Integrand size = 57, antiderivative size = 26 \[ \int \frac {e^{-x} \left (27+45 x+21 x^2+3 x^3+e^x \left (-20 x^2-12 x^3-2 x^4\right )\right )}{9 x^2+6 x^3+x^4} \, dx=4-\frac {3 e^{-x}}{x}-2 x+\frac {2}{3+x}-5 \log (5) \]
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Time = 0.65 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81, number of steps used = 27, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.123, Rules used = {1608, 27, 6874, 2208, 2209, 2230, 697} \[ \int \frac {e^{-x} \left (27+45 x+21 x^2+3 x^3+e^x \left (-20 x^2-12 x^3-2 x^4\right )\right )}{9 x^2+6 x^3+x^4} \, dx=-2 x+\frac {2}{x+3}-\frac {3 e^{-x}}{x} \]
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Rule 27
Rule 697
Rule 1608
Rule 2208
Rule 2209
Rule 2230
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (27+45 x+21 x^2+3 x^3+e^x \left (-20 x^2-12 x^3-2 x^4\right )\right )}{x^2 \left (9+6 x+x^2\right )} \, dx \\ & = \int \frac {e^{-x} \left (27+45 x+21 x^2+3 x^3+e^x \left (-20 x^2-12 x^3-2 x^4\right )\right )}{x^2 (3+x)^2} \, dx \\ & = \int \left (\frac {21 e^{-x}}{(3+x)^2}+\frac {27 e^{-x}}{x^2 (3+x)^2}+\frac {45 e^{-x}}{x (3+x)^2}+\frac {3 e^{-x} x}{(3+x)^2}-\frac {2 \left (10+6 x+x^2\right )}{(3+x)^2}\right ) \, dx \\ & = -\left (2 \int \frac {10+6 x+x^2}{(3+x)^2} \, dx\right )+3 \int \frac {e^{-x} x}{(3+x)^2} \, dx+21 \int \frac {e^{-x}}{(3+x)^2} \, dx+27 \int \frac {e^{-x}}{x^2 (3+x)^2} \, dx+45 \int \frac {e^{-x}}{x (3+x)^2} \, dx \\ & = -\frac {21 e^{-x}}{3+x}-2 \int \left (1+\frac {1}{(3+x)^2}\right ) \, dx+3 \int \left (-\frac {3 e^{-x}}{(3+x)^2}+\frac {e^{-x}}{3+x}\right ) \, dx-21 \int \frac {e^{-x}}{3+x} \, dx+27 \int \left (\frac {e^{-x}}{9 x^2}-\frac {2 e^{-x}}{27 x}+\frac {e^{-x}}{9 (3+x)^2}+\frac {2 e^{-x}}{27 (3+x)}\right ) \, dx+45 \int \left (\frac {e^{-x}}{9 x}-\frac {e^{-x}}{3 (3+x)^2}-\frac {e^{-x}}{9 (3+x)}\right ) \, dx \\ & = -2 x+\frac {2}{3+x}-\frac {21 e^{-x}}{3+x}-21 e^3 \operatorname {ExpIntegralEi}(-3-x)-2 \int \frac {e^{-x}}{x} \, dx+2 \int \frac {e^{-x}}{3+x} \, dx+3 \int \frac {e^{-x}}{x^2} \, dx+3 \int \frac {e^{-x}}{(3+x)^2} \, dx+3 \int \frac {e^{-x}}{3+x} \, dx+5 \int \frac {e^{-x}}{x} \, dx-5 \int \frac {e^{-x}}{3+x} \, dx-9 \int \frac {e^{-x}}{(3+x)^2} \, dx-15 \int \frac {e^{-x}}{(3+x)^2} \, dx \\ & = -\frac {3 e^{-x}}{x}-2 x+\frac {2}{3+x}-21 e^3 \operatorname {ExpIntegralEi}(-3-x)+3 \operatorname {ExpIntegralEi}(-x)-3 \int \frac {e^{-x}}{x} \, dx-3 \int \frac {e^{-x}}{3+x} \, dx+9 \int \frac {e^{-x}}{3+x} \, dx+15 \int \frac {e^{-x}}{3+x} \, dx \\ & = -\frac {3 e^{-x}}{x}-2 x+\frac {2}{3+x} \\ \end{align*}
Time = 3.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-x} \left (27+45 x+21 x^2+3 x^3+e^x \left (-20 x^2-12 x^3-2 x^4\right )\right )}{9 x^2+6 x^3+x^4} \, dx=-\frac {3 e^{-x}}{x}-2 x+\frac {2}{3+x} \]
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Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {2}{3+x}-2 x -\frac {3 \,{\mathrm e}^{-x}}{x}\) | \(21\) |
parts | \(\frac {2}{3+x}-2 x -\frac {3 \,{\mathrm e}^{-x}}{x}\) | \(21\) |
norman | \(\frac {\left (-9+20 \,{\mathrm e}^{x} x -3 x -2 \,{\mathrm e}^{x} x^{3}\right ) {\mathrm e}^{-x}}{\left (3+x \right ) x}\) | \(31\) |
parallelrisch | \(\frac {\left (-9+20 \,{\mathrm e}^{x} x -3 x -2 \,{\mathrm e}^{x} x^{3}\right ) {\mathrm e}^{-x}}{\left (3+x \right ) x}\) | \(31\) |
default | \(\frac {2}{3+x}-2 x -\frac {3 \,{\mathrm e}^{-x} \left (3+2 x \right )}{\left (3+x \right ) x}+\frac {3 \,{\mathrm e}^{-x}}{3+x}\) | \(42\) |
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {e^{-x} \left (27+45 x+21 x^2+3 x^3+e^x \left (-20 x^2-12 x^3-2 x^4\right )\right )}{9 x^2+6 x^3+x^4} \, dx=-\frac {{\left (2 \, {\left (x^{3} + 3 \, x^{2} - x\right )} e^{x} + 3 \, x + 9\right )} e^{\left (-x\right )}}{x^{2} + 3 \, x} \]
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Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.54 \[ \int \frac {e^{-x} \left (27+45 x+21 x^2+3 x^3+e^x \left (-20 x^2-12 x^3-2 x^4\right )\right )}{9 x^2+6 x^3+x^4} \, dx=- 2 x + \frac {2}{x + 3} - \frac {3 e^{- x}}{x} \]
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {e^{-x} \left (27+45 x+21 x^2+3 x^3+e^x \left (-20 x^2-12 x^3-2 x^4\right )\right )}{9 x^2+6 x^3+x^4} \, dx=-\frac {2 \, x^{3} + 6 \, x^{2} + 3 \, {\left (x + 3\right )} e^{\left (-x\right )} - 2 \, x}{x^{2} + 3 \, x} \]
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Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {e^{-x} \left (27+45 x+21 x^2+3 x^3+e^x \left (-20 x^2-12 x^3-2 x^4\right )\right )}{9 x^2+6 x^3+x^4} \, dx=-\frac {2 \, x^{3} + 6 \, x^{2} + 3 \, x e^{\left (-x\right )} - 2 \, x + 9 \, e^{\left (-x\right )}}{x^{2} + 3 \, x} \]
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Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-x} \left (27+45 x+21 x^2+3 x^3+e^x \left (-20 x^2-12 x^3-2 x^4\right )\right )}{9 x^2+6 x^3+x^4} \, dx=-2\,x-\frac {9\,{\mathrm {e}}^{-x}+x\,\left (3\,{\mathrm {e}}^{-x}-2\right )}{x\,\left (x+3\right )} \]
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