Integrand size = 40, antiderivative size = 16 \[ \int \frac {3+6 x^2+e^x x^2+\left (12 x^2+e^x \left (2 x^2+x^3\right )\right ) \log (x)}{x} \, dx=\left (6+e^x+\frac {3}{x^2}\right ) x^2 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {14, 2326, 2341} \[ \int \frac {3+6 x^2+e^x x^2+\left (12 x^2+e^x \left (2 x^2+x^3\right )\right ) \log (x)}{x} \, dx=e^x x^2 \log (x)+6 x^2 \log (x)+3 \log (x) \]
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Rule 14
Rule 2326
Rule 2341
Rubi steps \begin{align*} \text {integral}& = \int \left (e^x x (1+2 \log (x)+x \log (x))+\frac {3 \left (1+2 x^2+4 x^2 \log (x)\right )}{x}\right ) \, dx \\ & = 3 \int \frac {1+2 x^2+4 x^2 \log (x)}{x} \, dx+\int e^x x (1+2 \log (x)+x \log (x)) \, dx \\ & = e^x x^2 \log (x)+3 \int \left (\frac {1+2 x^2}{x}+4 x \log (x)\right ) \, dx \\ & = e^x x^2 \log (x)+3 \int \frac {1+2 x^2}{x} \, dx+12 \int x \log (x) \, dx \\ & = -3 x^2+6 x^2 \log (x)+e^x x^2 \log (x)+3 \int \left (\frac {1}{x}+2 x\right ) \, dx \\ & = 3 \log (x)+6 x^2 \log (x)+e^x x^2 \log (x) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {3+6 x^2+e^x x^2+\left (12 x^2+e^x \left (2 x^2+x^3\right )\right ) \log (x)}{x} \, dx=\left (3+\left (6+e^x\right ) x^2\right ) \log (x) \]
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Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31
method | result | size |
default | \(x^{2} {\mathrm e}^{x} \ln \left (x \right )+6 x^{2} \ln \left (x \right )+3 \ln \left (x \right )\) | \(21\) |
norman | \(x^{2} {\mathrm e}^{x} \ln \left (x \right )+6 x^{2} \ln \left (x \right )+3 \ln \left (x \right )\) | \(21\) |
risch | \(\left ({\mathrm e}^{x} x^{2}+6 x^{2}\right ) \ln \left (x \right )+3 \ln \left (x \right )\) | \(21\) |
parallelrisch | \(x^{2} {\mathrm e}^{x} \ln \left (x \right )+6 x^{2} \ln \left (x \right )+3 \ln \left (x \right )\) | \(21\) |
parts | \(x^{2} {\mathrm e}^{x} \ln \left (x \right )+6 x^{2} \ln \left (x \right )+3 \ln \left (x \right )\) | \(21\) |
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none
Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {3+6 x^2+e^x x^2+\left (12 x^2+e^x \left (2 x^2+x^3\right )\right ) \log (x)}{x} \, dx={\left (x^{2} e^{x} + 6 \, x^{2} + 3\right )} \log \left (x\right ) \]
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Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {3+6 x^2+e^x x^2+\left (12 x^2+e^x \left (2 x^2+x^3\right )\right ) \log (x)}{x} \, dx=x^{2} e^{x} \log {\left (x \right )} + 6 x^{2} \log {\left (x \right )} + 3 \log {\left (x \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {3+6 x^2+e^x x^2+\left (12 x^2+e^x \left (2 x^2+x^3\right )\right ) \log (x)}{x} \, dx=6 \, x^{2} \log \left (x\right ) + {\left (x^{2} \log \left (x\right ) - x + 1\right )} e^{x} + {\left (x - 1\right )} e^{x} + 3 \, \log \left (x\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {3+6 x^2+e^x x^2+\left (12 x^2+e^x \left (2 x^2+x^3\right )\right ) \log (x)}{x} \, dx=x^{2} e^{x} \log \left (x\right ) + 6 \, x^{2} \log \left (x\right ) + 3 \, \log \left (x\right ) \]
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Time = 14.57 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {3+6 x^2+e^x x^2+\left (12 x^2+e^x \left (2 x^2+x^3\right )\right ) \log (x)}{x} \, dx=\ln \left (x\right )\,\left (x^2\,{\mathrm {e}}^x+6\,x^2+3\right ) \]
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