Integrand size = 136, antiderivative size = 31 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=2 \left (-4+\frac {x}{1+\frac {x+\frac {4}{5 \left (\frac {2 e^x}{5}+x\right )^2}}{x}}\right ) \]
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\[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (16 e^{4 x} x+160 e^{3 x} x^2+125 x^2 \left (8+5 x^3\right )+200 e^x x \left (3+x+5 x^3\right )+40 e^{2 x} \left (2+2 x+15 x^3\right )\right )}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx \\ & = \int \left (1+\frac {20 x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3}-\frac {100 \left (1+2 x-2 e^x x^2-5 x^3+2 e^x x^3+5 x^4\right )}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}\right ) \, dx \\ & = x+20 \int \frac {x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3} \, dx-100 \int \frac {1+2 x-2 e^x x^2-5 x^3+2 e^x x^3+5 x^4}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx \\ & = x+20 \int \frac {x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3} \, dx-100 \int \left (\frac {1}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}+\frac {2 x}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}-\frac {2 e^x x^2}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}-\frac {5 x^3}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}+\frac {2 e^x x^3}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}+\frac {5 x^4}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2}\right ) \, dx \\ & = x+20 \int \frac {x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3} \, dx-100 \int \frac {1}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx-200 \int \frac {x}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx+200 \int \frac {e^x x^2}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx-200 \int \frac {e^x x^3}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx+500 \int \frac {x^3}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx-500 \int \frac {x^4}{\left (10+4 e^{2 x} x+20 e^x x^2+25 x^3\right )^2} \, dx \\ \end{align*}
Time = 8.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=x-\frac {10 x}{10+4 e^{2 x} x+20 e^x x^2+25 x^3} \]
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Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
method | result | size |
risch | \(x -\frac {10 x}{4 x \,{\mathrm e}^{2 x}+20 \,{\mathrm e}^{x} x^{2}+25 x^{3}+10}\) | \(29\) |
parallelrisch | \(\frac {100 x^{4}+80 \,{\mathrm e}^{x} x^{3}+16 \,{\mathrm e}^{2 x} x^{2}}{16 x \,{\mathrm e}^{2 x}+80 \,{\mathrm e}^{x} x^{2}+100 x^{3}+40}\) | \(48\) |
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Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )}}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=x - \frac {10 x}{25 x^{3} + 20 x^{2} e^{x} + 4 x e^{2 x} + 10} \]
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Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )}}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \]
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Time = 0.41 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\frac {25 \, x^{4} + 20 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (2 \, x\right )} - 10 \, x}{25 \, x^{3} + 20 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )} + 10} \]
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Timed out. \[ \int \frac {16 e^{4 x} x^2+1000 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+80 x^2+600 x^4\right )+e^x \left (600 x^2+200 x^3+1000 x^5\right )}{100+16 e^{4 x} x^2+500 x^3+160 e^{3 x} x^3+625 x^6+e^{2 x} \left (80 x+600 x^4\right )+e^x \left (400 x^2+1000 x^5\right )} \, dx=\int \frac {{\mathrm {e}}^{2\,x}\,\left (600\,x^4+80\,x^2+80\,x\right )+{\mathrm {e}}^x\,\left (1000\,x^5+200\,x^3+600\,x^2\right )+16\,x^2\,{\mathrm {e}}^{4\,x}+160\,x^3\,{\mathrm {e}}^{3\,x}+1000\,x^3+625\,x^6}{{\mathrm {e}}^{2\,x}\,\left (600\,x^4+80\,x\right )+{\mathrm {e}}^x\,\left (1000\,x^5+400\,x^2\right )+16\,x^2\,{\mathrm {e}}^{4\,x}+160\,x^3\,{\mathrm {e}}^{3\,x}+500\,x^3+625\,x^6+100} \,d x \]
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