Integrand size = 57, antiderivative size = 30 \[ \int \frac {-20+40 x-29 x^2+2 x^3-x^4+e^{x^2} \left (2 x^3-4 x^4+2 x^5\right )}{x^2-2 x^3+x^4} \, dx=e^{x^2}+5 \left (-3+\frac {4}{x}\right )-\frac {(9-x) x}{1-x} \]
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1608, 27, 6874, 46, 2240, 45} \[ \int \frac {-20+40 x-29 x^2+2 x^3-x^4+e^{x^2} \left (2 x^3-4 x^4+2 x^5\right )}{x^2-2 x^3+x^4} \, dx=e^{x^2}-x-\frac {8}{1-x}+\frac {20}{x} \]
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Rule 27
Rule 45
Rule 46
Rule 1608
Rule 2240
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-20+40 x-29 x^2+2 x^3-x^4+e^{x^2} \left (2 x^3-4 x^4+2 x^5\right )}{x^2 \left (1-2 x+x^2\right )} \, dx \\ & = \int \frac {-20+40 x-29 x^2+2 x^3-x^4+e^{x^2} \left (2 x^3-4 x^4+2 x^5\right )}{(-1+x)^2 x^2} \, dx \\ & = \int \left (-\frac {29}{(-1+x)^2}-\frac {20}{(-1+x)^2 x^2}+\frac {40}{(-1+x)^2 x}+2 e^{x^2} x+\frac {2 x}{(-1+x)^2}-\frac {x^2}{(-1+x)^2}\right ) \, dx \\ & = -\frac {29}{1-x}+2 \int e^{x^2} x \, dx+2 \int \frac {x}{(-1+x)^2} \, dx-20 \int \frac {1}{(-1+x)^2 x^2} \, dx+40 \int \frac {1}{(-1+x)^2 x} \, dx-\int \frac {x^2}{(-1+x)^2} \, dx \\ & = e^{x^2}-\frac {29}{1-x}+2 \int \left (\frac {1}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx-20 \int \left (\frac {1}{(-1+x)^2}-\frac {2}{-1+x}+\frac {1}{x^2}+\frac {2}{x}\right ) \, dx+40 \int \left (\frac {1}{1-x}+\frac {1}{(-1+x)^2}+\frac {1}{x}\right ) \, dx-\int \left (1+\frac {1}{(-1+x)^2}+\frac {2}{-1+x}\right ) \, dx \\ & = e^{x^2}-\frac {8}{1-x}+\frac {20}{x}-x \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \frac {-20+40 x-29 x^2+2 x^3-x^4+e^{x^2} \left (2 x^3-4 x^4+2 x^5\right )}{x^2-2 x^3+x^4} \, dx=e^{x^2}+\frac {8}{-1+x}+\frac {20}{x}-x \]
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Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70
method | result | size |
parts | \({\mathrm e}^{x^{2}}+\frac {20}{x}+\frac {8}{-1+x}-x\) | \(21\) |
risch | \(-x +\frac {28 x -20}{x \left (-1+x \right )}+{\mathrm e}^{x^{2}}\) | \(23\) |
parallelrisch | \(-\frac {x^{3}-x^{2} {\mathrm e}^{x^{2}}+20+{\mathrm e}^{x^{2}} x -29 x}{x \left (-1+x \right )}\) | \(34\) |
norman | \(\frac {-20+x^{2} {\mathrm e}^{x^{2}}+29 x -x^{3}-{\mathrm e}^{x^{2}} x}{x \left (-1+x \right )}\) | \(35\) |
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Time = 0.34 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-20+40 x-29 x^2+2 x^3-x^4+e^{x^2} \left (2 x^3-4 x^4+2 x^5\right )}{x^2-2 x^3+x^4} \, dx=-\frac {x^{3} - x^{2} - {\left (x^{2} - x\right )} e^{\left (x^{2}\right )} - 28 \, x + 20}{x^{2} - x} \]
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Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.50 \[ \int \frac {-20+40 x-29 x^2+2 x^3-x^4+e^{x^2} \left (2 x^3-4 x^4+2 x^5\right )}{x^2-2 x^3+x^4} \, dx=- x - \frac {20 - 28 x}{x^{2} - x} + e^{x^{2}} \]
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Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-20+40 x-29 x^2+2 x^3-x^4+e^{x^2} \left (2 x^3-4 x^4+2 x^5\right )}{x^2-2 x^3+x^4} \, dx=-x + \frac {20 \, {\left (2 \, x - 1\right )}}{x^{2} - x} - \frac {12}{x - 1} + e^{\left (x^{2}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-20+40 x-29 x^2+2 x^3-x^4+e^{x^2} \left (2 x^3-4 x^4+2 x^5\right )}{x^2-2 x^3+x^4} \, dx=-\frac {x^{3} - x^{2} e^{\left (x^{2}\right )} - x^{2} + x e^{\left (x^{2}\right )} - 28 \, x + 20}{x^{2} - x} \]
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Time = 12.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {-20+40 x-29 x^2+2 x^3-x^4+e^{x^2} \left (2 x^3-4 x^4+2 x^5\right )}{x^2-2 x^3+x^4} \, dx={\mathrm {e}}^{x^2}-x+\frac {28\,x-20}{x\,\left (x-1\right )} \]
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