Integrand size = 10, antiderivative size = 45 \[ \int x^2 \log ^3(c x) \, dx=-\frac {2 x^3}{27}+\frac {2}{9} x^3 \log (c x)-\frac {1}{3} x^3 \log ^2(c x)+\frac {1}{3} x^3 \log ^3(c x) \]
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Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2342, 2341} \[ \int x^2 \log ^3(c x) \, dx=\frac {1}{3} x^3 \log ^3(c x)-\frac {1}{3} x^3 \log ^2(c x)+\frac {2}{9} x^3 \log (c x)-\frac {2 x^3}{27} \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.00 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int x^2 \log ^3(c x) \, dx=-\frac {2 x^3}{27}+\frac {2}{9} x^3 \log (c x)-\frac {1}{3} x^3 \log ^2(c x)+\frac {1}{3} x^3 \log ^3(c x) \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84
method | result | size |
norman | \(-\frac {2 x^{3}}{27}+\frac {2 x^{3} \ln \left (x c \right )}{9}-\frac {x^{3} \ln \left (x c \right )^{2}}{3}+\frac {x^{3} \ln \left (x c \right )^{3}}{3}\) | \(38\) |
risch | \(-\frac {2 x^{3}}{27}+\frac {2 x^{3} \ln \left (x c \right )}{9}-\frac {x^{3} \ln \left (x c \right )^{2}}{3}+\frac {x^{3} \ln \left (x c \right )^{3}}{3}\) | \(38\) |
parallelrisch | \(-\frac {2 x^{3}}{27}+\frac {2 x^{3} \ln \left (x c \right )}{9}-\frac {x^{3} \ln \left (x c \right )^{2}}{3}+\frac {x^{3} \ln \left (x c \right )^{3}}{3}\) | \(38\) |
parts | \(\frac {x^{3} \ln \left (x c \right )^{3}}{3}-\frac {\frac {x^{3} c^{3} \ln \left (x c \right )^{2}}{3}-\frac {2 x^{3} c^{3} \ln \left (x c \right )}{9}+\frac {2 x^{3} c^{3}}{27}}{c^{3}}\) | \(53\) |
derivativedivides | \(\frac {\frac {x^{3} c^{3} \ln \left (x c \right )^{3}}{3}-\frac {x^{3} c^{3} \ln \left (x c \right )^{2}}{3}+\frac {2 x^{3} c^{3} \ln \left (x c \right )}{9}-\frac {2 x^{3} c^{3}}{27}}{c^{3}}\) | \(54\) |
default | \(\frac {\frac {x^{3} c^{3} \ln \left (x c \right )^{3}}{3}-\frac {x^{3} c^{3} \ln \left (x c \right )^{2}}{3}+\frac {2 x^{3} c^{3} \ln \left (x c \right )}{9}-\frac {2 x^{3} c^{3}}{27}}{c^{3}}\) | \(54\) |
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Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int x^2 \log ^3(c x) \, dx=\frac {1}{3} \, x^{3} \log \left (c x\right )^{3} - \frac {1}{3} \, x^{3} \log \left (c x\right )^{2} + \frac {2}{9} \, x^{3} \log \left (c x\right ) - \frac {2}{27} \, x^{3} \]
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Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int x^2 \log ^3(c x) \, dx=\frac {x^{3} \log {\left (c x \right )}^{3}}{3} - \frac {x^{3} \log {\left (c x \right )}^{2}}{3} + \frac {2 x^{3} \log {\left (c x \right )}}{9} - \frac {2 x^{3}}{27} \]
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Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int x^2 \log ^3(c x) \, dx=\frac {1}{27} \, {\left (9 \, \log \left (c x\right )^{3} - 9 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 2\right )} x^{3} \]
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Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int x^2 \log ^3(c x) \, dx=\frac {1}{3} \, x^{3} \log \left (c x\right )^{3} - \frac {1}{3} \, x^{3} \log \left (c x\right )^{2} + \frac {2}{9} \, x^{3} \log \left (c x\right ) - \frac {2}{27} \, x^{3} \]
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int x^2 \log ^3(c x) \, dx=\frac {x^3\,\left (9\,{\ln \left (c\,x\right )}^3-9\,{\ln \left (c\,x\right )}^2+6\,\ln \left (c\,x\right )-2\right )}{27} \]
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