Integrand size = 34, antiderivative size = 30 \[ \int \frac {1996 x^3+e^2 (-15968+9624 x)}{996004 x^3-800396 x^4+160801 x^5} \, dx=\frac {\frac {4 e^2}{x^2}+x}{100 (5-2 x)+\frac {1}{2} (-2-x)} \]
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Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {1608, 27, 1634} \[ \int \frac {1996 x^3+e^2 (-15968+9624 x)}{996004 x^3-800396 x^4+160801 x^5} \, dx=\frac {4 e^2}{499 x^2}+\frac {802 e^2}{249001 x}+\frac {2 \left (248502998+64481201 e^2\right )}{99849401 (998-401 x)} \]
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Rule 27
Rule 1608
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \int \frac {1996 x^3+e^2 (-15968+9624 x)}{x^3 \left (996004-800396 x+160801 x^2\right )} \, dx \\ & = \int \frac {1996 x^3+e^2 (-15968+9624 x)}{x^3 (-998+401 x)^2} \, dx \\ & = \int \left (-\frac {8 e^2}{499 x^3}-\frac {802 e^2}{249001 x^2}+\frac {2 \left (248502998+64481201 e^2\right )}{249001 (-998+401 x)^2}\right ) \, dx \\ & = \frac {2 \left (248502998+64481201 e^2\right )}{99849401 (998-401 x)}+\frac {4 e^2}{499 x^2}+\frac {802 e^2}{249001 x} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1996 x^3+e^2 (-15968+9624 x)}{996004 x^3-800396 x^4+160801 x^5} \, dx=-\frac {4 \left (802 e^2+499 x^2\right )}{401 x^2 (-998+401 x)} \]
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Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
norman | \(\frac {-\frac {1996 x^{2}}{401}-8 \,{\mathrm e}^{2}}{x^{2} \left (401 x -998\right )}\) | \(22\) |
gosper | \(-\frac {4 \left (499 x^{2}+802 \,{\mathrm e}^{2}\right )}{401 x^{2} \left (401 x -998\right )}\) | \(23\) |
risch | \(\frac {-\frac {1996 x^{2}}{401}-8 \,{\mathrm e}^{2}}{x^{2} \left (401 x -998\right )}\) | \(23\) |
parallelrisch | \(-\frac {1996 x^{2}+3208 \,{\mathrm e}^{2}}{401 x^{2} \left (401 x -998\right )}\) | \(23\) |
default | \(\frac {4 \,{\mathrm e}^{2}}{499 x^{2}}+\frac {802 \,{\mathrm e}^{2}}{249001 x}-\frac {4 \left (\frac {499}{401}+\frac {160801 \,{\mathrm e}^{2}}{498002}\right )}{401 x -998}\) | \(31\) |
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Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1996 x^3+e^2 (-15968+9624 x)}{996004 x^3-800396 x^4+160801 x^5} \, dx=-\frac {4 \, {\left (499 \, x^{2} + 802 \, e^{2}\right )}}{401 \, {\left (401 \, x^{3} - 998 \, x^{2}\right )}} \]
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Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {1996 x^3+e^2 (-15968+9624 x)}{996004 x^3-800396 x^4+160801 x^5} \, dx=\frac {- 1996 x^{2} - 3208 e^{2}}{160801 x^{3} - 400198 x^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1996 x^3+e^2 (-15968+9624 x)}{996004 x^3-800396 x^4+160801 x^5} \, dx=-\frac {4 \, {\left (499 \, x^{2} + 802 \, e^{2}\right )}}{401 \, {\left (401 \, x^{3} - 998 \, x^{2}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {1996 x^3+e^2 (-15968+9624 x)}{996004 x^3-800396 x^4+160801 x^5} \, dx=-\frac {2 \, {\left (64481201 \, e^{2} + 248502998\right )}}{99849401 \, {\left (401 \, x - 998\right )}} + \frac {2 \, {\left (401 \, x e^{2} + 998 \, e^{2}\right )}}{249001 \, x^{2}} \]
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Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {1996 x^3+e^2 (-15968+9624 x)}{996004 x^3-800396 x^4+160801 x^5} \, dx=\frac {\frac {1996\,x^2}{401}+8\,{\mathrm {e}}^2}{998\,x^2-401\,x^3} \]
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