Integrand size = 75, antiderivative size = 29 \[ \int \frac {3+2 x+3 x^2+e \left (-6+2 x-32 x^2+6 x^3\right )+e^2 \left (3-4 x+32 x^2-24 x^3+3 x^4\right )}{1+e (-2+2 x)+e^2 \left (1-2 x+x^2\right )} \, dx=x \left (3+x+x^2 \left (1+\frac {10 x}{x-\frac {x}{e}-x^2}\right )\right ) \]
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Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2011, 27, 1864} \[ \int \frac {3+2 x+3 x^2+e \left (-6+2 x-32 x^2+6 x^3\right )+e^2 \left (3-4 x+32 x^2-24 x^3+3 x^4\right )}{1+e (-2+2 x)+e^2 \left (1-2 x+x^2\right )} \, dx=x^3-9 x^2+\frac {(10-7 e) x}{e}+\frac {10 (1-e)^3}{e^2 (e x-e+1)} \]
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Rule 27
Rule 1864
Rule 2011
Rubi steps \begin{align*} \text {integral}& = \int \frac {3+2 x+3 x^2+e \left (-6+2 x-32 x^2+6 x^3\right )+e^2 \left (3-4 x+32 x^2-24 x^3+3 x^4\right )}{(-1+e)^2+2 (1-e) e x+e^2 x^2} \, dx \\ & = \int \frac {3+2 x+3 x^2+e \left (-6+2 x-32 x^2+6 x^3\right )+e^2 \left (3-4 x+32 x^2-24 x^3+3 x^4\right )}{(1-e+e x)^2} \, dx \\ & = \int \left (\frac {10-7 e}{e}-18 x+3 x^2+\frac {10 (-1+e)^3}{e (1-e+e x)^2}\right ) \, dx \\ & = \frac {(10-7 e) x}{e}-9 x^2+x^3+\frac {10 (1-e)^3}{e^2 (1-e+e x)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {3+2 x+3 x^2+e \left (-6+2 x-32 x^2+6 x^3\right )+e^2 \left (3-4 x+32 x^2-24 x^3+3 x^4\right )}{1+e (-2+2 x)+e^2 \left (1-2 x+x^2\right )} \, dx=-\frac {10 (-1+e)^3}{e^2 (1+e (-1+x))}+\frac {(10-7 e) x}{e}-9 x^2+x^3 \]
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Time = 0.60 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97
method | result | size |
norman | \(\frac {x^{4} {\mathrm e}+\left (-10 \,{\mathrm e}+1\right ) x^{3}+\left (2 \,{\mathrm e}+1\right ) x^{2}-3 \left ({\mathrm e}^{2}-2 \,{\mathrm e}+1\right ) {\mathrm e}^{-1}}{x \,{\mathrm e}-{\mathrm e}+1}\) | \(57\) |
gosper | \(\frac {\left (x^{4} {\mathrm e}^{2}-10 x^{3} {\mathrm e}^{2}+2 x^{2} {\mathrm e}^{2}+x^{3} {\mathrm e}+x^{2} {\mathrm e}-3 \,{\mathrm e}^{2}+6 \,{\mathrm e}-3\right ) {\mathrm e}^{-1}}{x \,{\mathrm e}-{\mathrm e}+1}\) | \(68\) |
parallelrisch | \(\frac {\left (x^{4} {\mathrm e}^{2}-10 x^{3} {\mathrm e}^{2}+2 x^{2} {\mathrm e}^{2}+x^{3} {\mathrm e}+x^{2} {\mathrm e}-3 \,{\mathrm e}^{2}+6 \,{\mathrm e}-3\right ) {\mathrm e}^{-1}}{x \,{\mathrm e}-{\mathrm e}+1}\) | \(68\) |
risch | \({\mathrm e} \,{\mathrm e}^{-1} x^{3}-9 \,{\mathrm e} \,{\mathrm e}^{-1} x^{2}-7 \,{\mathrm e} \,{\mathrm e}^{-1} x +10 \,{\mathrm e}^{-1} x -\frac {10 \,{\mathrm e}^{-2} {\mathrm e}^{3}}{x \,{\mathrm e}-{\mathrm e}+1}+\frac {30 \,{\mathrm e}^{-2} {\mathrm e}^{2}}{x \,{\mathrm e}-{\mathrm e}+1}-\frac {30 \,{\mathrm e}^{-2} {\mathrm e}}{x \,{\mathrm e}-{\mathrm e}+1}+\frac {10 \,{\mathrm e}^{-2}}{x \,{\mathrm e}-{\mathrm e}+1}\) | \(101\) |
meijerg | \(-3 \,{\mathrm e}^{-3} \left ({\mathrm e}-1\right )^{3} \left (-\frac {x \,{\mathrm e} \left (-\frac {5 x^{3} {\mathrm e}^{3}}{\left ({\mathrm e}-1\right )^{3}}-\frac {10 x^{2} {\mathrm e}^{2}}{\left ({\mathrm e}-1\right )^{2}}-\frac {30 x \,{\mathrm e}}{{\mathrm e}-1}+60\right )}{15 \left ({\mathrm e}-1\right ) \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )}-4 \ln \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )\right )+{\mathrm e}^{-4} \left ({\mathrm e}-1\right )^{2} \left (-24 \,{\mathrm e}^{2}+6 \,{\mathrm e}\right ) \left (\frac {x \,{\mathrm e} \left (-\frac {2 x^{2} {\mathrm e}^{2}}{\left ({\mathrm e}-1\right )^{2}}-\frac {6 x \,{\mathrm e}}{{\mathrm e}-1}+12\right )}{4 \left ({\mathrm e}-1\right ) \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )}+3 \ln \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )\right )-{\mathrm e}^{-3} \left (32 \,{\mathrm e}^{2}-32 \,{\mathrm e}+3\right ) \left ({\mathrm e}-1\right ) \left (-\frac {x \,{\mathrm e} \left (-\frac {3 x \,{\mathrm e}}{{\mathrm e}-1}+6\right )}{3 \left ({\mathrm e}-1\right ) \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )}-2 \ln \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )\right )+{\mathrm e}^{-2} \left (-4 \,{\mathrm e}^{2}+2 \,{\mathrm e}+2\right ) \left (\frac {x \,{\mathrm e}}{\left ({\mathrm e}-1\right ) \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )}+\ln \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )\right )+\frac {3 \,{\mathrm e}^{2} x}{\left ({\mathrm e}-1\right )^{2} \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )}-\frac {6 \,{\mathrm e} x}{\left ({\mathrm e}-1\right )^{2} \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )}+\frac {3 x}{\left ({\mathrm e}-1\right )^{2} \left (1-\frac {x \,{\mathrm e}}{{\mathrm e}-1}\right )}\) | \(384\) |
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Time = 0.34 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {3+2 x+3 x^2+e \left (-6+2 x-32 x^2+6 x^3\right )+e^2 \left (3-4 x+32 x^2-24 x^3+3 x^4\right )}{1+e (-2+2 x)+e^2 \left (1-2 x+x^2\right )} \, dx=\frac {{\left (x^{4} - 10 \, x^{3} + 2 \, x^{2} + 7 \, x - 10\right )} e^{3} + {\left (x^{3} + x^{2} - 17 \, x + 30\right )} e^{2} + 10 \, {\left (x - 3\right )} e + 10}{{\left (x - 1\right )} e^{3} + e^{2}} \]
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Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {3+2 x+3 x^2+e \left (-6+2 x-32 x^2+6 x^3\right )+e^2 \left (3-4 x+32 x^2-24 x^3+3 x^4\right )}{1+e (-2+2 x)+e^2 \left (1-2 x+x^2\right )} \, dx=x^{3} - 9 x^{2} + x \left (-7 + \frac {10}{e}\right ) + \frac {- 10 e^{3} - 30 e + 10 + 30 e^{2}}{x e^{3} - e^{3} + e^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {3+2 x+3 x^2+e \left (-6+2 x-32 x^2+6 x^3\right )+e^2 \left (3-4 x+32 x^2-24 x^3+3 x^4\right )}{1+e (-2+2 x)+e^2 \left (1-2 x+x^2\right )} \, dx={\left (x^{3} e - 9 \, x^{2} e - x {\left (7 \, e - 10\right )}\right )} e^{\left (-1\right )} - \frac {10 \, {\left (e^{3} - 3 \, e^{2} + 3 \, e - 1\right )}}{x e^{3} - e^{3} + e^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {3+2 x+3 x^2+e \left (-6+2 x-32 x^2+6 x^3\right )+e^2 \left (3-4 x+32 x^2-24 x^3+3 x^4\right )}{1+e (-2+2 x)+e^2 \left (1-2 x+x^2\right )} \, dx={\left (x^{3} e^{6} - 9 \, x^{2} e^{6} - 7 \, x e^{6} + 10 \, x e^{5}\right )} e^{\left (-6\right )} - \frac {10 \, {\left (e^{3} - 3 \, e^{2} + 3 \, e - 1\right )} e^{\left (-2\right )}}{x e - e + 1} \]
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Time = 13.73 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.76 \[ \int \frac {3+2 x+3 x^2+e \left (-6+2 x-32 x^2+6 x^3\right )+e^2 \left (3-4 x+32 x^2-24 x^3+3 x^4\right )}{1+e (-2+2 x)+e^2 \left (1-2 x+x^2\right )} \, dx=x^2\,\left (3\,{\mathrm {e}}^{-1}\,\left (\mathrm {e}-1\right )-3\,{\mathrm {e}}^{-1}\,\left (4\,\mathrm {e}-1\right )\right )+x\,\left ({\mathrm {e}}^{-2}\,\left (32\,{\mathrm {e}}^2-32\,\mathrm {e}+3\right )-3\,{\mathrm {e}}^{-2}\,{\left (\mathrm {e}-1\right )}^2+2\,{\mathrm {e}}^{-1}\,\left (\mathrm {e}-1\right )\,\left (6\,{\mathrm {e}}^{-1}\,\left (\mathrm {e}-1\right )-6\,{\mathrm {e}}^{-1}\,\left (4\,\mathrm {e}-1\right )\right )\right )+x^3-\frac {10\,{\mathrm {e}}^{-1}\,\left (3\,\mathrm {e}-3\,{\mathrm {e}}^2+{\mathrm {e}}^3-1\right )}{\mathrm {e}-{\mathrm {e}}^2+x\,{\mathrm {e}}^2} \]
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