3.1740 \(\int \frac{x^2}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx\)

Optimal. Leaf size=134 \[ -\frac{105 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{105 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^5}-\frac{35 b x^2 \sqrt{a+\frac{b}{x}}}{4 a^4}+\frac{7 x^3 \sqrt{a+\frac{b}{x}}}{a^3}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(105*b^2*Sqrt[a + b/x]*x)/(8*a^5) - (35*b*Sqrt[a + b/x]*x^2)/(4*a^4) - (2*x^3)/(
3*a*(a + b/x)^(3/2)) - (6*x^3)/(a^2*Sqrt[a + b/x]) + (7*Sqrt[a + b/x]*x^3)/a^3 -
(105*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(11/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.197683, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{105 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{105 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^5}-\frac{35 b x^2 \sqrt{a+\frac{b}{x}}}{4 a^4}+\frac{7 x^3 \sqrt{a+\frac{b}{x}}}{a^3}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x)^(5/2),x]

[Out]

(105*b^2*Sqrt[a + b/x]*x)/(8*a^5) - (35*b*Sqrt[a + b/x]*x^2)/(4*a^4) - (2*x^3)/(
3*a*(a + b/x)^(3/2)) - (6*x^3)/(a^2*Sqrt[a + b/x]) + (7*Sqrt[a + b/x]*x^3)/a^3 -
(105*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(11/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.1459, size = 117, normalized size = 0.87 \[ - \frac{2 x^{3}}{3 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{6 x^{3}}{a^{2} \sqrt{a + \frac{b}{x}}} + \frac{7 x^{3} \sqrt{a + \frac{b}{x}}}{a^{3}} - \frac{35 b x^{2} \sqrt{a + \frac{b}{x}}}{4 a^{4}} + \frac{105 b^{2} x \sqrt{a + \frac{b}{x}}}{8 a^{5}} - \frac{105 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{8 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**3/(3*a*(a + b/x)**(3/2)) - 6*x**3/(a**2*sqrt(a + b/x)) + 7*x**3*sqrt(a + b
/x)/a**3 - 35*b*x**2*sqrt(a + b/x)/(4*a**4) + 105*b**2*x*sqrt(a + b/x)/(8*a**5)
- 105*b**3*atanh(sqrt(a + b/x)/sqrt(a))/(8*a**(11/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.193822, size = 106, normalized size = 0.79 \[ \frac{x \sqrt{a+\frac{b}{x}} \left (8 a^4 x^4-18 a^3 b x^3+63 a^2 b^2 x^2+420 a b^3 x+315 b^4\right )}{24 a^5 (a x+b)^2}-\frac{105 b^3 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{16 a^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*x*(315*b^4 + 420*a*b^3*x + 63*a^2*b^2*x^2 - 18*a^3*b*x^3 + 8*a^4*
x^4))/(24*a^5*(b + a*x)^2) - (105*b^3*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x]
)/(16*a^(11/2))

_______________________________________________________________________________________

Maple [B]  time = 0.023, size = 620, normalized size = 4.6 \[{\frac{x}{48\, \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 16\,{a}^{19/2} \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{3}-84\,{a}^{19/2}\sqrt{a{x}^{2}+bx}{x}^{4}b+48\,{a}^{17/2} \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}b-294\,{a}^{17/2}\sqrt{a{x}^{2}+bx}{x}^{3}{b}^{2}+672\,{a}^{17/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}{b}^{2}+48\,{a}^{15/2} \left ( a{x}^{2}+bx \right ) ^{3/2}x{b}^{2}-378\,{a}^{15/2}\sqrt{a{x}^{2}+bx}{x}^{2}{b}^{3}-384\,{a}^{15/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}x{b}^{2}+2016\,{a}^{15/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}{b}^{3}+16\,{a}^{13/2} \left ( a{x}^{2}+bx \right ) ^{3/2}{b}^{3}-210\,{a}^{13/2}\sqrt{a{x}^{2}+bx}x{b}^{4}-352\,{b}^{3}{a}^{13/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}+2016\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }x{b}^{4}-42\,{a}^{11/2}\sqrt{a{x}^{2}+bx}{b}^{5}+672\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }{b}^{5}+21\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{8}{b}^{3}-336\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{8}{b}^{3}+63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{7}{b}^{4}-1008\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{7}{b}^{4}+63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{6}{b}^{5}-1008\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{6}{b}^{5}+21\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{5}{b}^{6}-336\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{5}{b}^{6} \right ){a}^{-{\frac{21}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x)^(5/2),x)

[Out]

1/48*((a*x+b)/x)^(1/2)*x/a^(21/2)*(16*a^(19/2)*(a*x^2+b*x)^(3/2)*x^3-84*a^(19/2)
*(a*x^2+b*x)^(1/2)*x^4*b+48*a^(17/2)*(a*x^2+b*x)^(3/2)*x^2*b-294*a^(17/2)*(a*x^2
+b*x)^(1/2)*x^3*b^2+672*a^(17/2)*(x*(a*x+b))^(1/2)*x^3*b^2+48*a^(15/2)*(a*x^2+b*
x)^(3/2)*x*b^2-378*a^(15/2)*(a*x^2+b*x)^(1/2)*x^2*b^3-384*a^(15/2)*(x*(a*x+b))^(
3/2)*x*b^2+2016*a^(15/2)*(x*(a*x+b))^(1/2)*x^2*b^3+16*a^(13/2)*(a*x^2+b*x)^(3/2)
*b^3-210*a^(13/2)*(a*x^2+b*x)^(1/2)*x*b^4-352*b^3*a^(13/2)*(x*(a*x+b))^(3/2)+201
6*a^(13/2)*(x*(a*x+b))^(1/2)*x*b^4-42*a^(11/2)*(a*x^2+b*x)^(1/2)*b^5+672*a^(11/2
)*(x*(a*x+b))^(1/2)*b^5+21*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))
*x^3*a^8*b^3-336*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*a^8*b
^3+63*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^7*b^4-1008*ln(
1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^7*b^4+63*ln(1/2*(2*(a*x
^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a^6*b^5-1008*ln(1/2*(2*(x*(a*x+b))^(1/
2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a^6*b^5+21*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*
a*x+b)/a^(1/2))*a^5*b^6-336*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2)
)*a^5*b^6)/(x*(a*x+b))^(1/2)/(a*x+b)^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.256672, size = 1, normalized size = 0.01 \[ \left [\frac{315 \,{\left (a b^{3} x + b^{4}\right )} \sqrt{\frac{a x + b}{x}} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (8 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 63 \, a^{2} b^{2} x^{2} + 420 \, a b^{3} x + 315 \, b^{4}\right )} \sqrt{a}}{48 \,{\left (a^{6} x + a^{5} b\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}, \frac{315 \,{\left (a b^{3} x + b^{4}\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (8 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 63 \, a^{2} b^{2} x^{2} + 420 \, a b^{3} x + 315 \, b^{4}\right )} \sqrt{-a}}{24 \,{\left (a^{6} x + a^{5} b\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(315*(a*b^3*x + b^4)*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a
*x + b)*sqrt(a)) + 2*(8*a^4*x^4 - 18*a^3*b*x^3 + 63*a^2*b^2*x^2 + 420*a*b^3*x +
315*b^4)*sqrt(a))/((a^6*x + a^5*b)*sqrt(a)*sqrt((a*x + b)/x)), 1/24*(315*(a*b^3*
x + b^4)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) + (8*a^4*x^4 -
18*a^3*b*x^3 + 63*a^2*b^2*x^2 + 420*a*b^3*x + 315*b^4)*sqrt(-a))/((a^6*x + a^5*
b)*sqrt(-a)*sqrt((a*x + b)/x))]

_______________________________________________________________________________________

Sympy [A]  time = 39.3637, size = 532, normalized size = 3.97 \[ \frac{8 a^{\frac{133}{2}} b^{128} x^{72}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{18 a^{\frac{131}{2}} b^{129} x^{71}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{63 a^{\frac{129}{2}} b^{130} x^{70}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{420 a^{\frac{127}{2}} b^{131} x^{69}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{315 a^{\frac{125}{2}} b^{132} x^{68}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{315 a^{63} b^{\frac{263}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{315 a^{62} b^{\frac{265}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*a**(133/2)*b**128*x**72/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) +
24*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) - 18*a**(131/2)*b**129*x**
71/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/
2)*x**(135/2)*sqrt(a*x/b + 1)) + 63*a**(129/2)*b**130*x**70/(24*a**(137/2)*b**(2
57/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/
b + 1)) + 420*a**(127/2)*b**131*x**69/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(
a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) + 315*a**(125/
2)*b**132*x**68/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(13
5/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) - 315*a**63*b**(263/2)*x**(137/2)*sq
rt(a*x/b + 1)*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(24*a**(137/2)*b**(257/2)*x**(137/2
)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) - 315*a
**62*b**(265/2)*x**(135/2)*sqrt(a*x/b + 1)*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(24*a*
*(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(13
5/2)*sqrt(a*x/b + 1))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.264804, size = 203, normalized size = 1.51 \[ \frac{1}{24} \, b{\left (\frac{315 \, b^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{5}} + \frac{16 \, a^{4} b^{2} + \frac{144 \,{\left (a x + b\right )} a^{3} b^{2}}{x} - \frac{693 \,{\left (a x + b\right )}^{2} a^{2} b^{2}}{x^{2}} + \frac{840 \,{\left (a x + b\right )}^{3} a b^{2}}{x^{3}} - \frac{315 \,{\left (a x + b\right )}^{4} b^{2}}{x^{4}}}{{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )}^{3} a^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*b*(315*b^2*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^5) + (16*a^4*b^2
+ 144*(a*x + b)*a^3*b^2/x - 693*(a*x + b)^2*a^2*b^2/x^2 + 840*(a*x + b)^3*a*b^2/
x^3 - 315*(a*x + b)^4*b^2/x^4)/((a*sqrt((a*x + b)/x) - (a*x + b)*sqrt((a*x + b)/
x)/x)^3*a^5))