Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{x}{2 a \left (a x^2+b\right )} \]
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Rubi [A] time = 0.013133, 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, 288, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{x}{2 a \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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Rule 263
Rule 288
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^2} \, dx &=\int \frac{x^2}{\left (b+a x^2\right )^2} \, dx\\ &=-\frac{x}{2 a \left (b+a x^2\right )}+\frac{\int \frac{1}{b+a x^2} \, dx}{2 a}\\ &=-\frac{x}{2 a \left (b+a x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0197144, size = 45, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{x}{2 a \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 36, normalized size = 0.8 \begin{align*} -{\frac{x}{2\,a \left ( a{x}^{2}+b \right ) }}+{\frac{1}{2\,a}\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 = 1.45832, size = 263, normalized size = 5.84 \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^{3} b x^{2} + a^{2} b^{2}\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^{3} b x^{2} + a^{2} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.456773, size = 78, normalized size = 1.73 \begin{align*} - \frac{x}{2 a^{2} x^{2} + 2 a b} - \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (- a b \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (a b \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19658, size = 47, normalized size = 1.04 \begin{align*} \frac{\arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a} - \frac{x}{2 \,{\left (a x^{2} + b\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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