3.1739 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^7} \, dx\)
Optimal. Leaf size=116 \[ -\frac{2 a^5}{b^6 \sqrt{a+\frac{b}{x}}}-\frac{10 a^4 \sqrt{a+\frac{b}{x}}}{b^6}+\frac{20 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^6}-\frac{4 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{b^6}+\frac{10 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^6}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^6} \]
[Out]
(-2*a^5)/(b^6*Sqrt[a + b/x]) - (10*a^4*Sqrt[a + b/x])/b^6 + (20*a^3*(a + b/x)^(3
/2))/(3*b^6) - (4*a^2*(a + b/x)^(5/2))/b^6 + (10*a*(a + b/x)^(7/2))/(7*b^6) - (2
*(a + b/x)^(9/2))/(9*b^6)
_______________________________________________________________________________________
Rubi [A] time = 0.137062, 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
\[ -\frac{2 a^5}{b^6 \sqrt{a+\frac{b}{x}}}-\frac{10 a^4 \sqrt{a+\frac{b}{x}}}{b^6}+\frac{20 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^6}-\frac{4 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{b^6}+\frac{10 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^6}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^6} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^(3/2)*x^7),x]
[Out]
(-2*a^5)/(b^6*Sqrt[a + b/x]) - (10*a^4*Sqrt[a + b/x])/b^6 + (20*a^3*(a + b/x)^(3
/2))/(3*b^6) - (4*a^2*(a + b/x)^(5/2))/b^6 + (10*a*(a + b/x)^(7/2))/(7*b^6) - (2
*(a + b/x)^(9/2))/(9*b^6)
_______________________________________________________________________________________
Rubi in Sympy [A] time = 19.4454, size = 100, normalized size = 0.86 \[ - \frac{2 a^{5}}{b^{6} \sqrt{a + \frac{b}{x}}} - \frac{10 a^{4} \sqrt{a + \frac{b}{x}}}{b^{6}} + \frac{20 a^{3} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{6}} - \frac{4 a^{2} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{b^{6}} + \frac{10 a \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{6}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{9 b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(3/2)/x**7,x)
[Out]
-2*a**5/(b**6*sqrt(a + b/x)) - 10*a**4*sqrt(a + b/x)/b**6 + 20*a**3*(a + b/x)**(
3/2)/(3*b**6) - 4*a**2*(a + b/x)**(5/2)/b**6 + 10*a*(a + b/x)**(7/2)/(7*b**6) -
2*(a + b/x)**(9/2)/(9*b**6)
_______________________________________________________________________________________
Mathematica [A] time = 0.0564802, size = 80, normalized size = 0.69 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (256 a^5 x^5+128 a^4 b x^4-32 a^3 b^2 x^3+16 a^2 b^3 x^2-10 a b^4 x+7 b^5\right )}{63 b^6 x^4 (a x+b)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^(3/2)*x^7),x]
[Out]
(-2*Sqrt[a + b/x]*(7*b^5 - 10*a*b^4*x + 16*a^2*b^3*x^2 - 32*a^3*b^2*x^3 + 128*a^
4*b*x^4 + 256*a^5*x^5))/(63*b^6*x^4*(b + a*x))
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 77, normalized size = 0.7 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 256\,{a}^{5}{x}^{5}+128\,{a}^{4}b{x}^{4}-32\,{a}^{3}{b}^{2}{x}^{3}+16\,{a}^{2}{b}^{3}{x}^{2}-10\,a{b}^{4}x+7\,{b}^{5} \right ) }{63\,{x}^{6}{b}^{6}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(3/2)/x^7,x)
[Out]
-2/63*(a*x+b)*(256*a^5*x^5+128*a^4*b*x^4-32*a^3*b^2*x^3+16*a^2*b^3*x^2-10*a*b^4*
x+7*b^5)/x^6/b^6/((a*x+b)/x)^(3/2)
_______________________________________________________________________________________
Maxima [A] time = 1.44032, size = 132, normalized size = 1.14 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}}}{9 \, b^{6}} + \frac{10 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a}{7 \, b^{6}} - \frac{4 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a^{2}}{b^{6}} + \frac{20 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{3}}{3 \, b^{6}} - \frac{10 \, \sqrt{a + \frac{b}{x}} a^{4}}{b^{6}} - \frac{2 \, a^{5}}{\sqrt{a + \frac{b}{x}} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*x^7),x, algorithm="maxima")
[Out]
-2/9*(a + b/x)^(9/2)/b^6 + 10/7*(a + b/x)^(7/2)*a/b^6 - 4*(a + b/x)^(5/2)*a^2/b^
6 + 20/3*(a + b/x)^(3/2)*a^3/b^6 - 10*sqrt(a + b/x)*a^4/b^6 - 2*a^5/(sqrt(a + b/
x)*b^6)
_______________________________________________________________________________________
Fricas [A] time = 0.238616, size = 96, normalized size = 0.83 \[ -\frac{2 \,{\left (256 \, a^{5} x^{5} + 128 \, a^{4} b x^{4} - 32 \, a^{3} b^{2} x^{3} + 16 \, a^{2} b^{3} x^{2} - 10 \, a b^{4} x + 7 \, b^{5}\right )}}{63 \, b^{6} x^{5} \sqrt{\frac{a x + b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*x^7),x, algorithm="fricas")
[Out]
-2/63*(256*a^5*x^5 + 128*a^4*b*x^4 - 32*a^3*b^2*x^3 + 16*a^2*b^3*x^2 - 10*a*b^4*
x + 7*b^5)/(b^6*x^5*sqrt((a*x + b)/x))
_______________________________________________________________________________________
Sympy [A] time = 37.5092, size = 9534, normalized size = 82.19 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(3/2)/x**7,x)
[Out]
-512*a**(47/2)*b**(91/2)*x**19*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 9
45*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*
b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(2
9/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405
405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19
/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x*
*(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a*
*(9/2)*b**66*x**(9/2)) - 7424*a**(45/2)*b**(93/2)*x**18*sqrt(a*x/b + 1)/(63*a**(
39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**
(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189
189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25
/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*
x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) +
28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2
)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 50112*a**(43/2)*b**(95/2)*x**1
7*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2)
+ 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(3
1/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57
*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2)
+ 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a
**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**6
4*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 2088
00*a**(41/2)*b**(97/2)*x**16*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945
*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b*
*54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/
2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 40540
5*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2
)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(
15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(
9/2)*b**66*x**(9/2)) - 600300*a**(39/2)*b**(99/2)*x**15*sqrt(a*x/b + 1)/(63*a**(
39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**
(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189
189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25
/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*
x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) +
28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2
)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 1260630*a**(37/2)*b**(101/2)*x
**14*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/
2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a*
*(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b*
*57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23
/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 8599
5*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b
**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 1
996008*a**(35/2)*b**(103/2)*x**13*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2)
+ 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/
2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x*
*(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) +
405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**
(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63
*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63
*a**(9/2)*b**66*x**(9/2)) - 2423850*a**(33/2)*b**(105/2)*x**12*sqrt(a*x/b + 1)/(
63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b*
*53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2
) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405
*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)
*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(
17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a
**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 2273076*a**(31/2)*b**(1
07/2)*x**11*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*
x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 8
5995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(2
7/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59
*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2)
+ 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(
13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/
2)) - 1644214*a**(29/2)*b**(109/2)*x**10*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**
(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*
a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b
**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(2
5/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189
189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2
)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/
2) + 63*a**(9/2)*b**66*x**(9/2)) - 910624*a**(27/2)*b**(111/2)*x**9*sqrt(a*x/b +
1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/
2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**
(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 4
05405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(
21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62
*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) +
945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 383994*a**(25/2)*b
**(113/2)*x**8*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**
52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2)
+ 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a*
*(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b*
*59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19
/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a
**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**
(9/2)) - 127764*a**(23/2)*b**(115/2)*x**7*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x*
*(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665
*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*
b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(
25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 18
9189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/
2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11
/2) + 63*a**(9/2)*b**66*x**(9/2)) - 41202*a**(21/2)*b**(117/2)*x**6*sqrt(a*x/b +
1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/
2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**
(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 4
05405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(
21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62
*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) +
945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 17928*a**(19/2)*b*
*(119/2)*x**5*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**5
2*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) +
85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**
(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**
59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/
2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a*
*(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(
9/2)) - 9006*a**(17/2)*b**(121/2)*x**4*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(3
9/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a*
*(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**
56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/
2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 18918
9*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*
b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2)
+ 63*a**(9/2)*b**66*x**(9/2)) - 3660*a**(15/2)*b**(123/2)*x**3*sqrt(a*x/b + 1)/
(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b
**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/
2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 40540
5*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2
)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**
(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*
a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 1026*a**(13/2)*b**(125
/2)*x**2*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**
(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 8599
5*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2
)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x*
*(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) +
85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/
2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2))
- 176*a**(11/2)*b**(127/2)*x*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 94
5*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b
**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29
/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 4054
05*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/
2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**
(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**
(9/2)*b**66*x**(9/2)) - 14*a**(9/2)*b**(129/2)*sqrt(a*x/b + 1)/(63*a**(39/2)*b**
51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) +
28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(2
9/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58
*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2)
+ 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a*
*(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x
**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 512*a**24*b**45*x**(39/2)/(63*a**(39/2)
*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2
) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a
**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b
**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(2
1/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 2866
5*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**
65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 7680*a**23*b**46*x**(37/2)/(63*a**(
39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**
(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189
189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25
/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*
x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) +
28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2
)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 53760*a**22*b**47*x**(35/2)/(6
3*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**
53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2)
+ 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*
a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*
b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(1
7/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a*
*(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 232960*a**21*b**48*x**(3
3/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35
/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x*
*(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) +
405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**
(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**6
2*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) +
945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 698880*a**20*b**4
9*x**(31/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615
*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b
**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(2
7/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315
315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/
2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(
13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 1537536*a*
*19*b**50*x**(29/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2
) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**
(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**
57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/
2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995
*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b*
*64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 25
62560*a**18*b**51*x**(27/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*
x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 8
5995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(2
7/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59
*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2)
+ 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(
13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/
2)) + 3294720*a**17*b**52*x**(25/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2
)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(3
3/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 3153
15*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/
2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x
**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6
615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**6
6*x**(9/2)) + 3294720*a**16*b**53*x**(23/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*
a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**
54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2
) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405
*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)
*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(1
5/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9
/2)*b**66*x**(9/2)) + 2562560*a**15*b**54*x**(21/2)/(63*a**(39/2)*b**51*x**(39/2
) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(3
3/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*
x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2)
+ 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a
**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**
63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) +
63*a**(9/2)*b**66*x**(9/2)) + 1537536*a**14*b**55*x**(19/2)/(63*a**(39/2)*b**51*
x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 286
65*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2
)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x*
*(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) +
189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(1
5/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(
11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 698880*a**13*b**56*x**(17/2)/(63*a**(39/2)
*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2
) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a
**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b
**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(2
1/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 2866
5*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**
65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 232960*a**12*b**57*x**(15/2)/(63*a*
*(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x
**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 1
89189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(
25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**6
0*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2)
+ 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11
/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 53760*a**11*b**58*x**(13/2)/
(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b
**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/
2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 40540
5*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2
)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**
(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*
a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 7680*a**10*b**59*x**(1
1/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35
/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x*
*(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) +
405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**
(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**6
2*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) +
945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 512*a**9*b**60*x*
*(9/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(
35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*
x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2)
+ 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a
**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b*
*62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2)
+ 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.261277, size = 211, normalized size = 1.82 \[ -\frac{2}{63} \, b{\left (\frac{63 \, a^{5}}{b^{7} \sqrt{\frac{a x + b}{x}}} + \frac{315 \, a^{4} b^{56} \sqrt{\frac{a x + b}{x}} - \frac{210 \,{\left (a x + b\right )} a^{3} b^{56} \sqrt{\frac{a x + b}{x}}}{x} + \frac{126 \,{\left (a x + b\right )}^{2} a^{2} b^{56} \sqrt{\frac{a x + b}{x}}}{x^{2}} - \frac{45 \,{\left (a x + b\right )}^{3} a b^{56} \sqrt{\frac{a x + b}{x}}}{x^{3}} + \frac{7 \,{\left (a x + b\right )}^{4} b^{56} \sqrt{\frac{a x + b}{x}}}{x^{4}}}{b^{63}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*x^7),x, algorithm="giac")
[Out]
-2/63*b*(63*a^5/(b^7*sqrt((a*x + b)/x)) + (315*a^4*b^56*sqrt((a*x + b)/x) - 210*
(a*x + b)*a^3*b^56*sqrt((a*x + b)/x)/x + 126*(a*x + b)^2*a^2*b^56*sqrt((a*x + b)
/x)/x^2 - 45*(a*x + b)^3*a*b^56*sqrt((a*x + b)/x)/x^3 + 7*(a*x + b)^4*b^56*sqrt(
(a*x + b)/x)/x^4)/b^63)