Optimal. Leaf size=27 \[ \frac{x^2}{2 a}-\frac{b \log \left (a x^2+b\right )}{2 a^2} \]
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Rubi [A] time = 0.0182291, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{x^2}{2 a}-\frac{b \log \left (a x^2+b\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{b+a x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{b+a x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a}-\frac{b}{a (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2 a}-\frac{b \log \left (b+a x^2\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0042536, size = 27, normalized size = 1. \[ \frac{x^2}{2 a}-\frac{b \log \left (a x^2+b\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 24, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,a}}-{\frac{b\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938308, size = 31, normalized size = 1.15 \begin{align*} \frac{x^{2}}{2 \, a} - \frac{b \log \left (a x^{2} + b\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56157, size = 49, normalized size = 1.81 \begin{align*} \frac{a x^{2} - b \log \left (a x^{2} + b\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.293708, size = 20, normalized size = 0.74 \begin{align*} \frac{x^{2}}{2 a} - \frac{b \log{\left (a x^{2} + b \right )}}{2 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09707, size = 32, normalized size = 1.19 \begin{align*} \frac{x^{2}}{2 \, a} - \frac{b \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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