3.1749 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^7} \, dx\)

Optimal. Leaf size=116 \[ -\frac{2 a^5}{3 b^6 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{10 a^4}{b^6 \sqrt{a+\frac{b}{x}}}+\frac{20 a^3 \sqrt{a+\frac{b}{x}}}{b^6}-\frac{20 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^6}+\frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{b^6}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^6} \]

[Out]

(-2*a^5)/(3*b^6*(a + b/x)^(3/2)) + (10*a^4)/(b^6*Sqrt[a + b/x]) + (20*a^3*Sqrt[a
+ b/x])/b^6 - (20*a^2*(a + b/x)^(3/2))/(3*b^6) + (2*a*(a + b/x)^(5/2))/b^6 - (2
*(a + b/x)^(7/2))/(7*b^6)

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Rubi [A]  time = 0.135187, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 a^5}{3 b^6 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{10 a^4}{b^6 \sqrt{a+\frac{b}{x}}}+\frac{20 a^3 \sqrt{a+\frac{b}{x}}}{b^6}-\frac{20 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^6}+\frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{b^6}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^6} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(5/2)*x^7),x]

[Out]

(-2*a^5)/(3*b^6*(a + b/x)^(3/2)) + (10*a^4)/(b^6*Sqrt[a + b/x]) + (20*a^3*Sqrt[a
+ b/x])/b^6 - (20*a^2*(a + b/x)^(3/2))/(3*b^6) + (2*a*(a + b/x)^(5/2))/b^6 - (2
*(a + b/x)^(7/2))/(7*b^6)

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Rubi in Sympy [A]  time = 19.4184, size = 100, normalized size = 0.86 \[ - \frac{2 a^{5}}{3 b^{6} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{10 a^{4}}{b^{6} \sqrt{a + \frac{b}{x}}} + \frac{20 a^{3} \sqrt{a + \frac{b}{x}}}{b^{6}} - \frac{20 a^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{6}} + \frac{2 a \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{b^{6}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(5/2)/x**7,x)

[Out]

-2*a**5/(3*b**6*(a + b/x)**(3/2)) + 10*a**4/(b**6*sqrt(a + b/x)) + 20*a**3*sqrt(
a + b/x)/b**6 - 20*a**2*(a + b/x)**(3/2)/(3*b**6) + 2*a*(a + b/x)**(5/2)/b**6 -
2*(a + b/x)**(7/2)/(7*b**6)

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Mathematica [A]  time = 0.0610729, size = 80, normalized size = 0.69 \[ \frac{2 \sqrt{a+\frac{b}{x}} \left (256 a^5 x^5+384 a^4 b x^4+96 a^3 b^2 x^3-16 a^2 b^3 x^2+6 a b^4 x-3 b^5\right )}{21 b^6 x^3 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(5/2)*x^7),x]

[Out]

(2*Sqrt[a + b/x]*(-3*b^5 + 6*a*b^4*x - 16*a^2*b^3*x^2 + 96*a^3*b^2*x^3 + 384*a^4
*b*x^4 + 256*a^5*x^5))/(21*b^6*x^3*(b + a*x)^2)

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Maple [A]  time = 0.009, size = 77, normalized size = 0.7 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 256\,{a}^{5}{x}^{5}+384\,{a}^{4}b{x}^{4}+96\,{a}^{3}{b}^{2}{x}^{3}-16\,{a}^{2}{b}^{3}{x}^{2}+6\,a{b}^{4}x-3\,{b}^{5} \right ) }{21\,{x}^{6}{b}^{6}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x^7,x)

[Out]

2/21*(a*x+b)*(256*a^5*x^5+384*a^4*b*x^4+96*a^3*b^2*x^3-16*a^2*b^3*x^2+6*a*b^4*x-
3*b^5)/x^6/b^6/((a*x+b)/x)^(5/2)

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Maxima [A]  time = 1.46381, size = 132, normalized size = 1.14 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}}}{7 \, b^{6}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a}{b^{6}} - \frac{20 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{2}}{3 \, b^{6}} + \frac{20 \, \sqrt{a + \frac{b}{x}} a^{3}}{b^{6}} + \frac{10 \, a^{4}}{\sqrt{a + \frac{b}{x}} b^{6}} - \frac{2 \, a^{5}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^7),x, algorithm="maxima")

[Out]

-2/7*(a + b/x)^(7/2)/b^6 + 2*(a + b/x)^(5/2)*a/b^6 - 20/3*(a + b/x)^(3/2)*a^2/b^
6 + 20*sqrt(a + b/x)*a^3/b^6 + 10*a^4/(sqrt(a + b/x)*b^6) - 2/3*a^5/((a + b/x)^(
3/2)*b^6)

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Fricas [A]  time = 0.24142, size = 112, normalized size = 0.97 \[ \frac{2 \,{\left (256 \, a^{5} x^{5} + 384 \, a^{4} b x^{4} + 96 \, a^{3} b^{2} x^{3} - 16 \, a^{2} b^{3} x^{2} + 6 \, a b^{4} x - 3 \, b^{5}\right )}}{21 \,{\left (a b^{6} x^{5} + b^{7} x^{4}\right )} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^7),x, algorithm="fricas")

[Out]

2/21*(256*a^5*x^5 + 384*a^4*b*x^4 + 96*a^3*b^2*x^3 - 16*a^2*b^3*x^2 + 6*a*b^4*x
- 3*b^5)/((a*b^6*x^5 + b^7*x^4)*sqrt((a*x + b)/x))

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Sympy [A]  time = 43.1632, size = 9263, normalized size = 79.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(5/2)/x**7,x)

[Out]

512*a**(43/2)*b**(91/2)*x**18*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 31
5*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b*
*54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2
) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135
*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*
b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/
2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*
b**66*x**(7/2)) + 7424*a**(41/2)*b**(93/2)*x**17*sqrt(a*x/b + 1)/(21*a**(37/2)*b
**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2)
+ 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(2
7/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58
*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2)
+ 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(
13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(
9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 50112*a**(39/2)*b**(95/2)*x**16*sqrt(a*x/b
+ 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33
/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**
(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 13
5135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(1
9/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x
**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315
*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 208800*a**(37/2)*b**(97
/2)*x**15*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x*
*(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 2866
5*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)
*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**
(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28
665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*
b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 60
0300*a**(35/2)*b**(99/2)*x**14*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 3
15*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b
**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/
2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 13513
5*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)
*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13
/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)
*b**66*x**(7/2)) + 1260630*a**(33/2)*b**(101/2)*x**13*sqrt(a*x/b + 1)/(21*a**(37
/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(3
3/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*
a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*
b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(
19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555
*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65
*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 1995994*a**(31/2)*b**(103/2)*x**12*sqr
t(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 220
5*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b
**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25
/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 1051
05*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)
*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/
2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 2423668*a**(29/
2)*b**(105/2)*x**11*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2
)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31
/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105
*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)
*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(
17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*
a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(
7/2)) + 2271984*a**(27/2)*b**(107/2)*x**10*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x
**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555
*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b
**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(2
3/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 630
63*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*
b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) +
21*a**(7/2)*b**66*x**(7/2)) + 1640210*a**(25/2)*b**(109/2)*x**9*sqrt(a*x/b + 1)
/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*
b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/
2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135
*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)
*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(1
5/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**
(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 900614*a**(23/2)*b**(111/2)
*x**8*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35
/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a*
*(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**
57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/
2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*
a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**6
4*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 365976
*a**(21/2)*b**(113/2)*x**7*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a
**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54
*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) +
105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a*
*(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**
61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2)
+ 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**
66*x**(7/2)) + 103740*a**(19/2)*b**(115/2)*x**6*sqrt(a*x/b + 1)/(21*a**(37/2)*b*
*51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) +
9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27
/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*
x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2)
+ 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(1
3/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9
/2) + 21*a**(7/2)*b**66*x**(7/2)) + 17178*a**(17/2)*b**(117/2)*x**5*sqrt(a*x/b +
1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/
2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(
29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135
135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19
/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x*
*(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*
a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 90*a**(15/2)*b**(119/2)*
x**4*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/
2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**
(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**5
7*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2
) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a
**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64
*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 1004*a*
*(13/2)*b**(121/2)*x**3*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(
35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x*
*(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 10
5105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(2
1/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*
x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2
205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*
x**(7/2)) - 344*a**(11/2)*b**(123/2)*x**2*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x*
*(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*
a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b*
*56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23
/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 6306
3*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b
**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) +
21*a**(7/2)*b**66*x**(7/2)) - 66*a**(9/2)*b**(125/2)*x*sqrt(a*x/b + 1)/(21*a**(3
7/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(
33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063
*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)
*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**
(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 955
5*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**6
5*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 6*a**(7/2)*b**(127/2)*sqrt(a*x/b + 1)
/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*
b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/
2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135
*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)
*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(1
5/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**
(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 512*a**22*b**45*x**(37/2)/(
21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b*
*53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2)
+ 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a
**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b
**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/
2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9
/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 7680*a**21*b**46*x**(35/2)/(2
1*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**
53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2)
+ 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a*
*(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b*
*60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2
) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/
2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 53760*a**20*b**47*x**(33/2)/(2
1*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**
53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2)
+ 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a*
*(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b*
*60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2
) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/
2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 232960*a**19*b**48*x**(31/2)/(
21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b*
*53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2)
+ 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a
**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b
**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/
2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9
/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 698880*a**18*b**49*x**(29/2)/
(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b
**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2
) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*
a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*
b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15
/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(
9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 1537536*a**17*b**50*x**(27/2
)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)
*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29
/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 13513
5*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2
)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(
15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a*
*(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 2562560*a**16*b**51*x**(25
/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/
2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(
29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135
135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19
/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x*
*(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*
a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 3294720*a**15*b**52*x**(
23/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(3
3/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x*
*(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 1
35135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(
19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*
x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 31
5*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 3294720*a**14*b**53*x*
*(21/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**
(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*
x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) +
135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a*
*(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**6
2*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) +
315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 2562560*a**13*b**54*
x**(19/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a
**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**5
5*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2)
+ 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*
a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b*
*62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2)
+ 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 1537536*a**12*b**5
5*x**(17/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205
*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b*
*55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/
2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 10510
5*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*
b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2
) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 698880*a**11*b**
56*x**(15/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 220
5*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b
**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25
/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 1051
05*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)
*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/
2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 232960*a**10*b*
*57*x**(13/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 22
05*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*
b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(2
5/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105
105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2
)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11
/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 53760*a**9*b**
58*x**(11/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 220
5*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b
**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25
/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 1051
05*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)
*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/
2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 7680*a**8*b**59
*x**(9/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a
**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**5
5*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2)
+ 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*
a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b*
*62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2)
+ 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 512*a**7*b**60*x**
(7/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(3
3/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x*
*(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 1
35135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(
19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*
x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 31
5*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2))

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GIAC/XCAS [A]  time = 0.271226, size = 203, normalized size = 1.75 \[ -\frac{2}{21} \, b{\left (\frac{7 \,{\left (a^{5} - \frac{15 \,{\left (a x + b\right )} a^{4}}{x}\right )} x}{{\left (a x + b\right )} b^{7} \sqrt{\frac{a x + b}{x}}} - \frac{210 \, a^{3} b^{42} \sqrt{\frac{a x + b}{x}} - \frac{70 \,{\left (a x + b\right )} a^{2} b^{42} \sqrt{\frac{a x + b}{x}}}{x} + \frac{21 \,{\left (a x + b\right )}^{2} a b^{42} \sqrt{\frac{a x + b}{x}}}{x^{2}} - \frac{3 \,{\left (a x + b\right )}^{3} b^{42} \sqrt{\frac{a x + b}{x}}}{x^{3}}}{b^{49}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^7),x, algorithm="giac")

[Out]

-2/21*b*(7*(a^5 - 15*(a*x + b)*a^4/x)*x/((a*x + b)*b^7*sqrt((a*x + b)/x)) - (210
*a^3*b^42*sqrt((a*x + b)/x) - 70*(a*x + b)*a^2*b^42*sqrt((a*x + b)/x)/x + 21*(a*
x + b)^2*a*b^42*sqrt((a*x + b)/x)/x^2 - 3*(a*x + b)^3*b^42*sqrt((a*x + b)/x)/x^3
)/b^49)