Optimal. Leaf size=17 \[ \frac{1}{2} \log \left (a+b x^2+c x^4\right ) \]
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Rubi [A] time = 0.0186135, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1247, 628} \[ \frac{1}{2} \log \left (a+b x^2+c x^4\right ) \]
Antiderivative was successfully verified.
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Rule 1247
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (b+2 c x^2\right )}{a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \log \left (a+b x^2+c x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0060359, size = 17, normalized size = 1. \[ \frac{1}{2} \log \left (a+b x^2+c x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 16, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13879, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{2} \, \log \left (c x^{4} + b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.995402, size = 38, normalized size = 2.24 \begin{align*} \frac{1}{2} \, \log \left (c x^{4} + b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.417882, size = 14, normalized size = 0.82 \begin{align*} \frac{\log{\left (a + b x^{2} + c x^{4} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13661, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{2} \, \log \left (c x^{4} + b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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