Integrand size = 54, antiderivative size = 32 \[ \int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{7500-7500 x+1875 x^2} \, dx=x-\frac {1}{3} x \left (\left (1+\frac {e}{25}\right )^2+\frac {2}{2-x}+\frac {2}{x}+x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {27, 12, 1864} \[ \int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{7500-7500 x+1875 x^2} \, dx=-\frac {x^2}{3}+\frac {(1250-e (50+e)) x}{1875}-\frac {4}{3 (2-x)} \]
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Rule 12
Rule 27
Rule 1864
Rubi steps \begin{align*} \text {integral}& = \int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{1875 (-2+x)^2} \, dx \\ & = \frac {\int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{(-2+x)^2} \, dx}{1875} \\ & = \frac {\int \left (1250 \left (1-\frac {e (50+e)}{1250}\right )-\frac {2500}{(-2+x)^2}-1250 x\right ) \, dx}{1875} \\ & = -\frac {4}{3 (2-x)}+\frac {(1250-e (50+e)) x}{1875}-\frac {x^2}{3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{7500-7500 x+1875 x^2} \, dx=\frac {4}{3 (-2+x)}-\frac {1}{3} (-2+x)^2-\frac {2 x}{3}-\frac {2 e x}{75}-\frac {e^2 x}{1875} \]
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Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {x^{2}}{3}+\frac {2 x}{3}-\frac {{\mathrm e}^{2} x}{1875}-\frac {2 x \,{\mathrm e}}{75}+\frac {4}{3 \left (-2+x \right )}\) | \(27\) |
risch | \(-\frac {x^{2}}{3}+\frac {2 x}{3}-\frac {{\mathrm e}^{2} x}{1875}-\frac {2 x \,{\mathrm e}}{75}+\frac {4}{3 \left (-2+x \right )}\) | \(27\) |
norman | \(\frac {\left (-\frac {{\mathrm e}^{2}}{1875}-\frac {2 \,{\mathrm e}}{75}+\frac {4}{3}\right ) x^{2}-\frac {x^{3}}{3}-\frac {4}{3}+\frac {4 \,{\mathrm e}^{2}}{1875}+\frac {8 \,{\mathrm e}}{75}}{-2+x}\) | \(40\) |
gosper | \(-\frac {x^{2} {\mathrm e}^{2}+50 x^{2} {\mathrm e}+625 x^{3}-4 \,{\mathrm e}^{2}-2500 x^{2}-200 \,{\mathrm e}+2500}{1875 \left (-2+x \right )}\) | \(45\) |
parallelrisch | \(-\frac {x^{2} {\mathrm e}^{2}+50 x^{2} {\mathrm e}+625 x^{3}-4 \,{\mathrm e}^{2}-2500 x^{2}-200 \,{\mathrm e}+2500}{1875 \left (-2+x \right )}\) | \(45\) |
meijerg | \(\frac {x}{3-\frac {3 x}{2}}-2 \left (-\frac {{\mathrm e}^{2}}{1875}-\frac {2 \,{\mathrm e}}{75}+\frac {10}{3}\right ) \left (-\frac {x \left (-\frac {3 x}{2}+6\right )}{6 \left (1-\frac {x}{2}\right )}-2 \ln \left (1-\frac {x}{2}\right )\right )-2 \left (-\frac {2 \,{\mathrm e}^{2}}{1875}-\frac {4 \,{\mathrm e}}{75}+\frac {8}{3}\right ) \left (\frac {x}{2-x}+\ln \left (1-\frac {x}{2}\right )\right )-\frac {x \left (-\frac {1}{2} x^{2}-3 x +12\right )}{3 \left (1-\frac {x}{2}\right )}-8 \ln \left (1-\frac {x}{2}\right )-\frac {{\mathrm e}^{2} x}{1875 \left (1-\frac {x}{2}\right )}-\frac {2 \,{\mathrm e} x}{75 \left (1-\frac {x}{2}\right )}\) | \(129\) |
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Time = 0.38 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{7500-7500 x+1875 x^2} \, dx=-\frac {625 \, x^{3} - 2500 \, x^{2} + {\left (x^{2} - 2 \, x\right )} e^{2} + 50 \, {\left (x^{2} - 2 \, x\right )} e + 2500 \, x - 2500}{1875 \, {\left (x - 2\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{7500-7500 x+1875 x^2} \, dx=- \frac {x^{2}}{3} - x \left (- \frac {2}{3} + \frac {e^{2}}{1875} + \frac {2 e}{75}\right ) + \frac {4}{3 x - 6} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{7500-7500 x+1875 x^2} \, dx=-\frac {1}{3} \, x^{2} - \frac {1}{1875} \, x {\left (e^{2} + 50 \, e - 1250\right )} + \frac {4}{3 \, {\left (x - 2\right )}} \]
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Time = 0.51 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{7500-7500 x+1875 x^2} \, dx=-\frac {1}{3} \, x^{2} - \frac {1}{1875} \, x e^{2} - \frac {2}{75} \, x e + \frac {2}{3} \, x + \frac {4}{3 \, {\left (x - 2\right )}} \]
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Time = 13.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{7500-7500 x+1875 x^2} \, dx=\frac {4}{3\,\left (x-2\right )}-x\,\left (\frac {2\,\mathrm {e}}{75}+\frac {{\mathrm {e}}^2}{1875}-\frac {2}{3}\right )-\frac {x^2}{3} \]
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