Integrand size = 91, antiderivative size = 27 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4 e^{\frac {e^{-x} x^2}{-e^x+\frac {6 x}{5}}} x \]
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\[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=\int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{\left (5 e^x-6 x\right )^2} \, dx \\ & = \int \left (4 e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}}+\frac {40 e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} (-1+x) x^2}{5 e^x-6 x}+\frac {120 e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} (-1+x) x^3}{\left (5 e^x-6 x\right )^2}\right ) \, dx \\ & = 4 \int e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}} \, dx+40 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} (-1+x) x^2}{5 e^x-6 x} \, dx+120 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} (-1+x) x^3}{\left (5 e^x-6 x\right )^2} \, dx \\ & = 4 \int e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}} \, dx+40 \int \left (-\frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^2}{5 e^x-6 x}+\frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^3}{5 e^x-6 x}\right ) \, dx+120 \int \left (-\frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^3}{\left (5 e^x-6 x\right )^2}+\frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^4}{\left (5 e^x-6 x\right )^2}\right ) \, dx \\ & = 4 \int e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}} \, dx-40 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^2}{5 e^x-6 x} \, dx+40 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^3}{5 e^x-6 x} \, dx-120 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^3}{\left (5 e^x-6 x\right )^2} \, dx+120 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^4}{\left (5 e^x-6 x\right )^2} \, dx \\ \end{align*}
Time = 4.56 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4 e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x \]
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Time = 1.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
risch | \(4 x \,{\mathrm e}^{\frac {5 x^{2}}{-5 \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{x} x}}\) | \(24\) |
parallelrisch | \(4 x \,{\mathrm e}^{-\frac {5 x^{2} {\mathrm e}^{-x}}{5 \,{\mathrm e}^{x}-6 x}}\) | \(24\) |
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Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4 \, x e^{\left (\frac {5 \, x^{2}}{6 \, x e^{x} - 5 \, e^{\left (2 \, x\right )}}\right )} \]
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Time = 2.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4 x e^{- \frac {5 x^{2}}{- 6 x e^{x} + 5 e^{2 x}}} \]
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\[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=\int { -\frac {4 \, {\left (30 \, x^{4} - 30 \, x^{3} + 60 \, x e^{\left (2 \, x\right )} - 2 \, {\left (25 \, x^{3} - 7 \, x^{2}\right )} e^{x} - 25 \, e^{\left (3 \, x\right )}\right )} e^{\left (\frac {5 \, x^{2}}{6 \, x e^{x} - 5 \, e^{\left (2 \, x\right )}}\right )}}{36 \, x^{2} e^{x} - 60 \, x e^{\left (2 \, x\right )} + 25 \, e^{\left (3 \, x\right )}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4 \, x e^{\left (-x + \frac {6 \, x^{2} e^{x} + 5 \, x^{2} - 5 \, x e^{\left (2 \, x\right )}}{6 \, x e^{x} - 5 \, e^{\left (2 \, x\right )}}\right )} \]
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Time = 10.49 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4\,x\,{\mathrm {e}}^{-\frac {5\,x^2}{5\,{\mathrm {e}}^{2\,x}-6\,x\,{\mathrm {e}}^x}} \]
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