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 = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 43
Rule 2395
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*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ \frac {1}{3} \left (-\frac {x^2}{4}+\frac {1}{2} x^2 \log (3 x+1)+\frac {x}{6}-\frac {1}{18} \log (3 x+1)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 24, normalized size = 0.60 \[ -\frac {1}{12} \, x^{2} + \frac {1}{54} \, {\left (9 \, x^{2} - 1\right )} \log \left (3 \, x + 1\right ) + \frac {1}{18} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 42, normalized size = 1.05 \[ \frac {1}{54} \, {\left (3 \, x + 1\right )}^{2} \log \left (3 \, x + 1\right ) - \frac {1}{108} \, {\left (3 \, x + 1\right )}^{2} - \frac {1}{27} \, {\left (3 \, x + 1\right )} \log \left (3 \, x + 1\right ) + \frac {1}{9} \, x + \frac {1}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 39, normalized size = 0.98 \[ -\frac {x^{2}}{12}+\frac {x}{18}+\frac {\left (3 x +1\right )^{2} \ln \left (3 x +1\right )}{54}-\frac {\left (3 x +1\right ) \ln \left (3 x +1\right )}{27}+\frac {1}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 28, normalized size = 0.70 \[ \frac {1}{6} \, x^{2} \log \left (3 \, x + 1\right ) - \frac {1}{12} \, x^{2} + \frac {1}{18} \, x - \frac {1}{54} \, \log \left (3 \, x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 22, normalized size = 0.55 \[ \frac {x}{18}+\frac {\ln \left (3\,x+1\right )\,\left (x^2-\frac {1}{9}\right )}{6}-\frac {x^2}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 27, normalized size = 0.68 \[ \frac {x^{2} \log {\left (3 x + 1 \right )}}{6} - \frac {x^{2}}{12} + \frac {x}{18} - \frac {\log {\left (3 x + 1 \right )}}{54} \]
Verification of antiderivative is not currently implemented for this CAS.
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