Integrand size = 83, antiderivative size = 28 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=5 \left (x+\frac {4 e^{3-(2+x)^2}}{\left (x+\frac {5 x^2}{4}\right )^2}\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(28)=56\).
Time = 2.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86, number of steps used = 45, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.072, Rules used = {6820, 12, 6874, 2274, 2266, 2236} \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=\frac {250 e^{-x^2-4 x-1}}{5 x+4}+\frac {500 e^{-x^2-4 x-1}}{(5 x+4)^2}-\frac {50 e^{-x^2-4 x-1}}{x}+\frac {20 e^{-x^2-4 x-1}}{x^2}+5 x \]
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Rule 12
Rule 2236
Rule 2266
Rule 2274
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {5 e^{-1-4 x-x^2} \left (-512-2304 x-1792 x^2-640 x^3+e^{1+4 x+x^2} x^3 (4+5 x)^3\right )}{x^3 (4+5 x)^3} \, dx \\ & = 5 \int \frac {e^{-1-4 x-x^2} \left (-512-2304 x-1792 x^2-640 x^3+e^{1+4 x+x^2} x^3 (4+5 x)^3\right )}{x^3 (4+5 x)^3} \, dx \\ & = 5 \int \left (1-\frac {640 e^{-1-4 x-x^2}}{(4+5 x)^3}-\frac {512 e^{-1-4 x-x^2}}{x^3 (4+5 x)^3}-\frac {2304 e^{-1-4 x-x^2}}{x^2 (4+5 x)^3}-\frac {1792 e^{-1-4 x-x^2}}{x (4+5 x)^3}\right ) \, dx \\ & = 5 x-2560 \int \frac {e^{-1-4 x-x^2}}{x^3 (4+5 x)^3} \, dx-3200 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^3} \, dx-8960 \int \frac {e^{-1-4 x-x^2}}{x (4+5 x)^3} \, dx-11520 \int \frac {e^{-1-4 x-x^2}}{x^2 (4+5 x)^3} \, dx \\ & = 5 x+\frac {320 e^{-1-4 x-x^2}}{(4+5 x)^2}+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+768 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx-2560 \int \left (\frac {e^{-1-4 x-x^2}}{64 x^3}-\frac {15 e^{-1-4 x-x^2}}{256 x^2}+\frac {75 e^{-1-4 x-x^2}}{512 x}-\frac {125 e^{-1-4 x-x^2}}{64 (4+5 x)^3}-\frac {375 e^{-1-4 x-x^2}}{256 (4+5 x)^2}-\frac {375 e^{-1-4 x-x^2}}{512 (4+5 x)}\right ) \, dx-8960 \int \left (\frac {e^{-1-4 x-x^2}}{64 x}-\frac {5 e^{-1-4 x-x^2}}{4 (4+5 x)^3}-\frac {5 e^{-1-4 x-x^2}}{16 (4+5 x)^2}-\frac {5 e^{-1-4 x-x^2}}{64 (4+5 x)}\right ) \, dx-11520 \int \left (\frac {e^{-1-4 x-x^2}}{64 x^2}-\frac {15 e^{-1-4 x-x^2}}{256 x}+\frac {25 e^{-1-4 x-x^2}}{16 (4+5 x)^3}+\frac {25 e^{-1-4 x-x^2}}{32 (4+5 x)^2}+\frac {75 e^{-1-4 x-x^2}}{256 (4+5 x)}\right ) \, dx \\ & = 5 x+\frac {320 e^{-1-4 x-x^2}}{(4+5 x)^2}-\frac {768 e^{-1-4 x-x^2}}{5 (4+5 x)}-40 \int \frac {e^{-1-4 x-x^2}}{x^3} \, dx-\frac {1536}{25} \int e^{-1-4 x-x^2} \, dx+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-140 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+150 \int \frac {e^{-1-4 x-x^2}}{x^2} \, dx-180 \int \frac {e^{-1-4 x-x^2}}{x^2} \, dx-\frac {9216}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-375 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+675 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+700 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+1875 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+2800 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx-3375 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+3750 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx+5000 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^3} \, dx-9000 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx+11200 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^3} \, dx-18000 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^3} \, dx \\ & = \frac {20 e^{-1-4 x-x^2}}{x^2}+\frac {30 e^{-1-4 x-x^2}}{x}+5 x+\frac {500 e^{-1-4 x-x^2}}{(4+5 x)^2}+\frac {1682 e^{-1-4 x-x^2}}{5 (4+5 x)}+40 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+80 \int \frac {e^{-1-4 x-x^2}}{x^2} \, dx+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-140 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-200 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-224 \int e^{-1-4 x-x^2} \, dx-2 \left (300 \int e^{-1-4 x-x^2} \, dx\right )+360 \int e^{-1-4 x-x^2} \, dx-\frac {9216}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-375 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-448 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-600 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+675 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+700 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+720 \int e^{-1-4 x-x^2} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1200 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx-1344 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1800 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+1875 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-2688 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx-3375 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+4320 \int \frac {e^{-1-4 x-x^2}}{(4+5 x)^2} \, dx+4320 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-\frac {1}{25} \left (1536 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx \\ & = \frac {20 e^{-1-4 x-x^2}}{x^2}-\frac {50 e^{-1-4 x-x^2}}{x}+5 x+\frac {500 e^{-1-4 x-x^2}}{(4+5 x)^2}+\frac {250 e^{-1-4 x-x^2}}{4+5 x}-\frac {768}{25} e^3 \sqrt {\pi } \text {erf}(2+x)+40 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+96 \int e^{-1-4 x-x^2} \, dx+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-140 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-160 \int e^{-1-4 x-x^2} \, dx-200 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+\frac {5376}{25} \int e^{-1-4 x-x^2} \, dx-320 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-\frac {1728}{5} \int e^{-1-4 x-x^2} \, dx-\frac {9216}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-375 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-448 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+576 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-600 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+675 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+700 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+\frac {32256}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1344 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1800 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+1875 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-\frac {10368}{5} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-3375 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+4320 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-\left (224 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx-2 \left (\left (300 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx\right )+\left (360 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx+\left (720 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx \\ & = \frac {20 e^{-1-4 x-x^2}}{x^2}-\frac {50 e^{-1-4 x-x^2}}{x}+5 x+\frac {500 e^{-1-4 x-x^2}}{(4+5 x)^2}+\frac {250 e^{-1-4 x-x^2}}{4+5 x}+\frac {2432}{25} e^3 \sqrt {\pi } \text {erf}(2+x)+40 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-140 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-200 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-320 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-\frac {9216}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-375 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-448 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+576 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-600 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+675 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+700 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+\frac {32256}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1344 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1800 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+1875 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-\frac {10368}{5} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-3375 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+4320 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+\left (96 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx-\left (160 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx+\frac {1}{25} \left (5376 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx-\frac {1}{5} \left (1728 e^3\right ) \int e^{-\frac {1}{4} (-4-2 x)^2} \, dx \\ & = \frac {20 e^{-1-4 x-x^2}}{x^2}-\frac {50 e^{-1-4 x-x^2}}{x}+5 x+\frac {500 e^{-1-4 x-x^2}}{(4+5 x)^2}+\frac {250 e^{-1-4 x-x^2}}{4+5 x}+40 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+128 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-140 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-200 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-320 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-\frac {9216}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-375 \int \frac {e^{-1-4 x-x^2}}{x} \, dx-448 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+576 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-600 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+675 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+700 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{x} \, dx+720 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+\frac {32256}{25} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1344 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-1800 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+1875 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-\frac {10368}{5} \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx-3375 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx+4320 \int \frac {e^{-1-4 x-x^2}}{4+5 x} \, dx \\ \end{align*}
Time = 8.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=\frac {5 e^{-1-4 x-x^2} \left (64+e^{1+4 x+x^2} x^3 (4+5 x)^2\right )}{x^2 (4+5 x)^2} \]
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Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
method | result | size |
risch | \(5 x +\frac {320 \,{\mathrm e}^{-x^{2}-4 x -1}}{x^{2} \left (4+5 x \right )^{2}}\) | \(28\) |
parts | \(5 x +\frac {320 \,{\mathrm e}^{-x^{2}-4 x -1}}{x^{2} \left (4+5 x \right )^{2}}\) | \(28\) |
norman | \(\frac {\left (320-240 \,{\mathrm e}^{x^{2}+4 x +1} x^{3}-128 x^{2} {\mathrm e}^{x^{2}+4 x +1}+125 \,{\mathrm e}^{x^{2}+4 x +1} x^{5}\right ) {\mathrm e}^{-x^{2}-4 x -1}}{x^{2} \left (4+5 x \right )^{2}}\) | \(67\) |
parallelrisch | \(\frac {\left (8000+3125 \,{\mathrm e}^{x^{2}+4 x +1} x^{5}-6000 \,{\mathrm e}^{x^{2}+4 x +1} x^{3}-3200 x^{2} {\mathrm e}^{x^{2}+4 x +1}\right ) {\mathrm e}^{-x^{2}-4 x -1}}{25 x^{2} \left (25 x^{2}+40 x +16\right )}\) | \(73\) |
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=\frac {5 \, {\left ({\left (25 \, x^{5} + 40 \, x^{4} + 16 \, x^{3}\right )} e^{\left (x^{2} + 4 \, x + 1\right )} + 64\right )} e^{\left (-x^{2} - 4 \, x - 1\right )}}{25 \, x^{4} + 40 \, x^{3} + 16 \, x^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=5 x + \frac {320 e^{- x^{2} - 4 x - 1}}{25 x^{4} + 40 x^{3} + 16 x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.96 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=5 \, x + \frac {96 \, {\left (5 \, x + 3\right )}}{25 \, x^{2} + 40 \, x + 16} - \frac {48 \, {\left (5 \, x + 2\right )}}{25 \, x^{2} + 40 \, x + 16} - \frac {80 \, {\left (3 \, x + 2\right )}}{25 \, x^{2} + 40 \, x + 16} + \frac {320 \, e^{\left (-x^{2} - 4 \, x\right )}}{25 \, x^{4} e + 40 \, x^{3} e + 16 \, x^{2} e} - \frac {32}{25 \, x^{2} + 40 \, x + 16} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=\frac {5 \, {\left (25 \, x^{5} e + 40 \, x^{4} e + 16 \, x^{3} e + 64 \, e^{\left (-x^{2} - 4 \, x\right )}\right )}}{25 \, x^{4} e + 40 \, x^{3} e + 16 \, x^{2} e} \]
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Time = 0.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {e^{-1-4 x-x^2} \left (-2560-11520 x-8960 x^2-3200 x^3+e^{1+4 x+x^2} \left (320 x^3+1200 x^4+1500 x^5+625 x^6\right )\right )}{64 x^3+240 x^4+300 x^5+125 x^6} \, dx=\frac {80\,x}{{\left (5\,x+4\right )}^2}+\frac {200\,x^2}{{\left (5\,x+4\right )}^2}+\frac {125\,x^3}{{\left (5\,x+4\right )}^2}+\frac {320\,{\mathrm {e}}^{-x^2-4\,x-1}}{x^2\,{\left (5\,x+4\right )}^2} \]
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