Optimal. Leaf size=36 \[ \frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{b \log \left (c^2 x^6+1\right )}{6 c} \]
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Rubi [A] time = 0.0206472, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5033, 260} \[ \frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{b \log \left (c^2 x^6+1\right )}{6 c} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 260
Rubi steps
\begin{align*} \int x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-(b c) \int \frac{x^5}{1+c^2 x^6} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{b \log \left (1+c^2 x^6\right )}{6 c}\\ \end{align*}
Mathematica [A] time = 0.007222, size = 41, normalized size = 1.14 \[ \frac{a x^3}{3}-\frac{b \log \left (c^2 x^6+1\right )}{6 c}+\frac{1}{3} b x^3 \tan ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 36, normalized size = 1. \begin{align*}{\frac{{x}^{3}a}{3}}+{\frac{b{x}^{3}\arctan \left ( c{x}^{3} \right ) }{3}}-{\frac{b\ln \left ({c}^{2}{x}^{6}+1 \right ) }{6\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00259, size = 51, normalized size = 1.42 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{{\left (2 \, c x^{3} \arctan \left (c x^{3}\right ) - \log \left (c^{2} x^{6} + 1\right )\right )} b}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.66065, size = 89, normalized size = 2.47 \begin{align*} \frac{2 \, b c x^{3} \arctan \left (c x^{3}\right ) + 2 \, a c x^{3} - b \log \left (c^{2} x^{6} + 1\right )}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 83.1445, size = 808, normalized size = 22.44 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.161, size = 54, normalized size = 1.5 \begin{align*} \frac{2 \, a c x^{3} +{\left (2 \, c x^{3} \arctan \left (c x^{3}\right ) - \log \left (c^{2} x^{6} + 1\right )\right )} b}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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