3.1676 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^{5/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} b^{3/2}}+\frac{\sqrt{x}}{b (a x+b)} \]

[Out]

Sqrt[x]/(b*(b + a*x)) + ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]]/(Sqrt[a]*b^(3/2))

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Rubi [A]  time = 0.048847, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} b^{3/2}}+\frac{\sqrt{x}}{b (a x+b)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^(5/2)),x]

[Out]

Sqrt[x]/(b*(b + a*x)) + ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]]/(Sqrt[a]*b^(3/2))

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Rubi in Sympy [A]  time = 8.59332, size = 37, normalized size = 0.82 \[ \frac{\sqrt{x}}{b \left (a x + b\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)/(b*(a*x + b)) + atan(sqrt(a)*sqrt(x)/sqrt(b))/(sqrt(a)*b**(3/2))

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Mathematica [A]  time = 0.0367219, size = 45, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} b^{3/2}}+\frac{\sqrt{x}}{b (a x+b)} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[x]/(b*(b + a*x)) + ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]]/(Sqrt[a]*b^(3/2))

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Maple [A]  time = 0.01, size = 36, normalized size = 0.8 \[{\frac{1}{b \left ( ax+b \right ) }\sqrt{x}}+{\frac{1}{b}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^(5/2),x)

[Out]

x^(1/2)/b/(a*x+b)+1/b/(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(1/((a + b/x)^2*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23615, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (a x + b\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (a x - b\right )}}{a x + b}\right ) + 2 \, \sqrt{-a b} \sqrt{x}}{2 \,{\left (a b x + b^{2}\right )} \sqrt{-a b}}, -\frac{{\left (a x + b\right )} \arctan \left (\frac{b}{\sqrt{a b} \sqrt{x}}\right ) - \sqrt{a b} \sqrt{x}}{{\left (a b x + b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*((a*x + b)*log((2*a*b*sqrt(x) + sqrt(-a*b)*(a*x - b))/(a*x + b)) + 2*sqrt(-
a*b)*sqrt(x))/((a*b*x + b^2)*sqrt(-a*b)), -((a*x + b)*arctan(b/(sqrt(a*b)*sqrt(x
))) - sqrt(a*b)*sqrt(x))/((a*b*x + b^2)*sqrt(a*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(1/(a+b/x)**2/x**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230426, size = 47, normalized size = 1.04 \[ \frac{\arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{\sqrt{x}}{{\left (a x + b\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b) + sqrt(x)/((a*x + b)*b)