Optimal. Leaf size=53 \[ \frac{\left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.0150325, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1088, 205} \[ \frac{\left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1088
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{\left (2 a b+2 b^2 x^2\right ) \int \frac{1}{2 a b+2 b^2 x^2} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0117462, size = 44, normalized size = 0.83 \[ \frac{\left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.167, size = 34, normalized size = 0.6 \begin{align*}{(b{x}^{2}+a)\arctan \left ({bx{\frac{1}{\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 = 1.255, size = 151, normalized size = 2.85 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{2 \, a b}, \frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{a b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.156914, size = 53, normalized size = 1. \begin{align*} - \frac{\sqrt{- \frac{1}{a b}} \log{\left (- a \sqrt{- \frac{1}{a b}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a b}} \log{\left (a \sqrt{- \frac{1}{a b}} + x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12336, size = 31, normalized size = 0.58 \begin{align*} \frac{\arctan \left (\frac{b x}{\sqrt{a b}}\right ) \mathrm{sgn}\left (b x^{2} + a\right )}{\sqrt{a b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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