2 2 2 2 2 2
(- b e + (- 6 a b d + 6 b c)e + (- 6 a d + 12 a b c d - 6 b c )e)
*
e
Ei(-------)
d x + c
/
3
6 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x))*(a+b*x), x)
(209)
e
-------
2 2 2 2 d x + c
(b d x + (b d e + 2 a d )x + b c e + 2 a c d - b c )%e
+
2 e
(- b e + (- 2 a d + 2 b c)e)Ei(-------)
d x + c
/
2
2 d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x)), x)
e
-------
d x + c e
(d x + c)%e - e Ei(-------)
d x + c
(210) ----------------------------------
d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x))/(a+b*x), x)
b e
- ---------
b d e x + a d e a d - b c e
Ei(------------------------------)%e - Ei(-------)
2 2 d x + c
(a d - b c d)x + a c d - b c
(211) -------------------------------------------------------------
b
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x))/(a+b*x)^2, x)
(212)
e
-------
2 2 d x + c
((a d - b c d)x + a c d - b c )%e
+
b e
- ---------
b d e x + a d e a d - b c
(- b d e x - a d e)Ei(------------------------------)%e
2 2
(a d - b c d)x + a c d - b c
/
2 2 2 3 2 3 2 2 2 2
(a b d - 2 a b c d + b c )x + a d - 2 a b c d + a b c
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x))/(a+b*x)^3, x)
(213)
2 3 3 2 2 4 2 3 3 2 2 2
((- a b d + b c d )e + a b d - 2 a b c d + b c d )x
+
2 3 3 2 3 4 2 3 2 2 2
((- a b d + b c d)e + 2 a d - 4 a b c d + 2 a b c d )x
+
2 2 2 2 3 3 2 2 2 2 3 3 4
(- a b c d + a b c d)e + 2 a c d - 5 a b c d + 4 a b c d - b c
*
e
-------
d x + c
%e
+
3 2 2 2 3 3 2 2
(b d e + (- 2 a b d + 2 b c d )e)x
+
2 2 2 2 3 2 2 2 2 2
(2 a b d e + (- 4 a b d + 4 a b c d )e)x + a b d e
+
3 3 2 2
(- 2 a d + 2 a b c d )e
*
b e
- ---------
b d e x + a d e a d - b c
Ei(------------------------------)%e
2 2
(a d - b c d)x + a c d - b c
/
4 2 4 3 3 3 2 4 2 2 5 3 6 4 2
(2 a b d - 8 a b c d + 12 a b c d - 8 a b c d + 2 b c )x
+
5 4 4 2 3 3 3 2 2 2 4 3 5 4 6 4
(4 a b d - 16 a b c d + 24 a b c d - 16 a b c d + 4 a b c )x + 2 a d
+
5 3 4 2 2 2 3 3 3 2 4 4
- 8 a b c d + 12 a b c d - 8 a b c d + 2 a b c
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(%e^(e/(c+d*x)^2)*(a+b*x)^3, x)
(214)
2 3 2
(- 8 a b d + 8 b c)e
+
3 3 2 2 2 2 3 3
(- 4 a d + 12 a b c d - 12 a b c d + 4 b c )e
*
+----+
| e
d |- --
| 2
+---+ \| d
\|%pi erf(--------)
d x + c
+
3 5 4 2 5 3 3 3 2 5 2
b d x + 4 a b d x + (b d e + 6 a b d )x
+
2 3 3 2 3 5 2 2 3 2
((8 a b d - 6 b c d )e + 4 a d )x + (8 a b c d - 7 b c d)e
+
3 4 2 2 3 2 3 2 3 4
4 a c d - 6 a b c d + 4 a b c d - b c d
*
e
-------------------
2 2 2
d x + 2 c d x + c
%e
+
3 2 2 3 2 2 3 2
(- b d e + (- 6 a b d + 12 a b c d - 6 b c d)e)
*
e
Ei(-------------------)
2 2 2
d x + 2 c d x + c
*
+----+
| e
|- --
| 2
\| d
/
+----+
5 | e
4 d |- --
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(%e^(e/(c+d*x)^2)*(a+b*x)^2, x)
(215)
+----+
| e
d |- --
| 2
2 2 2 2 2 2 +---+ \| d
(- 2 b e + (- 3 a d + 6 a b c d - 3 b c )e)\|%pi erf(--------)
d x + c
+
2 4 3 4 2 2 2 2 4 2 2 3
b d x + 3 a b d x + (2 b d e + 3 a d )x + 2 b c d e + 3 a c d
+
2 2 2 3
- 3 a b c d + b c d
*
e
-------------------
2 2 2
d x + 2 c d x + c
%e
+
2 2 e
(- 3 a b d + 3 b c d)e Ei(-------------------)
2 2 2
d x + 2 c d x + c
*
+----+
| e
|- --
| 2
\| d
/
+----+
4 | e
3 d |- --
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x)^2)*(a+b*x), x)
(216)
+----+
| e
d |- --
| 2
+---+ \| d
(- 2 a d + 2 b c)e\|%pi erf(--------)
d x + c
+
e
-------------------
2 2 2
3 2 3 2 2 d x + 2 c d x + c
(b d x + 2 a d x + 2 a c d - b c d)%e
+
e
- b d e Ei(-------------------)
2 2 2
d x + 2 c d x + c
*
+----+
| e
|- --
| 2
\| d
/
+----+
3 | e
2 d |- --
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x)^2), x)
+----+
| e e
d |- -- -------------------
| 2 2 2 2 +----+
+---+ \| d 2 d x + 2 c d x + c | e
- e\|%pi erf(--------) + (d x + c d)%e |- --
d x + c | 2
\| d
(217) ----------------------------------------------------------------
+----+
2 | e
d |- --
| 2
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x)^2)/(a+b*x), x)
e
---------------------
2 2 2
x %A d + 2 %A c d + c
++ %e
(218) | ----------------------- d%A
++ %A b + a
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x)^2)/(a+b*x)^2, x)
e
---------------------
2 2 2
x %A d + 2 %A c d + c
++ %e
(219) | ----------------------- d%A
++ 2 2 2
%A b + 2 %A a b + a
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x)^2)/(a+b*x)^3, x)
e
---------------------
2 2 2
x %A d + 2 %A c d + c
++ %e
(220) | --------------------------------- d%A
++ 3 3 2 2 2 3
%A b + 3 %A a b + 3 %A a b + a
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(%e^(e/(c+d*x)^3)*(a+b*x)^3, x)
(221)
3 6 4 2 6 3 2 6 2 3 3 3 6
b d x + 4 a b d x + 6 a b d x + (3 b d e + 4 a d )x
+
3 2 3 5 2 2 4 2 3 3 3 4 2
3 b c d e + 4 a c d - 6 a b c d + 4 a b c d - b c d
*
e
------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
+
2 3 3 2 e
(- 4 a b d + 4 b c d )e Ei(------------------------------)
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
*
+----+2
| e
|- --
3| 3
\| d
+
2 3 2 2 3 2
(6 a b d - 12 a b c d + 6 b c d)e
*
+----+
1 e | e
Gamma(-,- ------------------------------) |- --
3 3 3 2 2 2 3 3| 3
d x + 3 c d x + 3 c d x + c \| d
+
3 2 3 3 2 2 2 2 3 3
(3 b e + (4 a d - 12 a b c d + 12 a b c d - 4 b c )e)
*
2 e
Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
+----+2
6 | e
4 d |- --
3| 3
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec
integrate(%e^(e/(c+d*x)^3)*(a+b*x)^2, x)
(222)
2 5 3 5 2 2 5 2 4 2 3 2 3 2
(b d x + 3 a b d x + 3 a d x + 3 a c d - 3 a b c d + b c d )
*
e
------------------------------
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
%e
+
2 2 e
- b d e Ei(------------------------------)
3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
*
+----+2
| e
|- --
3| 3
\| d
+
+----+
2 2 1 e | e
(3 a b d - 3 b c d)e Gamma(-,- ------------------------------) |- --
3 3 3 2 2 2 3 3| 3
d x + 3 c d x + 3 c d x + c \| d
+
2 2 2 2 2 e
(3 a d - 6 a b c d + 3 b c )e Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
+----+2
5 | e
3 d |- --
3| 3
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x)^3)*(a+b*x), x)
(223)
e
------------------------------
3 3 2 2 2 3
4 2 4 3 2 2 d x + 3 c d x + 3 c d x + c
(b d x + 2 a d x + 2 a c d - b c d )%e
*
+----+2
| e
|- --
3| 3
\| d
+
+----+
1 e | e
b d e Gamma(-,- ------------------------------) |- --
3 3 3 2 2 2 3 3| 3
d x + 3 c d x + 3 c d x + c \| d
+
2 e
(2 a d - 2 b c)e Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
+----+2
4 | e
2 d |- --
3| 3
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x)^3), x)
(224)
e
------------------------------
3 3 2 2 2 3 +----+2
3 2 d x + 3 c d x + 3 c d x + c | e
(d x + c d )%e |- --
3| 3
\| d
+
2 e
e Gamma(-,- ------------------------------)
3 3 3 2 2 2 3
d x + 3 c d x + 3 c d x + c
/
+----+2
3 | e
d |- --
3| 3
\| d
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x)^3)/(a+b*x), x)
e
---------------------------------
3 3 2 2 2 3
x %A d + 3 %A c d + 3 %A c d + c
++ %e
(225) | ----------------------------------- d%A
++ %A b + a
Type: Union(Expression(Integer),...)
fricas
Time: 0 sec
integrate(%e^(e/(c+d*x)^3)/(a+b*x)^2, x)
e
---------------------------------
3 3 2 2 2 3
x %A d + 3 %A c d + 3 %A c d + c
++ %e
(226) | ----------------------------------- d%A
++ 2 2 2
%A b + 2 %A a b + a
Type: Union(Expression(Integer),...)
fricas
Time: 0.01 sec