Integrand size = 41, antiderivative size = 23 \[ \int \frac {e^{-21+4 x} \left (4-22 x+8 x^2\right )}{-200 x^2+300 x^3-150 x^4+25 x^5} \, dx=\frac {2 e^{-21+4 x} x}{25 (-x+(-1+x) x)^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(23)=46\).
Time = 0.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22, number of steps used = 13, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6820, 12, 6874, 2209, 2208} \[ \int \frac {e^{-21+4 x} \left (4-22 x+8 x^2\right )}{-200 x^2+300 x^3-150 x^4+25 x^5} \, dx=\frac {e^{4 x-21}}{50 x}+\frac {e^{4 x-21}}{50 (2-x)}+\frac {e^{4 x-21}}{25 (2-x)^2} \]
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Rule 12
Rule 2208
Rule 2209
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^{-21+4 x} \left (-2+11 x-4 x^2\right )}{25 (2-x)^3 x^2} \, dx \\ & = \frac {2}{25} \int \frac {e^{-21+4 x} \left (-2+11 x-4 x^2\right )}{(2-x)^3 x^2} \, dx \\ & = \frac {2}{25} \int \left (\frac {e^{-21+4 x}}{2-x}-\frac {e^{-21+4 x}}{(-2+x)^3}+\frac {9 e^{-21+4 x}}{4 (-2+x)^2}-\frac {e^{-21+4 x}}{4 x^2}+\frac {e^{-21+4 x}}{x}\right ) \, dx \\ & = -\left (\frac {1}{50} \int \frac {e^{-21+4 x}}{x^2} \, dx\right )+\frac {2}{25} \int \frac {e^{-21+4 x}}{2-x} \, dx-\frac {2}{25} \int \frac {e^{-21+4 x}}{(-2+x)^3} \, dx+\frac {2}{25} \int \frac {e^{-21+4 x}}{x} \, dx+\frac {9}{50} \int \frac {e^{-21+4 x}}{(-2+x)^2} \, dx \\ & = \frac {e^{-21+4 x}}{25 (2-x)^2}+\frac {9 e^{-21+4 x}}{50 (2-x)}+\frac {e^{-21+4 x}}{50 x}-\frac {2 \text {Ei}(-4 (2-x))}{25 e^{13}}+\frac {2 \text {Ei}(4 x)}{25 e^{21}}-\frac {2}{25} \int \frac {e^{-21+4 x}}{x} \, dx-\frac {4}{25} \int \frac {e^{-21+4 x}}{(-2+x)^2} \, dx+\frac {18}{25} \int \frac {e^{-21+4 x}}{-2+x} \, dx \\ & = \frac {e^{-21+4 x}}{25 (2-x)^2}+\frac {e^{-21+4 x}}{50 (2-x)}+\frac {e^{-21+4 x}}{50 x}+\frac {16 \text {Ei}(-4 (2-x))}{25 e^{13}}-\frac {16}{25} \int \frac {e^{-21+4 x}}{-2+x} \, dx \\ & = \frac {e^{-21+4 x}}{25 (2-x)^2}+\frac {e^{-21+4 x}}{50 (2-x)}+\frac {e^{-21+4 x}}{50 x} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-21+4 x} \left (4-22 x+8 x^2\right )}{-200 x^2+300 x^3-150 x^4+25 x^5} \, dx=\frac {2 e^{-21+4 x}}{25 (-2+x)^2 x} \]
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{4 x -21}}{25 x \left (-2+x \right )^{2}}\) | \(17\) |
norman | \(\frac {2 \,{\mathrm e}^{4 x -21}}{25 x \left (-2+x \right )^{2}}\) | \(19\) |
gosper | \(\frac {2 \,{\mathrm e}^{4 x -21}}{25 x \left (x^{2}-4 x +4\right )}\) | \(24\) |
parallelrisch | \(\frac {2 \,{\mathrm e}^{4 x -21}}{25 x \left (x^{2}-4 x +4\right )}\) | \(24\) |
derivativedivides | \(\frac {109 \,{\mathrm e}^{4 x -21} \left (16 \left (-x +\frac {21}{4}\right )^{2}+184 x -505\right )}{50 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}+\frac {31 \,{\mathrm e}^{4 x -21} \left (48 \left (-x +\frac {21}{4}\right )^{2}-472 x +21\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}-\frac {{\mathrm e}^{4 x -21} \left (4976 \left (-x +\frac {21}{4}\right )^{2}+25480 x -108927\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}\) | \(143\) |
default | \(\frac {109 \,{\mathrm e}^{4 x -21} \left (16 \left (-x +\frac {21}{4}\right )^{2}+184 x -505\right )}{50 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}+\frac {31 \,{\mathrm e}^{4 x -21} \left (48 \left (-x +\frac {21}{4}\right )^{2}-472 x +21\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}-\frac {{\mathrm e}^{4 x -21} \left (4976 \left (-x +\frac {21}{4}\right )^{2}+25480 x -108927\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}\) | \(143\) |
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-21+4 x} \left (4-22 x+8 x^2\right )}{-200 x^2+300 x^3-150 x^4+25 x^5} \, dx=\frac {2 \, e^{\left (4 \, x - 21\right )}}{25 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{-21+4 x} \left (4-22 x+8 x^2\right )}{-200 x^2+300 x^3-150 x^4+25 x^5} \, dx=\frac {2 e^{4 x - 21}}{25 x^{3} - 100 x^{2} + 100 x} \]
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-21+4 x} \left (4-22 x+8 x^2\right )}{-200 x^2+300 x^3-150 x^4+25 x^5} \, dx=\frac {2 \, e^{\left (4 \, x\right )}}{25 \, {\left (x^{3} e^{21} - 4 \, x^{2} e^{21} + 4 \, x e^{21}\right )}} \]
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-21+4 x} \left (4-22 x+8 x^2\right )}{-200 x^2+300 x^3-150 x^4+25 x^5} \, dx=\frac {2 \, e^{\left (4 \, x\right )}}{25 \, {\left (x^{3} e^{21} - 4 \, x^{2} e^{21} + 4 \, x e^{21}\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-21+4 x} \left (4-22 x+8 x^2\right )}{-200 x^2+300 x^3-150 x^4+25 x^5} \, dx=\frac {2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{-21}}{25\,\left (x^3-4\,x^2+4\,x\right )} \]
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