Optimal. Leaf size=35 \[ \frac{(A b-a B) \log \left (a+b x^3\right )}{3 b^2}+\frac{B x^3}{3 b} \]
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Rubi [A] time = 0.0327442, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 43} \[ \frac{(A b-a B) \log \left (a+b x^3\right )}{3 b^2}+\frac{B x^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{B}{b}+\frac{A b-a B}{b (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{B x^3}{3 b}+\frac{(A b-a B) \log \left (a+b x^3\right )}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0122157, size = 31, normalized size = 0.89 \[ \frac{(A b-a B) \log \left (a+b x^3\right )+b B x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 40, normalized size = 1.1 \begin{align*}{\frac{B{x}^{3}}{3\,b}}+{\frac{\ln \left ( b{x}^{3}+a \right ) A}{3\,b}}-{\frac{\ln \left ( b{x}^{3}+a \right ) Ba}{3\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14153, size = 42, normalized size = 1.2 \begin{align*} \frac{B x^{3}}{3 \, b} - \frac{{\left (B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38283, size = 65, normalized size = 1.86 \begin{align*} \frac{B b x^{3} -{\left (B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.691102, size = 27, normalized size = 0.77 \begin{align*} \frac{B x^{3}}{3 b} - \frac{\left (- A b + B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11574, size = 43, normalized size = 1.23 \begin{align*} \frac{B x^{3}}{3 \, b} - \frac{{\left (B a - A b\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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