3.1672 \(\int \frac{x^{3/2}}{\left (a+\frac{b}{x}\right )^2} \, dx\)

Optimal. Leaf size=85 \[ -\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}+\frac{7 b^2 \sqrt{x}}{a^4}-\frac{7 b x^{3/2}}{3 a^3}+\frac{7 x^{5/2}}{5 a^2}-\frac{x^{7/2}}{a (a x+b)} \]

[Out]

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

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Rubi [A]  time = 0.0982764, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}+\frac{7 b^2 \sqrt{x}}{a^4}-\frac{7 b x^{3/2}}{3 a^3}+\frac{7 x^{5/2}}{5 a^2}-\frac{x^{7/2}}{a (a x+b)} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(a + b/x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 17.764, size = 78, normalized size = 0.92 \[ - \frac{x^{\frac{7}{2}}}{a \left (a x + b\right )} + \frac{7 x^{\frac{5}{2}}}{5 a^{2}} - \frac{7 b x^{\frac{3}{2}}}{3 a^{3}} + \frac{7 b^{2} \sqrt{x}}{a^{4}} - \frac{7 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)/(a+b/x)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0857948, size = 79, normalized size = 0.93 \[ \frac{\sqrt{x} \left (6 a^3 x^3-14 a^2 b x^2+70 a b^2 x+105 b^3\right )}{15 a^4 (a x+b)}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)/(a + b/x)^2,x]

[Out]

(Sqrt[x]*(105*b^3 + 70*a*b^2*x - 14*a^2*b*x^2 + 6*a^3*x^3))/(15*a^4*(b + a*x)) -
(7*b^(5/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(9/2)

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Maple [A]  time = 0.017, size = 71, normalized size = 0.8 \[{\frac{2}{5\,{a}^{2}}{x}^{{\frac{5}{2}}}}-{\frac{4\,b}{3\,{a}^{3}}{x}^{{\frac{3}{2}}}}+6\,{\frac{{b}^{2}\sqrt{x}}{{a}^{4}}}+{\frac{{b}^{3}}{{a}^{4} \left ( ax+b \right ) }\sqrt{x}}-7\,{\frac{{b}^{3}}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(a+b/x)^2,x)

[Out]

2/5*x^(5/2)/a^2-4/3*b*x^(3/2)/a^3+6*b^2*x^(1/2)/a^4+1/a^4*b^3*x^(1/2)/(a*x+b)-7/
a^4*b^3/(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^(3/2)/(a + b/x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240419, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (6 \, a^{3} x^{3} - 14 \, a^{2} b x^{2} + 70 \, a b^{2} x + 105 \, b^{3}\right )} \sqrt{x}}{30 \,{\left (a^{5} x + a^{4} b\right )}}, -\frac{105 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) -{\left (6 \, a^{3} x^{3} - 14 \, a^{2} b x^{2} + 70 \, a b^{2} x + 105 \, b^{3}\right )} \sqrt{x}}{15 \,{\left (a^{5} x + a^{4} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/(a + b/x)^2,x, algorithm="fricas")

[Out]

[1/30*(105*(a*b^2*x + b^3)*sqrt(-b/a)*log((a*x - 2*a*sqrt(x)*sqrt(-b/a) - b)/(a*
x + b)) + 2*(6*a^3*x^3 - 14*a^2*b*x^2 + 70*a*b^2*x + 105*b^3)*sqrt(x))/(a^5*x +
a^4*b), -1/15*(105*(a*b^2*x + b^3)*sqrt(b/a)*arctan(sqrt(x)/sqrt(b/a)) - (6*a^3*
x^3 - 14*a^2*b*x^2 + 70*a*b^2*x + 105*b^3)*sqrt(x))/(a^5*x + a^4*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**(3/2)/(a+b/x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22281, size = 103, normalized size = 1.21 \[ -\frac{7 \, b^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{b^{3} \sqrt{x}}{{\left (a x + b\right )} a^{4}} + \frac{2 \,{\left (3 \, a^{8} x^{\frac{5}{2}} - 10 \, a^{7} b x^{\frac{3}{2}} + 45 \, a^{6} b^{2} \sqrt{x}\right )}}{15 \, a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/(a + b/x)^2,x, algorithm="giac")

[Out]

-7*b^3*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) + b^3*sqrt(x)/((a*x + b)*a^4)
+ 2/15*(3*a^8*x^(5/2) - 10*a^7*b*x^(3/2) + 45*a^6*b^2*sqrt(x))/a^10