Optimal. Leaf size=45 \[ \frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{6} b c^3 \log \left (c^2-x^2\right )+\frac{1}{6} b c x^2 \]
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Rubi [A] time = 0.0327064, antiderivative size = 45, 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 = {6097, 263, 266, 43} \[ \frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{6} b c^3 \log \left (c^2-x^2\right )+\frac{1}{6} b c x^2 \]
Antiderivative was successfully verified.
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Rule 6097
Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 = 0.0079073, size = 50, normalized size = 1.11 \[ \frac{a x^3}{3}+\frac{1}{6} b c^3 \log \left (x^2-c^2\right )+\frac{1}{6} b c x^2+\frac{1}{3} b x^3 \tanh ^{-1}\left (\frac{c}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 67, normalized size = 1.5 \begin{align*}{\frac{{x}^{3}a}{3}}+{\frac{b{x}^{3}}{3}{\it Artanh} \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{3}b}{6}\ln \left ({\frac{c}{x}}-1 \right ) }+{\frac{bc{x}^{2}}{6}}-{\frac{{c}^{3}b}{3}\ln \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{3}b}{6}\ln \left ( 1+{\frac{c}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961072, size = 57, normalized size = 1.27 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{6} \,{\left (2 \, x^{3} \operatorname{artanh}\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 = 1.76935, size = 117, normalized size = 2.6 \begin{align*} \frac{1}{6} \, b c^{3} \log \left (-c^{2} + x^{2}\right ) + \frac{1}{6} \, b x^{3} \log \left (-\frac{c + x}{c - x}\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 = 1.10867, size = 49, normalized size = 1.09 \begin{align*} \frac{a x^{3}}{3} + \frac{b c^{3} \log{\left (- c + x \right )}}{3} + \frac{b c^{3} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{3} + \frac{b c x^{2}}{6} + \frac{b x^{3} \operatorname{atanh}{\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 = 1.1681, size = 66, normalized size = 1.47 \begin{align*} \frac{1}{6} \, b c^{3} \log \left (-c^{2} + x^{2}\right ) + \frac{1}{6} \, b x^{3} \log \left (-\frac{c + x}{c - x}\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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