3.1671 \(\int \frac{x^{5/2}}{\left (a+\frac{b}{x}\right )^2} \, dx\)

Optimal. Leaf size=98 \[ \frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{11/2}}-\frac{9 b^3 \sqrt{x}}{a^5}+\frac{3 b^2 x^{3/2}}{a^4}-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (a x+b)} \]

[Out]

(-9*b^3*Sqrt[x])/a^5 + (3*b^2*x^(3/2))/a^4 - (9*b*x^(5/2))/(5*a^3) + (9*x^(7/2))
/(7*a^2) - x^(9/2)/(a*(b + a*x)) + (9*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])
/a^(11/2)

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Rubi [A]  time = 0.121125, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{11/2}}-\frac{9 b^3 \sqrt{x}}{a^5}+\frac{3 b^2 x^{3/2}}{a^4}-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (a x+b)} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(a + b/x)^2,x]

[Out]

(-9*b^3*Sqrt[x])/a^5 + (3*b^2*x^(3/2))/a^4 - (9*b*x^(5/2))/(5*a^3) + (9*x^(7/2))
/(7*a^2) - x^(9/2)/(a*(b + a*x)) + (9*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])
/a^(11/2)

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Rubi in Sympy [A]  time = 22.2268, size = 92, normalized size = 0.94 \[ - \frac{x^{\frac{9}{2}}}{a \left (a x + b\right )} + \frac{9 x^{\frac{7}{2}}}{7 a^{2}} - \frac{9 b x^{\frac{5}{2}}}{5 a^{3}} + \frac{3 b^{2} x^{\frac{3}{2}}}{a^{4}} - \frac{9 b^{3} \sqrt{x}}{a^{5}} + \frac{9 b^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(9/2)/(a*(a*x + b)) + 9*x**(7/2)/(7*a**2) - 9*b*x**(5/2)/(5*a**3) + 3*b**2*x
**(3/2)/a**4 - 9*b**3*sqrt(x)/a**5 + 9*b**(7/2)*atan(sqrt(a)*sqrt(x)/sqrt(b))/a*
*(11/2)

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Mathematica [A]  time = 0.0989349, size = 90, normalized size = 0.92 \[ \frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{11/2}}+\frac{\sqrt{x} \left (10 a^4 x^4-18 a^3 b x^3+42 a^2 b^2 x^2-210 a b^3 x-315 b^4\right )}{35 a^5 (a x+b)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(-315*b^4 - 210*a*b^3*x + 42*a^2*b^2*x^2 - 18*a^3*b*x^3 + 10*a^4*x^4))/
(35*a^5*(b + a*x)) + (9*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(11/2)

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Maple [A]  time = 0.015, size = 83, normalized size = 0.9 \[{\frac{2}{7\,{a}^{2}}{x}^{{\frac{7}{2}}}}-{\frac{4\,b}{5\,{a}^{3}}{x}^{{\frac{5}{2}}}}+2\,{\frac{{b}^{2}{x}^{3/2}}{{a}^{4}}}-8\,{\frac{{b}^{3}\sqrt{x}}{{a}^{5}}}-{\frac{{b}^{4}}{{a}^{5} \left ( ax+b \right ) }\sqrt{x}}+9\,{\frac{{b}^{4}}{{a}^{5}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(a+b/x)^2,x)

[Out]

2/7*x^(7/2)/a^2-4/5*b*x^(5/2)/a^3+2*b^2*x^(3/2)/a^4-8*b^3*x^(1/2)/a^5-1/a^5*b^4*
x^(1/2)/(a*x+b)+9/a^5*b^4/(a*b)^(1/2)*arctan(a*x^(1/2)/(a*b)^(1/2))

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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^(5/2)/(a + b/x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240039, size = 1, normalized size = 0.01 \[ \left [\frac{315 \,{\left (a b^{3} x + b^{4}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (10 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a b^{3} x - 315 \, b^{4}\right )} \sqrt{x}}{70 \,{\left (a^{6} x + a^{5} b\right )}}, \frac{315 \,{\left (a b^{3} x + b^{4}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) +{\left (10 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a b^{3} x - 315 \, b^{4}\right )} \sqrt{x}}{35 \,{\left (a^{6} x + a^{5} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/70*(315*(a*b^3*x + b^4)*sqrt(-b/a)*log((a*x + 2*a*sqrt(x)*sqrt(-b/a) - b)/(a*
x + b)) + 2*(10*a^4*x^4 - 18*a^3*b*x^3 + 42*a^2*b^2*x^2 - 210*a*b^3*x - 315*b^4)
*sqrt(x))/(a^6*x + a^5*b), 1/35*(315*(a*b^3*x + b^4)*sqrt(b/a)*arctan(sqrt(x)/sq
rt(b/a)) + (10*a^4*x^4 - 18*a^3*b*x^3 + 42*a^2*b^2*x^2 - 210*a*b^3*x - 315*b^4)*
sqrt(x))/(a^6*x + a^5*b)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218882, size = 119, normalized size = 1.21 \[ \frac{9 \, b^{4} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{5}} - \frac{b^{4} \sqrt{x}}{{\left (a x + b\right )} a^{5}} + \frac{2 \,{\left (5 \, a^{12} x^{\frac{7}{2}} - 14 \, a^{11} b x^{\frac{5}{2}} + 35 \, a^{10} b^{2} x^{\frac{3}{2}} - 140 \, a^{9} b^{3} \sqrt{x}\right )}}{35 \, a^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*b^4*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) - b^4*sqrt(x)/((a*x + b)*a^5)
+ 2/35*(5*a^12*x^(7/2) - 14*a^11*b*x^(5/2) + 35*a^10*b^2*x^(3/2) - 140*a^9*b^3*s
qrt(x))/a^14