Optimal. Leaf size=15 \[ \frac{\log \left (b+c x^2\right )}{2 c} \]
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Rubi [A] time = 0.0089441, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 260} \[ \frac{\log \left (b+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 260
Rubi steps
\begin{align*} \int \frac{x^3}{b x^2+c x^4} \, dx &=\int \frac{x}{b+c x^2} \, dx\\ &=\frac{\log \left (b+c x^2\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.0023321, size = 15, normalized size = 1. \[ \frac{\log \left (b+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 14, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( c{x}^{2}+b \right ) }{2\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00259, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (c x^{2} + b\right )}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43033, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (c x^{2} + b\right )}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.113073, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (b + c x^{2} \right )}}{2 c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20426, size = 19, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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