Optimal. Leaf size=43 \[ \frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{6} b c^3 \log \left (c^2+x^2\right )+\frac{1}{6} b c x^2 \]
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Rubi [A] time = 0.029785, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5033, 263, 266, 43} \[ \frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{6} b c^3 \log \left (c^2+x^2\right )+\frac{1}{6} b c x^2 \]
Antiderivative was successfully verified.
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Rule 5033
Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{3} (b c) \int \frac{x}{1+\frac{c^2}{x^2}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{3} (b c) \int \frac{x^3}{c^2+x^2} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{x}{c^2+x} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \left (1-\frac{c^2}{c^2+x}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{6} b c x^2+\frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{6} b c^3 \log \left (c^2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0073186, size = 48, normalized size = 1.12 \[ \frac{a x^3}{3}-\frac{1}{6} b c^3 \log \left (c^2+x^2\right )+\frac{1}{6} b c x^2+\frac{1}{3} b x^3 \tan ^{-1}\left (\frac{c}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 55, normalized size = 1.3 \begin{align*}{\frac{{x}^{3}a}{3}}+{\frac{b{x}^{3}}{3}\arctan \left ({\frac{c}{x}} \right ) }-{\frac{{c}^{3}b}{6}\ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) }+{\frac{bc{x}^{2}}{6}}+{\frac{{c}^{3}b}{3}\ln \left ({\frac{c}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988883, size = 58, normalized size = 1.35 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{6} \,{\left (2 \, x^{3} \arctan \left (\frac{c}{x}\right ) -{\left (c^{2} \log \left (c^{2} + x^{2}\right ) - x^{2}\right )} c\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1818, size = 103, normalized size = 2.4 \begin{align*} \frac{1}{3} \, b x^{3} \arctan \left (\frac{c}{x}\right ) - \frac{1}{6} \, b c^{3} \log \left (c^{2} + x^{2}\right ) + \frac{1}{6} \, b c x^{2} + \frac{1}{3} \, a x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.5446, size = 41, normalized size = 0.95 \begin{align*} \frac{a x^{3}}{3} - \frac{b c^{3} \log{\left (c^{2} + x^{2} \right )}}{6} + \frac{b c x^{2}}{6} + \frac{b x^{3} \operatorname{atan}{\left (\frac{c}{x} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12529, size = 54, normalized size = 1.26 \begin{align*} \frac{1}{3} \, b x^{3} \arctan \left (\frac{c}{x}\right ) - \frac{1}{6} \, b c^{3} \log \left (c^{2} + x^{2}\right ) + \frac{1}{6} \, b c x^{2} + \frac{1}{3} \, a x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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