Optimal. Leaf size=39 \[ -\frac{x^3}{9}+\frac{x^2}{6}+\frac{1}{3} x^3 \log (x+1)-\frac{x}{3}+\frac{1}{3} \log (x+1) \]
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Rubi [A] time = 0.0150372, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2395, 43} \[ -\frac{x^3}{9}+\frac{x^2}{6}+\frac{1}{3} x^3 \log (x+1)-\frac{x}{3}+\frac{1}{3} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 2395
Rule 43
Rubi steps
\begin{align*} \int x^2 \log (1+x) \, dx &=\frac{1}{3} x^3 \log (1+x)-\frac{1}{3} \int \frac{x^3}{1+x} \, dx\\ &=\frac{1}{3} x^3 \log (1+x)-\frac{1}{3} \int \left (1+\frac{1}{-1-x}-x+x^2\right ) \, dx\\ &=-\frac{x}{3}+\frac{x^2}{6}-\frac{x^3}{9}+\frac{1}{3} \log (1+x)+\frac{1}{3} x^3 \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0092566, size = 28, normalized size = 0.72 \[ \frac{1}{18} \left (x \left (-2 x^2+3 x-6\right )+6 \left (x^3+1\right ) \log (x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 46, normalized size = 1.2 \begin{align*}{\frac{ \left ( 1+x \right ) ^{3}\ln \left ( 1+x \right ) }{3}}-{\frac{{x}^{3}}{9}}+{\frac{{x}^{2}}{6}}-{\frac{x}{3}}-{\frac{11}{18}}- \left ( 1+x \right ) ^{2}\ln \left ( 1+x \right ) + \left ( 1+x \right ) \ln \left ( 1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953137, size = 39, normalized size = 1. \begin{align*} \frac{1}{3} \, x^{3} \log \left (x + 1\right ) - \frac{1}{9} \, x^{3} + \frac{1}{6} \, x^{2} - \frac{1}{3} \, x + \frac{1}{3} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91516, size = 74, normalized size = 1.9 \begin{align*} -\frac{1}{9} \, x^{3} + \frac{1}{6} \, x^{2} + \frac{1}{3} \,{\left (x^{3} + 1\right )} \log \left (x + 1\right ) - \frac{1}{3} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.107155, size = 29, normalized size = 0.74 \begin{align*} \frac{x^{3} \log{\left (x + 1 \right )}}{3} - \frac{x^{3}}{9} + \frac{x^{2}}{6} - \frac{x}{3} + \frac{\log{\left (x + 1 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04495, size = 66, normalized size = 1.69 \begin{align*} \frac{1}{3} \,{\left (x + 1\right )}^{3} \log \left (x + 1\right ) - \frac{1}{9} \,{\left (x + 1\right )}^{3} -{\left (x + 1\right )}^{2} \log \left (x + 1\right ) + \frac{1}{2} \,{\left (x + 1\right )}^{2} +{\left (x + 1\right )} \log \left (x + 1\right ) - x - 1 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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