Integrand size = 84, antiderivative size = 27 \[ \int \frac {e^{2 x} \left (-200 x^7+50 e^4 x^7-25 x^8\right )+e^x \left (-120 x^2-80 x^3-10 x^4+e^4 \left (30 x^2+10 x^3\right )\right )}{4-20 e^x x^5+25 e^{2 x} x^{10}} \, dx=\frac {4-e^4+x}{-\frac {2 e^{-x}}{5 x^3}+x^2} \]
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\[ \int \frac {e^{2 x} \left (-200 x^7+50 e^4 x^7-25 x^8\right )+e^x \left (-120 x^2-80 x^3-10 x^4+e^4 \left (30 x^2+10 x^3\right )\right )}{4-20 e^x x^5+25 e^{2 x} x^{10}} \, dx=\int \frac {e^{2 x} \left (-200 x^7+50 e^4 x^7-25 x^8\right )+e^x \left (-120 x^2-80 x^3-10 x^4+e^4 \left (30 x^2+10 x^3\right )\right )}{4-20 e^x x^5+25 e^{2 x} x^{10}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {5 e^x x^2 \left (10 e^{4+x} x^5+2 e^4 (3+x)-5 e^x x^5 (8+x)-2 \left (12+8 x+x^2\right )\right )}{\left (2-5 e^x x^5\right )^2} \, dx \\ & = 5 \int \frac {e^x x^2 \left (10 e^{4+x} x^5+2 e^4 (3+x)-5 e^x x^5 (8+x)-2 \left (12+8 x+x^2\right )\right )}{\left (2-5 e^x x^5\right )^2} \, dx \\ & = 5 \int \left (-\frac {2 e^x x^2 (5+x) \left (4-e^4+x\right )}{\left (-2+5 e^x x^5\right )^2}-\frac {e^x x^2 \left (8-2 e^4+x\right )}{-2+5 e^x x^5}\right ) \, dx \\ & = -\left (5 \int \frac {e^x x^2 \left (8-2 e^4+x\right )}{-2+5 e^x x^5} \, dx\right )-10 \int \frac {e^x x^2 (5+x) \left (4-e^4+x\right )}{\left (-2+5 e^x x^5\right )^2} \, dx \\ & = -\left (5 \int \left (-\frac {2 e^x \left (-4+e^4\right ) x^2}{-2+5 e^x x^5}+\frac {e^x x^3}{-2+5 e^x x^5}\right ) \, dx\right )-10 \int \left (-\frac {5 e^x \left (-4+e^4\right ) x^2}{\left (-2+5 e^x x^5\right )^2}-\frac {e^x \left (-9+e^4\right ) x^3}{\left (-2+5 e^x x^5\right )^2}+\frac {e^x x^4}{\left (-2+5 e^x x^5\right )^2}\right ) \, dx \\ & = -\left (5 \int \frac {e^x x^3}{-2+5 e^x x^5} \, dx\right )-10 \int \frac {e^x x^4}{\left (-2+5 e^x x^5\right )^2} \, dx-\left (10 \left (4-e^4\right )\right ) \int \frac {e^x x^2}{-2+5 e^x x^5} \, dx-\left (50 \left (4-e^4\right )\right ) \int \frac {e^x x^2}{\left (-2+5 e^x x^5\right )^2} \, dx-\left (10 \left (9-e^4\right )\right ) \int \frac {e^x x^3}{\left (-2+5 e^x x^5\right )^2} \, dx \\ \end{align*}
Time = 9.67 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {e^{2 x} \left (-200 x^7+50 e^4 x^7-25 x^8\right )+e^x \left (-120 x^2-80 x^3-10 x^4+e^4 \left (30 x^2+10 x^3\right )\right )}{4-20 e^x x^5+25 e^{2 x} x^{10}} \, dx=-\frac {5 e^x \left (-4+e^4-x\right ) x^3}{-2+5 e^x x^5} \]
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Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22
method | result | size |
norman | \(\frac {\left (20-5 \,{\mathrm e}^{4}\right ) x^{3} {\mathrm e}^{x}+5 \,{\mathrm e}^{x} x^{4}}{5 x^{5} {\mathrm e}^{x}-2}\) | \(33\) |
risch | \(\frac {4-{\mathrm e}^{4}+x}{x^{2}}-\frac {2 \left ({\mathrm e}^{4}-x -4\right )}{x^{2} \left (5 x^{5} {\mathrm e}^{x}-2\right )}\) | \(36\) |
parallelrisch | \(-\frac {25 \,{\mathrm e}^{x} x^{3} {\mathrm e}^{4}-25 \,{\mathrm e}^{x} x^{4}-100 \,{\mathrm e}^{x} x^{3}}{5 \left (5 x^{5} {\mathrm e}^{x}-2\right )}\) | \(38\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{2 x} \left (-200 x^7+50 e^4 x^7-25 x^8\right )+e^x \left (-120 x^2-80 x^3-10 x^4+e^4 \left (30 x^2+10 x^3\right )\right )}{4-20 e^x x^5+25 e^{2 x} x^{10}} \, dx=\frac {5 \, {\left (x^{4} - x^{3} e^{4} + 4 \, x^{3}\right )} e^{x}}{5 \, x^{5} e^{x} - 2} \]
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Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {e^{2 x} \left (-200 x^7+50 e^4 x^7-25 x^8\right )+e^x \left (-120 x^2-80 x^3-10 x^4+e^4 \left (30 x^2+10 x^3\right )\right )}{4-20 e^x x^5+25 e^{2 x} x^{10}} \, dx=\frac {2 x - 2 e^{4} + 8}{5 x^{7} e^{x} - 2 x^{2}} - \frac {- x - 4 + e^{4}}{x^{2}} \]
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {e^{2 x} \left (-200 x^7+50 e^4 x^7-25 x^8\right )+e^x \left (-120 x^2-80 x^3-10 x^4+e^4 \left (30 x^2+10 x^3\right )\right )}{4-20 e^x x^5+25 e^{2 x} x^{10}} \, dx=\frac {5 \, {\left (x^{4} - x^{3} {\left (e^{4} - 4\right )}\right )} e^{x}}{5 \, x^{5} e^{x} - 2} \]
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{2 x} \left (-200 x^7+50 e^4 x^7-25 x^8\right )+e^x \left (-120 x^2-80 x^3-10 x^4+e^4 \left (30 x^2+10 x^3\right )\right )}{4-20 e^x x^5+25 e^{2 x} x^{10}} \, dx=\frac {5 \, {\left (x^{4} e^{x} - x^{3} e^{\left (x + 4\right )} + 4 \, x^{3} e^{x}\right )}}{5 \, x^{5} e^{x} - 2} \]
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Time = 8.91 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{2 x} \left (-200 x^7+50 e^4 x^7-25 x^8\right )+e^x \left (-120 x^2-80 x^3-10 x^4+e^4 \left (30 x^2+10 x^3\right )\right )}{4-20 e^x x^5+25 e^{2 x} x^{10}} \, dx=\frac {5\,x^3\,{\mathrm {e}}^x\,\left (x-{\mathrm {e}}^4+4\right )}{5\,x^5\,{\mathrm {e}}^x-2} \]
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