Integrand size = 84, antiderivative size = 24 \[ \int \frac {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} \left (8000 x-1000 x^2\right )+e^{8+x} \left (1875 x^2-250 x^3\right )}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx=\frac {125 e^8 x}{\left (2+\frac {1}{2+\frac {e^{-x} x}{2}}\right )^2} \]
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\[ \int \frac {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} \left (8000 x-1000 x^2\right )+e^{8+x} \left (1875 x^2-250 x^3\right )}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx=\int \frac {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} \left (8000 x-1000 x^2\right )+e^{8+x} \left (1875 x^2-250 x^3\right )}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {125 e^8 \left (80 e^{3 x}-8 e^{2 x} (-8+x) x+x^3-e^x x^2 (-15+2 x)\right )}{4 \left (5 e^x+x\right )^3} \, dx \\ & = \frac {1}{4} \left (125 e^8\right ) \int \frac {80 e^{3 x}-8 e^{2 x} (-8+x) x+x^3-e^x x^2 (-15+2 x)}{\left (5 e^x+x\right )^3} \, dx \\ & = \frac {1}{4} \left (125 e^8\right ) \int \left (\frac {16}{25}+\frac {2 (-1+x) x^3}{25 \left (5 e^x+x\right )^3}-\frac {8 (-2+x) x}{25 \left (5 e^x+x\right )}+\frac {x^2 (-5+6 x)}{25 \left (5 e^x+x\right )^2}\right ) \, dx \\ & = 20 e^8 x+\frac {1}{4} \left (5 e^8\right ) \int \frac {x^2 (-5+6 x)}{\left (5 e^x+x\right )^2} \, dx+\frac {1}{2} \left (5 e^8\right ) \int \frac {(-1+x) x^3}{\left (5 e^x+x\right )^3} \, dx-\left (10 e^8\right ) \int \frac {(-2+x) x}{5 e^x+x} \, dx \\ & = 20 e^8 x+\frac {1}{4} \left (5 e^8\right ) \int \left (-\frac {5 x^2}{\left (5 e^x+x\right )^2}+\frac {6 x^3}{\left (5 e^x+x\right )^2}\right ) \, dx+\frac {1}{2} \left (5 e^8\right ) \int \left (-\frac {x^3}{\left (5 e^x+x\right )^3}+\frac {x^4}{\left (5 e^x+x\right )^3}\right ) \, dx-\left (10 e^8\right ) \int \left (-\frac {2 x}{5 e^x+x}+\frac {x^2}{5 e^x+x}\right ) \, dx \\ & = 20 e^8 x-\frac {1}{2} \left (5 e^8\right ) \int \frac {x^3}{\left (5 e^x+x\right )^3} \, dx+\frac {1}{2} \left (5 e^8\right ) \int \frac {x^4}{\left (5 e^x+x\right )^3} \, dx-\frac {1}{4} \left (25 e^8\right ) \int \frac {x^2}{\left (5 e^x+x\right )^2} \, dx+\frac {1}{2} \left (15 e^8\right ) \int \frac {x^3}{\left (5 e^x+x\right )^2} \, dx-\left (10 e^8\right ) \int \frac {x^2}{5 e^x+x} \, dx+\left (20 e^8\right ) \int \frac {x}{5 e^x+x} \, dx \\ \end{align*}
Time = 2.60 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} \left (8000 x-1000 x^2\right )+e^{8+x} \left (1875 x^2-250 x^3\right )}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx=\frac {125 e^8 x \left (4 e^x+x\right )^2}{4 \left (5 e^x+x\right )^2} \]
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Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25
method | result | size |
risch | \(20 x \,{\mathrm e}^{8}+\frac {5 \left (9 x +40 \,{\mathrm e}^{x}\right ) x^{2} {\mathrm e}^{8}}{4 \left (5 \,{\mathrm e}^{x}+x \right )^{2}}\) | \(30\) |
norman | \(\frac {\frac {125 \,{\mathrm e}^{8} x^{3}}{4}+500 x \,{\mathrm e}^{8} {\mathrm e}^{2 x}+250 x^{2} {\mathrm e}^{8} {\mathrm e}^{x}}{\left (5 \,{\mathrm e}^{x}+x \right )^{2}}\) | \(42\) |
parallelrisch | \(\frac {3125 \,{\mathrm e}^{8} x^{3}+25000 x^{2} {\mathrm e}^{8} {\mathrm e}^{x}+50000 x \,{\mathrm e}^{8} {\mathrm e}^{2 x}}{100 x^{2}+1000 \,{\mathrm e}^{x} x +2500 \,{\mathrm e}^{2 x}}\) | \(52\) |
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} \left (8000 x-1000 x^2\right )+e^{8+x} \left (1875 x^2-250 x^3\right )}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx=\frac {125 \, {\left (x^{3} e^{24} + 8 \, x^{2} e^{\left (x + 24\right )} + 16 \, x e^{\left (2 \, x + 24\right )}\right )}}{4 \, {\left (x^{2} e^{16} + 10 \, x e^{\left (x + 16\right )} + 25 \, e^{\left (2 \, x + 16\right )}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} \left (8000 x-1000 x^2\right )+e^{8+x} \left (1875 x^2-250 x^3\right )}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx=20 x e^{8} + \frac {45 x^{3} e^{8} + 200 x^{2} e^{8} e^{x}}{4 x^{2} + 40 x e^{x} + 100 e^{2 x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} \left (8000 x-1000 x^2\right )+e^{8+x} \left (1875 x^2-250 x^3\right )}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx=\frac {125 \, {\left (x^{3} e^{8} + 8 \, x^{2} e^{\left (x + 8\right )} + 16 \, x e^{\left (2 \, x + 8\right )}\right )}}{4 \, {\left (x^{2} + 10 \, x e^{x} + 25 \, e^{\left (2 \, x\right )}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.75 \[ \int \frac {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} \left (8000 x-1000 x^2\right )+e^{8+x} \left (1875 x^2-250 x^3\right )}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx=\frac {5 \, {\left (25 \, {\left (x + 8\right )}^{3} e^{24} - 472 \, {\left (x + 8\right )}^{2} e^{24} + 200 \, {\left (x + 8\right )}^{2} e^{\left (x + 24\right )} + 2752 \, {\left (x + 8\right )} e^{24} + 400 \, {\left (x + 8\right )} e^{\left (2 \, x + 24\right )} - 1920 \, {\left (x + 8\right )} e^{\left (x + 24\right )} - 4608 \, e^{24} + 2560 \, e^{\left (x + 24\right )}\right )}}{4 \, {\left ({\left (x + 8\right )}^{2} e^{16} - 16 \, {\left (x + 8\right )} e^{16} + 10 \, {\left (x + 8\right )} e^{\left (x + 16\right )} + 64 \, e^{16} + 25 \, e^{\left (2 \, x + 16\right )} - 80 \, e^{\left (x + 16\right )}\right )}} \]
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Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {10000 e^{8+3 x}+125 e^8 x^3+e^{8+2 x} \left (8000 x-1000 x^2\right )+e^{8+x} \left (1875 x^2-250 x^3\right )}{500 e^{3 x}+300 e^{2 x} x+60 e^x x^2+4 x^3} \, dx=\frac {125\,x\,\left (16\,{\mathrm {e}}^{2\,x+8}+8\,x\,{\mathrm {e}}^{x+8}+x^2\,{\mathrm {e}}^8\right )}{4\,\left (25\,{\mathrm {e}}^{2\,x}+10\,x\,{\mathrm {e}}^x+x^2\right )} \]
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