Optimal. Leaf size=32 \[ \frac {1}{3} x^3 \log ^2(c x)-\frac {2}{9} x^3 \log (c x)+\frac {2 x^3}{27} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2305, 2304} \[ \frac {1}{3} x^3 \log ^2(c x)-\frac {2}{9} x^3 \log (c x)+\frac {2 x^3}{27} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int x^2 \log ^2(c x) \, dx &=\frac {1}{3} x^3 \log ^2(c x)-\frac {2}{3} \int x^2 \log (c x) \, dx\\ &=\frac {2 x^3}{27}-\frac {2}{9} x^3 \log (c x)+\frac {1}{3} x^3 \log ^2(c x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 32, normalized size = 1.00 \[ \frac {1}{3} x^3 \log ^2(c x)-\frac {2}{9} x^3 \log (c x)+\frac {2 x^3}{27} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 26, normalized size = 0.81 \[ \frac {1}{3} \, x^{3} \log \left (c x\right )^{2} - \frac {2}{9} \, x^{3} \log \left (c x\right ) + \frac {2}{27} \, x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 26, normalized size = 0.81 \[ \frac {1}{3} \, x^{3} \log \left (c x\right )^{2} - \frac {2}{9} \, x^{3} \log \left (c x\right ) + \frac {2}{27} \, x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 27, normalized size = 0.84 \[ \frac {x^{3} \ln \left (c x \right )^{2}}{3}-\frac {2 x^{3} \ln \left (c x \right )}{9}+\frac {2 x^{3}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 21, normalized size = 0.66 \[ \frac {1}{27} \, {\left (9 \, \log \left (c x\right )^{2} - 6 \, \log \left (c x\right ) + 2\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 21, normalized size = 0.66 \[ \frac {x^3\,\left (9\,{\ln \left (c\,x\right )}^2-6\,\ln \left (c\,x\right )+2\right )}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 29, normalized size = 0.91 \[ \frac {x^{3} \log {\left (c x \right )}^{2}}{3} - \frac {2 x^{3} \log {\left (c x \right )}}{9} + \frac {2 x^{3}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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