Optimal. Leaf size=24 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{c}} \]
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Rubi [A] time = 0.0117432, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2}{b x^2+c x^4} \, dx &=\int \frac{1}{b+c x^2} \, dx\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0042586, size = 24, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 16, normalized size = 0.7 \begin{align*}{\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \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 = 1.43387, size = 151, normalized size = 6.29 \begin{align*} \left [-\frac{\sqrt{-b c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-b c} x - b}{c x^{2} + b}\right )}{2 \, b c}, \frac{\sqrt{b c} \arctan \left (\frac{\sqrt{b c} x}{b}\right )}{b c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.136575, size = 53, normalized size = 2.21 \begin{align*} - \frac{\sqrt{- \frac{1}{b c}} \log{\left (- b \sqrt{- \frac{1}{b c}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{b c}} \log{\left (b \sqrt{- \frac{1}{b c}} + x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28577, size = 20, normalized size = 0.83 \begin{align*} \frac{\arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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