Optimal. Leaf size=40 \[ -\frac{x^2}{12}+\frac{1}{2} x^2 \log \left (\sqrt [3]{3 x+1}\right )+\frac{x}{18}-\frac{1}{54} \log (3 x+1) \]
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Rubi [A] time = 0.0159573, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2395, 43} \[ -\frac{x^2}{12}+\frac{1}{2} x^2 \log \left (\sqrt [3]{3 x+1}\right )+\frac{x}{18}-\frac{1}{54} \log (3 x+1) \]
Antiderivative was successfully verified.
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Rule 2395
Rule 43
Rubi steps
\begin{align*} \int x \log \left (\sqrt [3]{1+3 x}\right ) \, dx &=\frac{1}{2} x^2 \log \left (\sqrt [3]{1+3 x}\right )-\frac{1}{2} \int \frac{x^2}{1+3 x} \, dx\\ &=\frac{1}{2} x^2 \log \left (\sqrt [3]{1+3 x}\right )-\frac{1}{2} \int \left (-\frac{1}{9}+\frac{x}{3}+\frac{1}{9 (1+3 x)}\right ) \, dx\\ &=\frac{x}{18}-\frac{x^2}{12}+\frac{1}{2} x^2 \log \left (\sqrt [3]{1+3 x}\right )-\frac{1}{54} \log (1+3 x)\\ \end{align*}
Mathematica [A] time = 0.0083492, size = 40, normalized size = 1. \[ \frac{1}{3} \left (-\frac{x^2}{4}+\frac{1}{2} x^2 \log (3 x+1)+\frac{x}{6}-\frac{1}{18} \log (3 x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 39, normalized size = 1. \begin{align*}{\frac{\ln \left ( 1+3\,x \right ) \left ( 1+3\,x \right ) ^{2}}{54}}-{\frac{{x}^{2}}{12}}+{\frac{x}{18}}+{\frac{1}{36}}-{\frac{ \left ( 1+3\,x \right ) \ln \left ( 1+3\,x \right ) }{27}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04281, size = 38, normalized size = 0.95 \begin{align*} \frac{1}{6} \, x^{2} \log \left (3 \, x + 1\right ) - \frac{1}{12} \, x^{2} + \frac{1}{18} \, x - \frac{1}{54} \, \log \left (3 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86306, size = 70, normalized size = 1.75 \begin{align*} -\frac{1}{12} \, x^{2} + \frac{1}{54} \,{\left (9 \, x^{2} - 1\right )} \log \left (3 \, x + 1\right ) + \frac{1}{18} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.117948, size = 27, normalized size = 0.68 \begin{align*} \frac{x^{2} \log{\left (3 x + 1 \right )}}{6} - \frac{x^{2}}{12} + \frac{x}{18} - \frac{\log{\left (3 x + 1 \right )}}{54} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21409, size = 57, normalized size = 1.42 \begin{align*} \frac{1}{54} \,{\left (3 \, x + 1\right )}^{2} \log \left (3 \, x + 1\right ) - \frac{1}{108} \,{\left (3 \, x + 1\right )}^{2} - \frac{1}{27} \,{\left (3 \, x + 1\right )} \log \left (3 \, x + 1\right ) + \frac{1}{9} \, x + \frac{1}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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