Optimal. Leaf size=45 \[ \frac{1}{3} x^3 \log ^3(c x)-\frac{1}{3} x^3 \log ^2(c x)+\frac{2}{9} x^3 \log (c x)-\frac{2 x^3}{27} \]
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Rubi [A] time = 0.0314186, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2305, 2304} \[ \frac{1}{3} x^3 \log ^3(c x)-\frac{1}{3} x^3 \log ^2(c x)+\frac{2}{9} x^3 \log (c x)-\frac{2 x^3}{27} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x^2 \log ^3(c x) \, dx &=\frac{1}{3} x^3 \log ^3(c x)-\int x^2 \log ^2(c x) \, dx\\ &=-\frac{1}{3} x^3 \log ^2(c x)+\frac{1}{3} x^3 \log ^3(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)+\frac{1}{3} x^3 \log ^3(c x)\\ \end{align*}
Mathematica [A] time = 0.0014576, size = 45, normalized size = 1. \[ \frac{1}{3} x^3 \log ^3(c x)-\frac{1}{3} x^3 \log ^2(c x)+\frac{2}{9} x^3 \log (c x)-\frac{2 x^3}{27} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 38, normalized size = 0.8 \begin{align*} -{\frac{2\,{x}^{3}}{27}}+{\frac{2\,{x}^{3}\ln \left ( cx \right ) }{9}}-{\frac{{x}^{3} \left ( \ln \left ( cx \right ) \right ) ^{2}}{3}}+{\frac{{x}^{3} \left ( \ln \left ( cx \right ) \right ) ^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07407, size = 39, normalized size = 0.87 \begin{align*} \frac{1}{27} \,{\left (9 \, \log \left (c x\right )^{3} - 9 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 2\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.781229, size = 96, normalized size = 2.13 \begin{align*} \frac{1}{3} \, x^{3} \log \left (c x\right )^{3} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.121807, size = 41, normalized size = 0.91 \begin{align*} \frac{x^{3} \log{\left (c x \right )}^{3}}{3} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10201, size = 50, normalized size = 1.11 \begin{align*} \frac{1}{3} \, x^{3} \log \left (c x\right )^{3} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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