Optimal. Leaf size=34 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]
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Rubi [A] time = 0.0121051, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {325, 205} \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]
Antiderivative was successfully verified.
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Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx &=-\frac{1}{a x}-\frac{b \int \frac{1}{a+b x^2} \, dx}{a}\\ &=-\frac{1}{a x}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.012191, size = 34, normalized size = 1. \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 30, normalized size = 0.9 \begin{align*} -{\frac{1}{ax}}-{\frac{b}{a}\arctan \left ({bx{\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 = 1.27306, size = 173, normalized size = 5.09 \begin{align*} \left [\frac{x \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 2}{2 \, a x}, -\frac{x \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 1}{a x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.321033, size = 65, normalized size = 1.91 \begin{align*} \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x \right )}}{2} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.72229, size = 39, normalized size = 1.15 \begin{align*} -\frac{b \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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