Integrand size = 87, antiderivative size = 34 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=-1+x+\frac {e^x x}{\left (-x+25 x^2\right ) \left (-1+\frac {x^2}{2-x}\right )} \]
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Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {6820, 6874, 2208, 2209} \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=x+\frac {1225 e^x}{1224 (1-25 x)}-\frac {e^x}{72 (1-x)}+\frac {4 e^x}{153 (x+2)} \]
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Rule 2208
Rule 2209
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{\left (2-51 x+24 x^2+25 x^3\right )^2}\right ) \, dx \\ & = x+\int \frac {e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{\left (2-51 x+24 x^2+25 x^3\right )^2} \, dx \\ & = x+\int \left (-\frac {e^x}{72 (-1+x)^2}+\frac {e^x}{72 (-1+x)}-\frac {4 e^x}{153 (2+x)^2}+\frac {4 e^x}{153 (2+x)}+\frac {30625 e^x}{1224 (-1+25 x)^2}-\frac {1225 e^x}{1224 (-1+25 x)}\right ) \, dx \\ & = x-\frac {1}{72} \int \frac {e^x}{(-1+x)^2} \, dx+\frac {1}{72} \int \frac {e^x}{-1+x} \, dx-\frac {4}{153} \int \frac {e^x}{(2+x)^2} \, dx+\frac {4}{153} \int \frac {e^x}{2+x} \, dx-\frac {1225 \int \frac {e^x}{-1+25 x} \, dx}{1224}+\frac {30625 \int \frac {e^x}{(-1+25 x)^2} \, dx}{1224} \\ & = \frac {1225 e^x}{1224 (1-25 x)}-\frac {e^x}{72 (1-x)}+x+\frac {4 e^x}{153 (2+x)}+\frac {1}{72} e \operatorname {ExpIntegralEi}(-1+x)+\frac {4 \operatorname {ExpIntegralEi}(2+x)}{153 e^2}-\frac {49 \sqrt [25]{e} \operatorname {ExpIntegralEi}\left (\frac {1}{25} (-1+25 x)\right )}{1224}-\frac {1}{72} \int \frac {e^x}{-1+x} \, dx-\frac {4}{153} \int \frac {e^x}{2+x} \, dx+\frac {1225 \int \frac {e^x}{-1+25 x} \, dx}{1224} \\ & = \frac {1225 e^x}{1224 (1-25 x)}-\frac {e^x}{72 (1-x)}+x+\frac {4 e^x}{153 (2+x)} \\ \end{align*}
Time = 1.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=x+\frac {e^x (2-x)}{2-51 x+24 x^2+25 x^3} \]
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Time = 0.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79
method | result | size |
risch | \(x -\frac {\left (-2+x \right ) {\mathrm e}^{x}}{25 x^{3}+24 x^{2}-51 x +2}\) | \(27\) |
default | \(x +\frac {4 \,{\mathrm e}^{x}}{153 \left (2+x \right )}-\frac {49 \,{\mathrm e}^{x}}{1224 \left (x -\frac {1}{25}\right )}+\frac {{\mathrm e}^{x}}{-72+72 x}\) | \(30\) |
parts | \(x +\frac {4 \,{\mathrm e}^{x}}{153 \left (2+x \right )}-\frac {49 \,{\mathrm e}^{x}}{1224 \left (x -\frac {1}{25}\right )}+\frac {{\mathrm e}^{x}}{-72+72 x}\) | \(30\) |
norman | \(\frac {-\frac {1851 x^{2}}{25}+\frac {1274 x}{25}+25 x^{4}-{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}-\frac {48}{25}}{25 x^{3}+24 x^{2}-51 x +2}\) | \(43\) |
parallelrisch | \(\frac {625 x^{4}-48-1851 x^{2}-25 \,{\mathrm e}^{x} x +1274 x +50 \,{\mathrm e}^{x}}{625 x^{3}+600 x^{2}-1275 x +50}\) | \(44\) |
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Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=\frac {25 \, x^{4} + 24 \, x^{3} - 51 \, x^{2} - {\left (x - 2\right )} e^{x} + 2 \, x}{25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2} \]
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Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=x + \frac {\left (2 - x\right ) e^{x}}{25 x^{3} + 24 x^{2} - 51 x + 2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (32) = 64\).
Time = 0.24 (sec) , antiderivative size = 229, normalized size of antiderivative = 6.74 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=x - \frac {{\left (x - 2\right )} e^{x}}{25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2} - \frac {285481771 \, x^{2} - 240455729 \, x + 9161458}{6242400 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} + \frac {4580729 \, x^{2} - 7021771 \, x + 273542}{130050 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} + \frac {329 \, {\left (136771 \, x^{2} - 51929 \, x + 1858\right )}}{1040400 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} - \frac {4075 \, x^{2} + 4687 \, x - 5294}{62424 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} - \frac {899 \, {\left (2275 \, x^{2} + 1255 \, x - 62\right )}}{83232 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} - \frac {587 \, {\left (929 \, x^{2} - 4579 \, x + 182\right )}}{62424 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} + \frac {775 \, x^{2} + 3019 \, x - 326}{1224 \, {\left (25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=\frac {25 \, x^{4} + 24 \, x^{3} - 51 \, x^{2} - x e^{x} + 2 \, x + 2 \, e^{x}}{25 \, x^{3} + 24 \, x^{2} - 51 \, x + 2} \]
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Time = 16.69 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6+e^x \left (104-200 x-27 x^2+76 x^3-25 x^4\right )}{4-204 x+2697 x^2-2348 x^3-1974 x^4+1200 x^5+625 x^6} \, dx=x+\frac {2\,{\mathrm {e}}^x-x\,{\mathrm {e}}^x}{\left (25\,x-1\right )\,\left (x-1\right )\,\left (x+2\right )} \]
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