Integrand size = 55, antiderivative size = 27 \[ \int \frac {e^{-x} \left (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 \left (324-9 x-14 x^2-x^3\right )\right )}{81+18 x+x^2} \, dx=e^{-x} (3-x) \left (-e^2-2 x+\frac {25}{9+x}\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(27)=54\).
Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {27, 2230, 2225, 2207, 2208, 2209} \[ \int \frac {e^{-x} \left (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 \left (324-9 x-14 x^2-x^3\right )\right )}{81+18 x+x^2} \, dx=2 e^{-x} x^2+4 e^{-x} x-\left (10-e^2\right ) e^{-x} x+4 e^{-x}+\frac {300 e^{-x}}{x+9}-\left (19+4 e^2\right ) e^{-x}-\left (10-e^2\right ) e^{-x} \]
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Rule 27
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 \left (324-9 x-14 x^2-x^3\right )\right )}{(9+x)^2} \, dx \\ & = \int \left (19 e^{-x} \left (1+\frac {4 e^2}{19}\right )-e^{-x} \left (-10+e^2\right ) x-2 e^{-x} x^2-\frac {300 e^{-x}}{(9+x)^2}-\frac {300 e^{-x}}{9+x}\right ) \, dx \\ & = -\left (2 \int e^{-x} x^2 \, dx\right )-300 \int \frac {e^{-x}}{(9+x)^2} \, dx-300 \int \frac {e^{-x}}{9+x} \, dx+\left (10-e^2\right ) \int e^{-x} x \, dx+\left (19+4 e^2\right ) \int e^{-x} \, dx \\ & = -e^{-x} \left (19+4 e^2\right )-e^{-x} \left (10-e^2\right ) x+2 e^{-x} x^2+\frac {300 e^{-x}}{9+x}-300 e^9 \text {Ei}(-9-x)-4 \int e^{-x} x \, dx+300 \int \frac {e^{-x}}{9+x} \, dx+\left (10-e^2\right ) \int e^{-x} \, dx \\ & = -e^{-x} \left (10-e^2\right )-e^{-x} \left (19+4 e^2\right )+4 e^{-x} x-e^{-x} \left (10-e^2\right ) x+2 e^{-x} x^2+\frac {300 e^{-x}}{9+x}-4 \int e^{-x} \, dx \\ & = 4 e^{-x}-e^{-x} \left (10-e^2\right )-e^{-x} \left (19+4 e^2\right )+4 e^{-x} x-e^{-x} \left (10-e^2\right ) x+2 e^{-x} x^2+\frac {300 e^{-x}}{9+x} \\ \end{align*}
Time = 2.81 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-x} \left (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 \left (324-9 x-14 x^2-x^3\right )\right )}{81+18 x+x^2} \, dx=\frac {e^{-x} (-3+x) \left (-25+18 x+2 x^2+e^2 (9+x)\right )}{9+x} \]
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Time = 0.63 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41
method | result | size |
norman | \(\frac {\left (\left ({\mathrm e}^{2}+12\right ) x^{2}+\left (6 \,{\mathrm e}^{2}-79\right ) x +2 x^{3}+75-27 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-x}}{x +9}\) | \(38\) |
gosper | \(\frac {\left (x^{2} {\mathrm e}^{2}+2 x^{3}+6 \,{\mathrm e}^{2} x +12 x^{2}-27 \,{\mathrm e}^{2}-79 x +75\right ) {\mathrm e}^{-x}}{x +9}\) | \(41\) |
risch | \(\frac {\left (x^{2} {\mathrm e}^{2}+2 x^{3}+6 \,{\mathrm e}^{2} x +12 x^{2}-27 \,{\mathrm e}^{2}-79 x +75\right ) {\mathrm e}^{-x}}{x +9}\) | \(41\) |
parallelrisch | \(\frac {\left (x^{2} {\mathrm e}^{2}+2 x^{3}+6 \,{\mathrm e}^{2} x +12 x^{2}-27 \,{\mathrm e}^{2}-79 x +75\right ) {\mathrm e}^{-x}}{x +9}\) | \(41\) |
default | \(\frac {300 \,{\mathrm e}^{-x}}{x +9}-37 \,{\mathrm e}^{-x}+26 \left (x -17\right ) {\mathrm e}^{-x}+2 \left (x^{2}-16 x +227\right ) {\mathrm e}^{-x}+324 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{-x}}{x +9}+{\mathrm e}^{9} \operatorname {Ei}_{1}\left (x +9\right )\right )-14 \,{\mathrm e}^{2} \left (-{\mathrm e}^{-x}-\frac {81 \,{\mathrm e}^{-x}}{x +9}+99 \,{\mathrm e}^{9} \operatorname {Ei}_{1}\left (x +9\right )\right )-{\mathrm e}^{2} \left (-\left (x -17\right ) {\mathrm e}^{-x}+\frac {729 \,{\mathrm e}^{-x}}{x +9}-972 \,{\mathrm e}^{9} \operatorname {Ei}_{1}\left (x +9\right )\right )-9 \,{\mathrm e}^{2} \left (\frac {9 \,{\mathrm e}^{-x}}{x +9}-10 \,{\mathrm e}^{9} \operatorname {Ei}_{1}\left (x +9\right )\right )\) | \(156\) |
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Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-x} \left (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 \left (324-9 x-14 x^2-x^3\right )\right )}{81+18 x+x^2} \, dx=\frac {{\left (2 \, x^{3} + 12 \, x^{2} + {\left (x^{2} + 6 \, x - 27\right )} e^{2} - 79 \, x + 75\right )} e^{\left (-x\right )}}{x + 9} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-x} \left (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 \left (324-9 x-14 x^2-x^3\right )\right )}{81+18 x+x^2} \, dx=\frac {\left (2 x^{3} + x^{2} e^{2} + 12 x^{2} - 79 x + 6 x e^{2} - 27 e^{2} + 75\right ) e^{- x}}{x + 9} \]
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\[ \int \frac {e^{-x} \left (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 \left (324-9 x-14 x^2-x^3\right )\right )}{81+18 x+x^2} \, dx=\int { -\frac {{\left (2 \, x^{4} + 26 \, x^{3} - 37 \, x^{2} + {\left (x^{3} + 14 \, x^{2} + 9 \, x - 324\right )} e^{2} - 852 \, x + 1461\right )} e^{\left (-x\right )}}{x^{2} + 18 \, x + 81} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \frac {e^{-x} \left (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 \left (324-9 x-14 x^2-x^3\right )\right )}{81+18 x+x^2} \, dx=\frac {2 \, x^{3} e^{\left (-x\right )} + 12 \, x^{2} e^{\left (-x\right )} + x^{2} e^{\left (-x + 2\right )} - 79 \, x e^{\left (-x\right )} + 6 \, x e^{\left (-x + 2\right )} + 75 \, e^{\left (-x\right )} - 27 \, e^{\left (-x + 2\right )}}{x + 9} \]
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Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-x} \left (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 \left (324-9 x-14 x^2-x^3\right )\right )}{81+18 x+x^2} \, dx=\frac {{\mathrm {e}}^{-x}\,\left (x-3\right )\,\left (18\,x+9\,{\mathrm {e}}^2+x\,{\mathrm {e}}^2+2\,x^2-25\right )}{x+9} \]
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