Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{x}{2 b \left (a x^2+b\right )} \]
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Rubi [A] time = 0.0114527, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{x}{2 b \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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Rule 263
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^4} \, dx &=\int \frac{1}{\left (b+a x^2\right )^2} \, dx\\ &=\frac{x}{2 b \left (b+a x^2\right )}+\frac{\int \frac{1}{b+a x^2} \, dx}{2 b}\\ &=\frac{x}{2 b \left (b+a x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0236597, size = 45, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{x}{2 b \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 0.8 \begin{align*}{\frac{x}{2\,b \left ( a{x}^{2}+b \right ) }}+{\frac{1}{2\,b}\arctan \left ({ax{\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 = 1.45185, size = 261, normalized size = 5.8 \begin{align*} \left [\frac{2 \, a b x -{\left (a x^{2} + b\right )} \sqrt{-a b} \log \left (\frac{a x^{2} - 2 \, \sqrt{-a b} x - b}{a x^{2} + b}\right )}{4 \,{\left (a^{2} b^{2} x^{2} + a b^{3}\right )}}, \frac{a b x +{\left (a x^{2} + b\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{b}\right )}{2 \,{\left (a^{2} b^{2} x^{2} + a b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.49636, size = 78, normalized size = 1.73 \begin{align*} \frac{x}{2 a b x^{2} + 2 b^{2}} - \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (- b^{2} \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (b^{2} \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15798, size = 47, normalized size = 1.04 \begin{align*} \frac{\arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b} + \frac{x}{2 \,{\left (a x^{2} + b\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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