Optimal. Leaf size=45 \[ \frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{b \log \left (c^2 x^2+1\right )}{6 c^3}-\frac{b x^2}{6 c} \]
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Rubi [A] time = 0.0318549, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4852, 266, 43} \[ \frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{b \log \left (c^2 x^2+1\right )}{6 c^3}-\frac{b x^2}{6 c} \]
Antiderivative was successfully verified.
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Rule 4852
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{3} (b c) \int \frac{x^3}{1+c^2 x^2} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{6} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{b x^2}{6 c}+\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{b \log \left (1+c^2 x^2\right )}{6 c^3}\\ \end{align*}
Mathematica [A] time = 0.0082896, size = 50, normalized size = 1.11 \[ \frac{a x^3}{3}+\frac{b \log \left (c^2 x^2+1\right )}{6 c^3}-\frac{b x^2}{6 c}+\frac{1}{3} b x^3 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 43, normalized size = 1. \begin{align*}{\frac{{x}^{3}a}{3}}+{\frac{b{x}^{3}\arctan \left ( cx \right ) }{3}}-{\frac{b{x}^{2}}{6\,c}}+{\frac{b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{6\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982194, size = 62, normalized size = 1.38 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{6} \,{\left (2 \, x^{3} \arctan \left (c x\right ) - c{\left (\frac{x^{2}}{c^{2}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30553, size = 111, normalized size = 2.47 \begin{align*} \frac{2 \, b c^{3} x^{3} \arctan \left (c x\right ) + 2 \, a c^{3} x^{3} - b c^{2} x^{2} + b \log \left (c^{2} x^{2} + 1\right )}{6 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.765492, size = 49, normalized size = 1.09 \begin{align*} \begin{cases} \frac{a x^{3}}{3} + \frac{b x^{3} \operatorname{atan}{\left (c x \right )}}{3} - \frac{b x^{2}}{6 c} + \frac{b \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{6 c^{3}} & \text{for}\: c \neq 0 \\\frac{a x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20281, size = 66, normalized size = 1.47 \begin{align*} \frac{2 \, b c^{3} x^{3} \arctan \left (c x\right ) + 2 \, a c^{3} x^{3} - b c^{2} x^{2} + b \log \left (c^{2} x^{2} + 1\right )}{6 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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