Integrand size = 8, antiderivative size = 19 \[ \int x^2 \log (c x) \, dx=-\frac {x^3}{9}+\frac {1}{3} x^3 \log (c x) \]
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Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2341} \[ \int x^2 \log (c x) \, dx=\frac {1}{3} x^3 \log (c x)-\frac {x^3}{9} \]
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Rule 2341
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3}{9}+\frac {1}{3} x^3 \log (c x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int x^2 \log (c x) \, dx=-\frac {x^3}{9}+\frac {1}{3} x^3 \log (c x) \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
norman | \(-\frac {x^{3}}{9}+\frac {x^{3} \ln \left (x c \right )}{3}\) | \(16\) |
risch | \(-\frac {x^{3}}{9}+\frac {x^{3} \ln \left (x c \right )}{3}\) | \(16\) |
parallelrisch | \(-\frac {x^{3}}{9}+\frac {x^{3} \ln \left (x c \right )}{3}\) | \(16\) |
parts | \(-\frac {x^{3}}{9}+\frac {x^{3} \ln \left (x c \right )}{3}\) | \(16\) |
derivativedivides | \(\frac {\frac {x^{3} c^{3} \ln \left (x c \right )}{3}-\frac {x^{3} c^{3}}{9}}{c^{3}}\) | \(26\) |
default | \(\frac {\frac {x^{3} c^{3} \ln \left (x c \right )}{3}-\frac {x^{3} c^{3}}{9}}{c^{3}}\) | \(26\) |
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none
Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int x^2 \log (c x) \, dx=\frac {1}{3} \, x^{3} \log \left (c x\right ) - \frac {1}{9} \, x^{3} \]
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Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int x^2 \log (c x) \, dx=\frac {x^{3} \log {\left (c x \right )}}{3} - \frac {x^{3}}{9} \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int x^2 \log (c x) \, dx=\frac {1}{3} \, x^{3} \log \left (c x\right ) - \frac {1}{9} \, x^{3} \]
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none
Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int x^2 \log (c x) \, dx=\frac {1}{3} \, x^{3} \log \left (c x\right ) - \frac {1}{9} \, x^{3} \]
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Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int x^2 \log (c x) \, dx=\frac {x^3\,\left (\ln \left (c\,x\right )-\frac {1}{3}\right )}{3} \]
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