Integrand size = 44, antiderivative size = 22 \[ \int \frac {1}{36} e^{-2 x} x^2 \left (-\frac {9 e^{2 x}}{x^2}+100 x-50 x^2+\frac {3 e^x (-80+40 x)}{x}\right ) \, dx=-\frac {x}{4}+\left (2-\frac {5}{6} e^{-x} x^2\right )^2 \]
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Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 21, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {12, 6820, 2227, 2207, 2225} \[ \int \frac {1}{36} e^{-2 x} x^2 \left (-\frac {9 e^{2 x}}{x^2}+100 x-50 x^2+\frac {3 e^x (-80+40 x)}{x}\right ) \, dx=\frac {25}{36} e^{-2 x} x^4-\frac {10}{3} e^{-x} x^2-\frac {x}{4} \]
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Rule 12
Rule 2207
Rule 2225
Rule 2227
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {1}{36} \int e^{-2 x} x^2 \left (-\frac {9 e^{2 x}}{x^2}+100 x-50 x^2+\frac {3 e^x (-80+40 x)}{x}\right ) \, dx \\ & = \frac {1}{36} \int \left (-9+120 e^{-x} (-2+x) x-50 e^{-2 x} (-2+x) x^3\right ) \, dx \\ & = -\frac {x}{4}-\frac {25}{18} \int e^{-2 x} (-2+x) x^3 \, dx+\frac {10}{3} \int e^{-x} (-2+x) x \, dx \\ & = -\frac {x}{4}-\frac {25}{18} \int \left (-2 e^{-2 x} x^3+e^{-2 x} x^4\right ) \, dx+\frac {10}{3} \int \left (-2 e^{-x} x+e^{-x} x^2\right ) \, dx \\ & = -\frac {x}{4}-\frac {25}{18} \int e^{-2 x} x^4 \, dx+\frac {25}{9} \int e^{-2 x} x^3 \, dx+\frac {10}{3} \int e^{-x} x^2 \, dx-\frac {20}{3} \int e^{-x} x \, dx \\ & = -\frac {x}{4}+\frac {20 e^{-x} x}{3}-\frac {10}{3} e^{-x} x^2-\frac {25}{18} e^{-2 x} x^3+\frac {25}{36} e^{-2 x} x^4-\frac {25}{9} \int e^{-2 x} x^3 \, dx+\frac {25}{6} \int e^{-2 x} x^2 \, dx-\frac {20}{3} \int e^{-x} \, dx+\frac {20}{3} \int e^{-x} x \, dx \\ & = \frac {20 e^{-x}}{3}-\frac {x}{4}-\frac {25}{12} e^{-2 x} x^2-\frac {10}{3} e^{-x} x^2+\frac {25}{36} e^{-2 x} x^4+\frac {25}{6} \int e^{-2 x} x \, dx-\frac {25}{6} \int e^{-2 x} x^2 \, dx+\frac {20}{3} \int e^{-x} \, dx \\ & = -\frac {x}{4}-\frac {25}{12} e^{-2 x} x-\frac {10}{3} e^{-x} x^2+\frac {25}{36} e^{-2 x} x^4+\frac {25}{12} \int e^{-2 x} \, dx-\frac {25}{6} \int e^{-2 x} x \, dx \\ & = -\frac {25}{24} e^{-2 x}-\frac {x}{4}-\frac {10}{3} e^{-x} x^2+\frac {25}{36} e^{-2 x} x^4-\frac {25}{12} \int e^{-2 x} \, dx \\ & = -\frac {x}{4}-\frac {10}{3} e^{-x} x^2+\frac {25}{36} e^{-2 x} x^4 \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {1}{36} e^{-2 x} x^2 \left (-\frac {9 e^{2 x}}{x^2}+100 x-50 x^2+\frac {3 e^x (-80+40 x)}{x}\right ) \, dx=\frac {1}{36} \left (-9 x-120 e^{-x} x^2+25 e^{-2 x} x^4\right ) \]
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Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77
method | result | size |
default | \(-\frac {x}{4}-\frac {10 x^{2} {\mathrm e}^{-x}}{3}+\frac {25 x^{4} {\mathrm e}^{-2 x}}{36}\) | \(39\) |
parts | \(-\frac {x}{4}-\frac {10 x^{2} {\mathrm e}^{-x}}{3}+\frac {25 x^{4} {\mathrm e}^{-2 x}}{36}\) | \(39\) |
parallelrisch | \(\frac {\left (-\frac {18 \,{\mathrm e}^{2 x}}{x}+50 x^{2}-80 \,{\mathrm e}^{\ln \left (x \right )+\ln \left (\frac {3}{x^{2}}\right )+x} x \right ) {\mathrm e}^{-2 x} x^{2}}{72}\) | \(52\) |
risch | \(-\frac {x}{4}+\frac {10 x^{2} {\mathrm e}^{-x} {\mathrm e}^{-\frac {i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}} {\mathrm e}^{-\frac {i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}}}{3}+\frac {25 x^{4} {\mathrm e}^{-2 x}}{36}\) | \(56\) |
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Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.68 \[ \int \frac {1}{36} e^{-2 x} x^2 \left (-\frac {9 e^{2 x}}{x^2}+100 x-50 x^2+\frac {3 e^x (-80+40 x)}{x}\right ) \, dx=\frac {1}{4} \, {\left (25 \, x^{2} - x e^{\left (2 \, x + \log \left (3\right ) + \log \left (\frac {3}{x^{2}}\right )\right )} - 40 \, x e^{\left (x + \frac {1}{2} \, \log \left (3\right ) + \frac {1}{2} \, \log \left (\frac {3}{x^{2}}\right )\right )}\right )} e^{\left (-2 \, x - \log \left (3\right ) - \log \left (\frac {3}{x^{2}}\right )\right )} \]
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Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{36} e^{-2 x} x^2 \left (-\frac {9 e^{2 x}}{x^2}+100 x-50 x^2+\frac {3 e^x (-80+40 x)}{x}\right ) \, dx=\frac {25 x^{4} e^{- 2 x}}{36} - \frac {10 x^{2} e^{- x}}{3} - \frac {x}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (30) = 60\).
Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.36 \[ \int \frac {1}{36} e^{-2 x} x^2 \left (-\frac {9 e^{2 x}}{x^2}+100 x-50 x^2+\frac {3 e^x (-80+40 x)}{x}\right ) \, dx=-\frac {10}{3} \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + \frac {20}{3} \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {25}{72} \, {\left (2 \, x^{4} + 4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x\right )} - \frac {25}{72} \, {\left (4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x\right )} - \frac {1}{4} \, x \]
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Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{36} e^{-2 x} x^2 \left (-\frac {9 e^{2 x}}{x^2}+100 x-50 x^2+\frac {3 e^x (-80+40 x)}{x}\right ) \, dx=\frac {25}{36} \, x^{4} e^{\left (-2 \, x\right )} - \frac {10}{3} \, x^{2} e^{\left (-x\right )} - \frac {1}{4} \, x \]
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Time = 14.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {1}{36} e^{-2 x} x^2 \left (-\frac {9 e^{2 x}}{x^2}+100 x-50 x^2+\frac {3 e^x (-80+40 x)}{x}\right ) \, dx=-\frac {x\,\left (120\,x\,{\mathrm {e}}^{-x}-25\,x^3\,{\mathrm {e}}^{-2\,x}+9\right )}{36} \]
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