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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]