Integrand size = 66, antiderivative size = 30 \[ \int \frac {3+2 e^{5+x} x^2-4 e^{2+2 x} x^2}{-3 x-2 x^2-8 e^2 x^2+2 e^{5+x} x^2-2 e^{2+2 x} x^2} \, dx=\log \left (4 \left (1+e^2 \left (4+e^{2 x}-e^{3+x}\right )+\frac {3}{2 x}\right )\right ) \]
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\[ \int \frac {3+2 e^{5+x} x^2-4 e^{2+2 x} x^2}{-3 x-2 x^2-8 e^2 x^2+2 e^{5+x} x^2-2 e^{2+2 x} x^2} \, dx=\int \frac {3+2 e^{5+x} x^2-4 e^{2+2 x} x^2}{-3 x-2 x^2-8 e^2 x^2+2 e^{5+x} x^2-2 e^{2+2 x} x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {3+2 e^{5+x} x^2-4 e^{2+2 x} x^2}{-3 x+2 e^{5+x} x^2-2 e^{2+2 x} x^2+\left (-2-8 e^2\right ) x^2} \, dx \\ & = \int \left (2+\frac {-3-6 x+2 e^{5+x} x^2-4 \left (1+4 e^2\right ) x^2}{x \left (3-2 e^{5+x} x+2 e^{2+2 x} x+2 \left (1+4 e^2\right ) x\right )}\right ) \, dx \\ & = 2 x+\int \frac {-3-6 x+2 e^{5+x} x^2-4 \left (1+4 e^2\right ) x^2}{x \left (3-2 e^{5+x} x+2 e^{2+2 x} x+2 \left (1+4 e^2\right ) x\right )} \, dx \\ & = 2 x+\int \left (\frac {6}{-3+2 e^{5+x} x-2 e^{2+2 x} x-2 \left (1+4 e^2\right ) x}+\frac {3}{x \left (-3+2 e^{5+x} x-2 e^{2+2 x} x-2 \left (1+4 e^2\right ) x\right )}+\frac {2 e^{5+x} x}{3-2 e^{5+x} x+2 e^{2+2 x} x+2 \left (1+4 e^2\right ) x}+\frac {4 \left (-1-4 e^2\right ) x}{3-2 e^{5+x} x+2 e^{2+2 x} x+2 \left (1+4 e^2\right ) x}\right ) \, dx \\ & = 2 x+2 \int \frac {e^{5+x} x}{3-2 e^{5+x} x+2 e^{2+2 x} x+2 \left (1+4 e^2\right ) x} \, dx+3 \int \frac {1}{x \left (-3+2 e^{5+x} x-2 e^{2+2 x} x-2 \left (1+4 e^2\right ) x\right )} \, dx+6 \int \frac {1}{-3+2 e^{5+x} x-2 e^{2+2 x} x-2 \left (1+4 e^2\right ) x} \, dx-\left (4 \left (1+4 e^2\right )\right ) \int \frac {x}{3-2 e^{5+x} x+2 e^{2+2 x} x+2 \left (1+4 e^2\right ) x} \, dx \\ \end{align*}
Time = 2.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {3+2 e^{5+x} x^2-4 e^{2+2 x} x^2}{-3 x-2 x^2-8 e^2 x^2+2 e^{5+x} x^2-2 e^{2+2 x} x^2} \, dx=-\log (x)+\log \left (3+2 x+8 e^2 x-2 e^{5+x} x+2 e^{2+2 x} x\right ) \]
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Time = 0.40 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\ln \left ({\mathrm e}^{2 x}-{\mathrm e}^{3+x}+\frac {\left (8 \,{\mathrm e}^{2} x +2 x +3\right ) {\mathrm e}^{-2}}{2 x}\right )\) | \(30\) |
norman | \(-\ln \left (x \right )+\ln \left (2 \,{\mathrm e}^{2} {\mathrm e}^{x} {\mathrm e}^{3} x -2 x \,{\mathrm e}^{2} {\mathrm e}^{2 x}-8 \,{\mathrm e}^{2} x -2 x -3\right )\) | \(35\) |
parallelrisch | \(-\ln \left (x \right )+\ln \left (\frac {\left (2 x \,{\mathrm e}^{2} {\mathrm e}^{2 x}-2 x \,{\mathrm e}^{2} {\mathrm e}^{3+x}+8 \,{\mathrm e}^{2} x +2 x +3\right ) {\mathrm e}^{-2}}{2}\right )\) | \(41\) |
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {3+2 e^{5+x} x^2-4 e^{2+2 x} x^2}{-3 x-2 x^2-8 e^2 x^2+2 e^{5+x} x^2-2 e^{2+2 x} x^2} \, dx=\log \left (\frac {8 \, x e^{10} + {\left (2 \, x + 3\right )} e^{8} + 2 \, x e^{\left (2 \, x + 10\right )} - 2 \, x e^{\left (x + 13\right )}}{x}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {3+2 e^{5+x} x^2-4 e^{2+2 x} x^2}{-3 x-2 x^2-8 e^2 x^2+2 e^{5+x} x^2-2 e^{2+2 x} x^2} \, dx=\log {\left (e^{2 x} - e^{3} \sqrt {e^{2 x}} + \frac {2 x + 8 x e^{2} + 3}{2 x e^{2}} \right )} \]
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Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {3+2 e^{5+x} x^2-4 e^{2+2 x} x^2}{-3 x-2 x^2-8 e^2 x^2+2 e^{5+x} x^2-2 e^{2+2 x} x^2} \, dx=\log \left (\frac {{\left (2 \, x {\left (4 \, e^{2} + 1\right )} + 2 \, x e^{\left (2 \, x + 2\right )} - 2 \, x e^{\left (x + 5\right )} + 3\right )} e^{\left (-2\right )}}{2 \, x}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).
Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {3+2 e^{5+x} x^2-4 e^{2+2 x} x^2}{-3 x-2 x^2-8 e^2 x^2+2 e^{5+x} x^2-2 e^{2+2 x} x^2} \, dx=\log \left (8 \, {\left (x + 5\right )} e^{10} + 2 \, {\left (x + 5\right )} e^{8} + 2 \, {\left (x + 5\right )} e^{\left (2 \, x + 10\right )} - 2 \, {\left (x + 5\right )} e^{\left (x + 13\right )} - 40 \, e^{10} - 7 \, e^{8} - 10 \, e^{\left (2 \, x + 10\right )} + 10 \, e^{\left (x + 13\right )}\right ) - \log \left (x\right ) \]
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Time = 0.42 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {3+2 e^{5+x} x^2-4 e^{2+2 x} x^2}{-3 x-2 x^2-8 e^2 x^2+2 e^{5+x} x^2-2 e^{2+2 x} x^2} \, dx=\ln \left (3\,{\mathrm {e}}^8+2\,x\,{\mathrm {e}}^8+8\,x\,{\mathrm {e}}^{10}-2\,x\,{\mathrm {e}}^{13}\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{10}\right )-\ln \left (x\right ) \]
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