Optimal. Leaf size=45 \[ \frac{\log \left (a+b x^3\right )}{3 (b c-a d)}-\frac{\log \left (c+d x^3\right )}{3 (b c-a d)} \]
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Rubi [A] time = 0.0308165, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {444, 36, 31} \[ \frac{\log \left (a+b x^3\right )}{3 (b c-a d)}-\frac{\log \left (c+d x^3\right )}{3 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 444
Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(a+b x) (c+d x)} \, dx,x,x^3\right )\\ &=\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,x^3\right )}{3 (b c-a d)}-\frac{d \operatorname{Subst}\left (\int \frac{1}{c+d x} \, dx,x,x^3\right )}{3 (b c-a d)}\\ &=\frac{\log \left (a+b x^3\right )}{3 (b c-a d)}-\frac{\log \left (c+d x^3\right )}{3 (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0175543, size = 31, normalized size = 0.69 \[ \frac{\log \left (a+b x^3\right )-\log \left (c+d x^3\right )}{3 b c-3 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 42, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( d{x}^{3}+c \right ) }{3\,ad-3\,bc}}-{\frac{\ln \left ( b{x}^{3}+a \right ) }{3\,ad-3\,bc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.924533, size = 55, normalized size = 1.22 \begin{align*} \frac{\log \left (b x^{3} + a\right )}{3 \,{\left (b c - a d\right )}} - \frac{\log \left (d x^{3} + c\right )}{3 \,{\left (b c - a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52765, size = 69, normalized size = 1.53 \begin{align*} \frac{\log \left (b x^{3} + a\right ) - \log \left (d x^{3} + c\right )}{3 \,{\left (b c - a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.50961, size = 138, normalized size = 3.07 \begin{align*} \frac{\log{\left (x^{3} + \frac{- \frac{a^{2} d^{2}}{a d - b c} + \frac{2 a b c d}{a d - b c} + a d - \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{3 \left (a d - b c\right )} - \frac{\log{\left (x^{3} + \frac{\frac{a^{2} d^{2}}{a d - b c} - \frac{2 a b c d}{a d - b c} + a d + \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{3 \left (a d - b c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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