Integrand size = 38, antiderivative size = 25 \[ \int \frac {x+e^5 \left (-9-23 x-7 x^2\right )+e^5 \left (-6 x-2 x^2\right ) \log (x)}{e^5 x} \, dx=-16+x+\frac {x}{e^5}-\log ^2(2)-(3+x)^2 (3+\log (x)) \]
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Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 14, 2350, 9} \[ \int \frac {x+e^5 \left (-9-23 x-7 x^2\right )+e^5 \left (-6 x-2 x^2\right ) \log (x)}{e^5 x} \, dx=-\frac {7 x^2}{2}+x^2 (-\log (x))-\left (23-\frac {1}{e^5}\right ) x+\frac {1}{2} (x+6)^2-6 x \log (x)-9 \log (x) \]
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Rule 9
Rule 12
Rule 14
Rule 2350
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {x+e^5 \left (-9-23 x-7 x^2\right )+e^5 \left (-6 x-2 x^2\right ) \log (x)}{x} \, dx}{e^5} \\ & = \frac {\int \left (\frac {-9 e^5+\left (1-23 e^5\right ) x-7 e^5 x^2}{x}-2 e^5 (3+x) \log (x)\right ) \, dx}{e^5} \\ & = -(2 \int (3+x) \log (x) \, dx)+\frac {\int \frac {-9 e^5+\left (1-23 e^5\right ) x-7 e^5 x^2}{x} \, dx}{e^5} \\ & = -6 x \log (x)-x^2 \log (x)+2 \int \frac {6+x}{2} \, dx+\frac {\int \left (1-23 e^5-\frac {9 e^5}{x}-7 e^5 x\right ) \, dx}{e^5} \\ & = -\left (\left (23-\frac {1}{e^5}\right ) x\right )-\frac {7 x^2}{2}+\frac {1}{2} (6+x)^2-9 \log (x)-6 x \log (x)-x^2 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {x+e^5 \left (-9-23 x-7 x^2\right )+e^5 \left (-6 x-2 x^2\right ) \log (x)}{e^5 x} \, dx=-17 x+\frac {x}{e^5}-3 x^2-9 \log (x)-6 x \log (x)-x^2 \log (x) \]
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Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
risch | \(-\left (6+x \right ) x \ln \left (x \right )-3 x^{2}-17 x +{\mathrm e}^{-5} x -9 \ln \left (x \right )\) | \(26\) |
parts | \(-x^{2} \ln \left (x \right )+{\mathrm e}^{-5} x -6 x \ln \left (x \right )-3 x^{2}-9 \ln \left (x \right )-17 x\) | \(32\) |
norman | \(-9 \ln \left (x \right )-3 x^{2}-6 x \ln \left (x \right )-x^{2} \ln \left (x \right )-{\mathrm e}^{-5} \left (-1+17 \,{\mathrm e}^{5}\right ) x\) | \(36\) |
parallelrisch | \({\mathrm e}^{-5} \left (-x^{2} {\mathrm e}^{5} \ln \left (x \right )-3 x^{2} {\mathrm e}^{5}-6 x \,{\mathrm e}^{5} \ln \left (x \right )-17 x \,{\mathrm e}^{5}-9 \,{\mathrm e}^{5} \ln \left (x \right )+x \right )\) | \(42\) |
default | \({\mathrm e}^{-5} \left (-2 \,{\mathrm e}^{5} \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )-6 \,{\mathrm e}^{5} \left (x \ln \left (x \right )-x \right )-\frac {7 x^{2} {\mathrm e}^{5}}{2}-23 x \,{\mathrm e}^{5}-9 \,{\mathrm e}^{5} \ln \left (x \right )+x \right )\) | \(55\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {x+e^5 \left (-9-23 x-7 x^2\right )+e^5 \left (-6 x-2 x^2\right ) \log (x)}{e^5 x} \, dx=-{\left ({\left (x^{2} + 6 \, x + 9\right )} e^{5} \log \left (x\right ) + {\left (3 \, x^{2} + 17 \, x\right )} e^{5} - x\right )} e^{\left (-5\right )} \]
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Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {x+e^5 \left (-9-23 x-7 x^2\right )+e^5 \left (-6 x-2 x^2\right ) \log (x)}{e^5 x} \, dx=\left (- x^{2} - 6 x\right ) \log {\left (x \right )} + \frac {- 3 x^{2} e^{5} - x \left (-1 + 17 e^{5}\right ) - 9 e^{5} \log {\left (x \right )}}{e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {x+e^5 \left (-9-23 x-7 x^2\right )+e^5 \left (-6 x-2 x^2\right ) \log (x)}{e^5 x} \, dx=-\frac {1}{2} \, {\left (7 \, x^{2} e^{5} + {\left (2 \, x^{2} \log \left (x\right ) - x^{2}\right )} e^{5} + 12 \, {\left (x \log \left (x\right ) - x\right )} e^{5} + 46 \, x e^{5} + 18 \, e^{5} \log \left (x\right ) - 2 \, x\right )} e^{\left (-5\right )} \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {x+e^5 \left (-9-23 x-7 x^2\right )+e^5 \left (-6 x-2 x^2\right ) \log (x)}{e^5 x} \, dx=-{\left (x^{2} e^{5} \log \left (x\right ) + 3 \, x^{2} e^{5} + 6 \, x e^{5} \log \left (x\right ) + 17 \, x e^{5} + 9 \, e^{5} \log \left (x\right ) - x\right )} e^{\left (-5\right )} \]
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Time = 10.92 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {x+e^5 \left (-9-23 x-7 x^2\right )+e^5 \left (-6 x-2 x^2\right ) \log (x)}{e^5 x} \, dx=x\,{\mathrm {e}}^{-5}-9\,\ln \left (x\right )-x^2\,\ln \left (x\right )-17\,x-6\,x\,\ln \left (x\right )-3\,x^2 \]
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