Integrand size = 46, antiderivative size = 22 \[ \int \frac {e^{2 x} \left (-16-24 x-12 x^2+4 x^3\right )}{128+128 x+8 x^2-20 x^3-2 x^4+x^5} \, dx=\frac {2 e^{2 x}}{(-4+x) \left (4+\frac {4}{x}+x\right )} \]
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Leaf count is larger than twice the leaf count of optimal. \(45\) vs. \(2(22)=44\).
Time = 0.31 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.05, number of steps used = 13, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {6820, 12, 6874, 2208, 2209} \[ \int \frac {e^{2 x} \left (-16-24 x-12 x^2+4 x^3\right )}{128+128 x+8 x^2-20 x^3-2 x^4+x^5} \, dx=-\frac {2 e^{2 x}}{9 (x+2)}+\frac {2 e^{2 x}}{3 (x+2)^2}-\frac {2 e^{2 x}}{9 (4-x)} \]
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Rule 12
Rule 2208
Rule 2209
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {4 e^{2 x} \left (-4-6 x-3 x^2+x^3\right )}{(4-x)^2 (2+x)^3} \, dx \\ & = 4 \int \frac {e^{2 x} \left (-4-6 x-3 x^2+x^3\right )}{(4-x)^2 (2+x)^3} \, dx \\ & = 4 \int \left (-\frac {e^{2 x}}{18 (-4+x)^2}+\frac {e^{2 x}}{9 (-4+x)}-\frac {e^{2 x}}{3 (2+x)^3}+\frac {7 e^{2 x}}{18 (2+x)^2}-\frac {e^{2 x}}{9 (2+x)}\right ) \, dx \\ & = -\left (\frac {2}{9} \int \frac {e^{2 x}}{(-4+x)^2} \, dx\right )+\frac {4}{9} \int \frac {e^{2 x}}{-4+x} \, dx-\frac {4}{9} \int \frac {e^{2 x}}{2+x} \, dx-\frac {4}{3} \int \frac {e^{2 x}}{(2+x)^3} \, dx+\frac {14}{9} \int \frac {e^{2 x}}{(2+x)^2} \, dx \\ & = -\frac {2 e^{2 x}}{9 (4-x)}+\frac {2 e^{2 x}}{3 (2+x)^2}-\frac {14 e^{2 x}}{9 (2+x)}+\frac {4}{9} e^8 \operatorname {ExpIntegralEi}(-2 (4-x))-\frac {4 \operatorname {ExpIntegralEi}(2 (2+x))}{9 e^4}-\frac {4}{9} \int \frac {e^{2 x}}{-4+x} \, dx-\frac {4}{3} \int \frac {e^{2 x}}{(2+x)^2} \, dx+\frac {28}{9} \int \frac {e^{2 x}}{2+x} \, dx \\ & = -\frac {2 e^{2 x}}{9 (4-x)}+\frac {2 e^{2 x}}{3 (2+x)^2}-\frac {2 e^{2 x}}{9 (2+x)}+\frac {8 \operatorname {ExpIntegralEi}(2 (2+x))}{3 e^4}-\frac {8}{3} \int \frac {e^{2 x}}{2+x} \, dx \\ & = -\frac {2 e^{2 x}}{9 (4-x)}+\frac {2 e^{2 x}}{3 (2+x)^2}-\frac {2 e^{2 x}}{9 (2+x)} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {e^{2 x} \left (-16-24 x-12 x^2+4 x^3\right )}{128+128 x+8 x^2-20 x^3-2 x^4+x^5} \, dx=\frac {2 e^{2 x} x}{(-4+x) (2+x)^2} \]
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Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(\frac {2 \,{\mathrm e}^{2 x} x}{x^{3}-12 x -16}\) | \(18\) |
norman | \(\frac {2 x \,{\mathrm e}^{2 x}}{\left (x -4\right ) \left (2+x \right )^{2}}\) | \(18\) |
risch | \(\frac {2 x \,{\mathrm e}^{2 x}}{\left (x -4\right ) \left (2+x \right )^{2}}\) | \(18\) |
parallelrisch | \(\frac {2 \,{\mathrm e}^{2 x} x}{x^{3}-12 x -16}\) | \(18\) |
default | \(-\frac {2 \,{\mathrm e}^{2 x}}{9 \left (2+x \right )}+\frac {2 \,{\mathrm e}^{2 x}}{9 \left (x -4\right )}+\frac {2 \,{\mathrm e}^{2 x}}{3 \left (2+x \right )^{2}}\) | \(35\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 x} \left (-16-24 x-12 x^2+4 x^3\right )}{128+128 x+8 x^2-20 x^3-2 x^4+x^5} \, dx=\frac {2 \, x e^{\left (2 \, x\right )}}{x^{3} - 12 \, x - 16} \]
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Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {e^{2 x} \left (-16-24 x-12 x^2+4 x^3\right )}{128+128 x+8 x^2-20 x^3-2 x^4+x^5} \, dx=\frac {2 x e^{2 x}}{x^{3} - 12 x - 16} \]
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Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 x} \left (-16-24 x-12 x^2+4 x^3\right )}{128+128 x+8 x^2-20 x^3-2 x^4+x^5} \, dx=\frac {2 \, x e^{\left (2 \, x\right )}}{x^{3} - 12 \, x - 16} \]
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 x} \left (-16-24 x-12 x^2+4 x^3\right )}{128+128 x+8 x^2-20 x^3-2 x^4+x^5} \, dx=\frac {2 \, x e^{\left (2 \, x\right )}}{x^{3} - 12 \, x - 16} \]
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Time = 15.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 x} \left (-16-24 x-12 x^2+4 x^3\right )}{128+128 x+8 x^2-20 x^3-2 x^4+x^5} \, dx=\frac {2\,x\,{\mathrm {e}}^{2\,x}}{{\left (x+2\right )}^2\,\left (x-4\right )} \]
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