Integrand size = 8, antiderivative size = 28 \[ \int x^2 \log ^2(x) \, dx=\frac {2 x^3}{27}-\frac {2}{9} x^3 \log (x)+\frac {1}{3} x^3 \log ^2(x) \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2342, 2341} \[ \int x^2 \log ^2(x) \, dx=\frac {2 x^3}{27}+\frac {1}{3} x^3 \log ^2(x)-\frac {2}{9} x^3 \log (x) \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \log ^2(x)-\frac {2}{3} \int x^2 \log (x) \, dx \\ & = \frac {2 x^3}{27}-\frac {2}{9} x^3 \log (x)+\frac {1}{3} x^3 \log ^2(x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int x^2 \log ^2(x) \, dx=\frac {2 x^3}{27}-\frac {2}{9} x^3 \log (x)+\frac {1}{3} x^3 \log ^2(x) \]
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Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x \right )}{9}+\frac {x^{3} \ln \left (x \right )^{2}}{3}\) | \(23\) |
norman | \(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x \right )}{9}+\frac {x^{3} \ln \left (x \right )^{2}}{3}\) | \(23\) |
risch | \(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x \right )}{9}+\frac {x^{3} \ln \left (x \right )^{2}}{3}\) | \(23\) |
parallelrisch | \(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x \right )}{9}+\frac {x^{3} \ln \left (x \right )^{2}}{3}\) | \(23\) |
parts | \(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x \right )}{9}+\frac {x^{3} \ln \left (x \right )^{2}}{3}\) | \(23\) |
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none
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int x^2 \log ^2(x) \, dx=\frac {1}{3} \, x^{3} \log \left (x\right )^{2} - \frac {2}{9} \, x^{3} \log \left (x\right ) + \frac {2}{27} \, x^{3} \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int x^2 \log ^2(x) \, dx=\frac {x^{3} \log {\left (x \right )}^{2}}{3} - \frac {2 x^{3} \log {\left (x \right )}}{9} + \frac {2 x^{3}}{27} \]
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none
Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int x^2 \log ^2(x) \, dx=\frac {1}{27} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} \]
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none
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int x^2 \log ^2(x) \, dx=\frac {1}{3} \, x^{3} \log \left (x\right )^{2} - \frac {2}{9} \, x^{3} \log \left (x\right ) + \frac {2}{27} \, x^{3} \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int x^2 \log ^2(x) \, dx=\frac {2\,x^3\,\left (\frac {9\,{\ln \left (x\right )}^2}{2}-3\,\ln \left (x\right )+1\right )}{27} \]
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