Integrand size = 157, antiderivative size = 25 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=e^x-\frac {9 e}{x \left (2 x+\frac {1}{5} (3+x+\log (x))\right )^2} \]
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\[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x x^2 (3+11 x)^3+225 e (5+33 x)+3 \left (75 e+e^x x^2 (3+11 x)^2\right ) \log (x)+3 e^x x^2 (3+11 x) \log ^2(x)+e^x x^2 \log ^3(x)}{x^2 (3+11 x+\log (x))^3} \, dx \\ & = \int \left (e^x+\frac {225 e (5+33 x+\log (x))}{x^2 (3+11 x+\log (x))^3}\right ) \, dx \\ & = (225 e) \int \frac {5+33 x+\log (x)}{x^2 (3+11 x+\log (x))^3} \, dx+\int e^x \, dx \\ & = e^x+(225 e) \int \left (\frac {2 (1+11 x)}{x^2 (3+11 x+\log (x))^3}+\frac {1}{x^2 (3+11 x+\log (x))^2}\right ) \, dx \\ & = e^x+(225 e) \int \frac {1}{x^2 (3+11 x+\log (x))^2} \, dx+(450 e) \int \frac {1+11 x}{x^2 (3+11 x+\log (x))^3} \, dx \\ & = e^x+(225 e) \int \frac {1}{x^2 (3+11 x+\log (x))^2} \, dx+(450 e) \int \left (\frac {1}{x^2 (3+11 x+\log (x))^3}+\frac {11}{x (3+11 x+\log (x))^3}\right ) \, dx \\ & = e^x+(225 e) \int \frac {1}{x^2 (3+11 x+\log (x))^2} \, dx+(450 e) \int \frac {1}{x^2 (3+11 x+\log (x))^3} \, dx+(4950 e) \int \frac {1}{x (3+11 x+\log (x))^3} \, dx \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=e^x-\frac {225 e}{x (3+11 x+\log (x))^2} \]
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Time = 1.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {225 \,{\mathrm e}}{x \left (\ln \left (x \right )+11 x +3\right )^{2}}+{\mathrm e}^{x}\) | \(20\) |
risch | \(-\frac {225 \,{\mathrm e}}{x \left (\ln \left (x \right )+11 x +3\right )^{2}}+{\mathrm e}^{x}\) | \(20\) |
parts | \(-\frac {225 \,{\mathrm e}}{x \left (\ln \left (x \right )+11 x +3\right )^{2}}+{\mathrm e}^{x}\) | \(20\) |
parallelrisch | \(-\frac {225 \,{\mathrm e}-66 \,{\mathrm e}^{x} x^{2}-121 \,{\mathrm e}^{x} x^{3}-9 \,{\mathrm e}^{x} x -x \,{\mathrm e}^{x} \ln \left (x \right )^{2}-6 x \,{\mathrm e}^{x} \ln \left (x \right )-22 x^{2} {\mathrm e}^{x} \ln \left (x \right )}{x \left (121 x^{2}+22 x \ln \left (x \right )+\ln \left (x \right )^{2}+66 x +6 \ln \left (x \right )+9\right )}\) | \(80\) |
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.24 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {x e^{x} \log \left (x\right )^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} e^{x} \log \left (x\right ) + {\left (121 \, x^{3} + 66 \, x^{2} + 9 \, x\right )} e^{x} - 225 \, e}{121 \, x^{3} + x \log \left (x\right )^{2} + 66 \, x^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 9 \, x} \]
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Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=e^{x} - \frac {225 e}{121 x^{3} + 66 x^{2} + x \log {\left (x \right )}^{2} + 9 x + \left (22 x^{2} + 6 x\right ) \log {\left (x \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.08 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {{\left (121 \, x^{3} + x \log \left (x\right )^{2} + 66 \, x^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 9 \, x\right )} e^{x} - 225 \, e}{121 \, x^{3} + x \log \left (x\right )^{2} + 66 \, x^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 9 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {121 \, x^{3} e^{x} + 22 \, x^{2} e^{x} \log \left (x\right ) + x e^{x} \log \left (x\right )^{2} + 66 \, x^{2} e^{x} + 6 \, x e^{x} \log \left (x\right ) + 9 \, x e^{x} - 225 \, e}{121 \, x^{3} + 22 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + 66 \, x^{2} + 6 \, x \log \left (x\right ) + 9 \, x} \]
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Timed out. \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\int \frac {{\mathrm {e}}^x\,\left (1331\,x^5+1089\,x^4+297\,x^3+27\,x^2\right )+\ln \left (x\right )\,\left (225\,\mathrm {e}+{\mathrm {e}}^x\,\left (363\,x^4+198\,x^3+27\,x^2\right )\right )+\mathrm {e}\,\left (7425\,x+1125\right )+{\mathrm {e}}^x\,{\ln \left (x\right )}^2\,\left (33\,x^3+9\,x^2\right )+x^2\,{\mathrm {e}}^x\,{\ln \left (x\right )}^3}{\ln \left (x\right )\,\left (363\,x^4+198\,x^3+27\,x^2\right )+{\ln \left (x\right )}^2\,\left (33\,x^3+9\,x^2\right )+x^2\,{\ln \left (x\right )}^3+27\,x^2+297\,x^3+1089\,x^4+1331\,x^5} \,d x \]
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