Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} b^{3/2}}+\frac{\sqrt{x}}{b (a x+b)} \]
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Rubi [A] time = 0.0157653, 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, Rules used = {263, 51, 63, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} b^{3/2}}+\frac{\sqrt{x}}{b (a x+b)} \]
Antiderivative was successfully verified.
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Rule 263
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^{5/2}} \, dx &=\int \frac{1}{\sqrt{x} (b+a x)^2} \, dx\\ &=\frac{\sqrt{x}}{b (b+a x)}+\frac{\int \frac{1}{\sqrt{x} (b+a x)} \, dx}{2 b}\\ &=\frac{\sqrt{x}}{b (b+a x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{b}\\ &=\frac{\sqrt{x}}{b (b+a x)}+\frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.017909, size = 45, normalized size = 1. \[ \frac{\sqrt{x}}{a b x+b^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 36, normalized size = 0.8 \begin{align*}{\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81712, size = 274, normalized size = 6.09 \begin{align*} \left [\frac{2 \, a b \sqrt{x} - \sqrt{-a b}{\left (a x + b\right )} \log \left (\frac{a x - b - 2 \, \sqrt{-a b} \sqrt{x}}{a x + b}\right )}{2 \,{\left (a^{2} b^{2} x + a b^{3}\right )}}, \frac{a b \sqrt{x} - \sqrt{a b}{\left (a x + b\right )} \arctan \left (\frac{\sqrt{a b}}{a \sqrt{x}}\right )}{a^{2} b^{2} x + a b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09055, size = 47, normalized size = 1.04 \begin{align*} \frac{\arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{\sqrt{x}}{{\left (a x + b\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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