Optimal. Leaf size=45 \[ \frac {1}{2} x^2 \log ^3(c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {3}{4} x^2 \log (c x)-\frac {3 x^2}{8} \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2305, 2304} \[ \frac {1}{2} x^2 \log ^3(c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {3}{4} x^2 \log (c x)-\frac {3 x^2}{8} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int x \log ^3(c x) \, dx &=\frac {1}{2} x^2 \log ^3(c x)-\frac {3}{2} \int x \log ^2(c x) \, dx\\ &=-\frac {3}{4} x^2 \log ^2(c x)+\frac {1}{2} x^2 \log ^3(c x)+\frac {3}{2} \int x \log (c x) \, dx\\ &=-\frac {3 x^2}{8}+\frac {3}{4} x^2 \log (c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {1}{2} x^2 \log ^3(c x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 45, normalized size = 1.00 \[ \frac {1}{2} x^2 \log ^3(c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {3}{4} x^2 \log (c x)-\frac {3 x^2}{8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 37, normalized size = 0.82 \[ \frac {1}{2} \, x^{2} \log \left (c x\right )^{3} - \frac {3}{4} \, x^{2} \log \left (c x\right )^{2} + \frac {3}{4} \, x^{2} \log \left (c x\right ) - \frac {3}{8} \, x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 37, normalized size = 0.82 \[ \frac {1}{2} \, x^{2} \log \left (c x\right )^{3} - \frac {3}{4} \, x^{2} \log \left (c x\right )^{2} + \frac {3}{4} \, x^{2} \log \left (c x\right ) - \frac {3}{8} \, x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 38, normalized size = 0.84 \[ \frac {x^{2} \ln \left (c x \right )^{3}}{2}-\frac {3 x^{2} \ln \left (c x \right )^{2}}{4}+\frac {3 x^{2} \ln \left (c x \right )}{4}-\frac {3 x^{2}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 29, normalized size = 0.64 \[ \frac {1}{8} \, {\left (4 \, \log \left (c x\right )^{3} - 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 3\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.48, size = 29, normalized size = 0.64 \[ \frac {x^2\,\left (4\,{\ln \left (c\,x\right )}^3-6\,{\ln \left (c\,x\right )}^2+6\,\ln \left (c\,x\right )-3\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 42, normalized size = 0.93 \[ \frac {x^{2} \log {\left (c x \right )}^{3}}{2} - \frac {3 x^{2} \log {\left (c x \right )}^{2}}{4} + \frac {3 x^{2} \log {\left (c x \right )}}{4} - \frac {3 x^{2}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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