Integrand size = 36, antiderivative size = 19 \[ \int \frac {-18+3 x^3+2 e^x x^2 \log (5 x)+e^x x^3 \log ^2(5 x)}{x^3} \, dx=\frac {9}{x^2}+3 x+e^x \log ^2(5 x) \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14, 2233} \[ \int \frac {-18+3 x^3+2 e^x x^2 \log (5 x)+e^x x^3 \log ^2(5 x)}{x^3} \, dx=\frac {9}{x^2}+3 x+e^x \log ^2(5 x) \]
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Rule 14
Rule 2233
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \left (-6+x^3\right )}{x^3}+\frac {e^x \log (5 x) (2+x \log (5 x))}{x}\right ) \, dx \\ & = 3 \int \frac {-6+x^3}{x^3} \, dx+\int \frac {e^x \log (5 x) (2+x \log (5 x))}{x} \, dx \\ & = e^x \log ^2(5 x)+3 \int \left (1-\frac {6}{x^3}\right ) \, dx \\ & = \frac {9}{x^2}+3 x+e^x \log ^2(5 x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-18+3 x^3+2 e^x x^2 \log (5 x)+e^x x^3 \log ^2(5 x)}{x^3} \, dx=\frac {9}{x^2}+3 x+e^x \log ^2(5 x) \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00
method | result | size |
default | \(3 x +\ln \left (5 x \right )^{2} {\mathrm e}^{x}+\frac {9}{x^{2}}\) | \(19\) |
parts | \(3 x +\ln \left (5 x \right )^{2} {\mathrm e}^{x}+\frac {9}{x^{2}}\) | \(19\) |
risch | \(\ln \left (5 x \right )^{2} {\mathrm e}^{x}+\frac {3 x^{3}+9}{x^{2}}\) | \(21\) |
parallelrisch | \(-\frac {-x^{2} {\mathrm e}^{x} \ln \left (5 x \right )^{2}-9-3 x^{3}}{x^{2}}\) | \(26\) |
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none
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {-18+3 x^3+2 e^x x^2 \log (5 x)+e^x x^3 \log ^2(5 x)}{x^3} \, dx=\frac {x^{2} e^{x} \log \left (5 \, x\right )^{2} + 3 \, x^{3} + 9}{x^{2}} \]
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Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-18+3 x^3+2 e^x x^2 \log (5 x)+e^x x^3 \log ^2(5 x)}{x^3} \, dx=3 x + e^{x} \log {\left (5 x \right )}^{2} + \frac {9}{x^{2}} \]
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none
Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {-18+3 x^3+2 e^x x^2 \log (5 x)+e^x x^3 \log ^2(5 x)}{x^3} \, dx={\left (\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{x} + 3 \, x + \frac {9}{x^{2}} \]
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none
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {-18+3 x^3+2 e^x x^2 \log (5 x)+e^x x^3 \log ^2(5 x)}{x^3} \, dx=\frac {x^{2} e^{x} \log \left (5 \, x\right )^{2} + 3 \, x^{3} + 9}{x^{2}} \]
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Time = 7.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-18+3 x^3+2 e^x x^2 \log (5 x)+e^x x^3 \log ^2(5 x)}{x^3} \, dx=3\,x+\frac {9}{x^2}+{\ln \left (5\,x\right )}^2\,{\mathrm {e}}^x \]
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