Optimal. Leaf size=92 \[ -\frac{a+b x^2}{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.0333592, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1112, 325, 205} \[ -\frac{a+b x^2}{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{\left (a b+b^2 x^2\right ) \int \frac{1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{a+b x^2}{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{a \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{a+b x^2}{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0135005, size = 56, normalized size = 0.61 \[ -\frac{\left (a+b x^2\right ) \left (\sqrt{b} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{a}\right )}{a^{3/2} x \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.171, size = 50, normalized size = 0.5 \begin{align*} -{\frac{b{x}^{2}+a}{ax} \left ( b\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) x+\sqrt{ab} \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\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 = 1.35675, size = 173, normalized size = 1.88 \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 [A] time = 0.363345, size = 65, normalized size = 0.71 \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 = 1.13449, size = 50, normalized size = 0.54 \begin{align*} -{\left (\frac{b \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{1}{a x}\right )} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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