Optimal. Leaf size=31 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{c}} \]
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Rubi [A] time = 0.0489377, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3, 2013, 620, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 3
Rule 2013
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx &=\int \frac{x}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0113331, size = 52, normalized size = 1.68 \[ \frac{x \sqrt{b+c x^2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right )}{\sqrt{c} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 44, normalized size = 1.4 \begin{align*}{x\sqrt{c{x}^{2}+b}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{\frac{1}{\sqrt{c}}}} \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 = 1.55389, size = 174, normalized size = 5.61 \begin{align*} \left [\frac{\log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right )}{2 \, \sqrt{c}}, -\frac{\sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right )}{c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17951, size = 53, normalized size = 1.71 \begin{align*} -\frac{\log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2}}\right )} \sqrt{c} - b \right |}\right )}{2 \, \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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