Integrand size = 83, antiderivative size = 25 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=25 \left (4-e^{x+4 x^2}+\frac {25 x}{-25+x}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(25)=50\).
Time = 0.33 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.36, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {6874, 2276, 2268, 37, 2266, 2235, 2274, 2272} \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=-1450 e^{4 x^2+x}+25 e^{8 x^2+2 x}+\frac {31250 e^{4 x^2+x}}{25-x}+\frac {25 (100-29 x)^2}{(25-x)^2} \]
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Rule 37
Rule 2235
Rule 2266
Rule 2268
Rule 2272
Rule 2274
Rule 2276
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (50 e^{2 x (1+4 x)} (1+8 x)-\frac {31250 (-100+29 x)}{(-25+x)^3}-\frac {50 e^{x+4 x^2} \left (1875+19175 x-6571 x^2+232 x^3\right )}{(-25+x)^2}\right ) \, dx \\ & = 50 \int e^{2 x (1+4 x)} (1+8 x) \, dx-50 \int \frac {e^{x+4 x^2} \left (1875+19175 x-6571 x^2+232 x^3\right )}{(-25+x)^2} \, dx-31250 \int \frac {-100+29 x}{(-25+x)^3} \, dx \\ & = \frac {25 (100-29 x)^2}{(25-x)^2}+50 \int e^{2 x+8 x^2} (1+8 x) \, dx-50 \int \left (5029 e^{x+4 x^2}-\frac {625 e^{x+4 x^2}}{(-25+x)^2}+\frac {125625 e^{x+4 x^2}}{-25+x}+232 e^{x+4 x^2} x\right ) \, dx \\ & = 25 e^{2 x+8 x^2}+\frac {25 (100-29 x)^2}{(25-x)^2}-11600 \int e^{x+4 x^2} x \, dx+31250 \int \frac {e^{x+4 x^2}}{(-25+x)^2} \, dx-251450 \int e^{x+4 x^2} \, dx-6281250 \int \frac {e^{x+4 x^2}}{-25+x} \, dx \\ & = -1450 e^{x+4 x^2}+25 e^{2 x+8 x^2}+\frac {25 (100-29 x)^2}{(25-x)^2}+\frac {31250 e^{x+4 x^2}}{25-x}+1450 \int e^{x+4 x^2} \, dx+250000 \int e^{x+4 x^2} \, dx-\frac {251450 \int e^{\frac {1}{16} (1+8 x)^2} \, dx}{\sqrt [16]{e}} \\ & = -1450 e^{x+4 x^2}+25 e^{2 x+8 x^2}+\frac {25 (100-29 x)^2}{(25-x)^2}+\frac {31250 e^{x+4 x^2}}{25-x}-\frac {125725 \sqrt {\pi } \text {erfi}\left (\frac {1}{4} (1+8 x)\right )}{2 \sqrt [16]{e}}+\frac {1450 \int e^{\frac {1}{16} (1+8 x)^2} \, dx}{\sqrt [16]{e}}+\frac {250000 \int e^{\frac {1}{16} (1+8 x)^2} \, dx}{\sqrt [16]{e}} \\ & = -1450 e^{x+4 x^2}+25 e^{2 x+8 x^2}+\frac {25 (100-29 x)^2}{(25-x)^2}+\frac {31250 e^{x+4 x^2}}{25-x} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(25)=50\).
Time = 10.83 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=\frac {25 \left (e^{2 x (1+4 x)} (-25+x)^2+625 (-825+58 x)-2 e^{x+4 x^2} \left (2500-825 x+29 x^2\right )\right )}{(-25+x)^2} \]
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Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96
method | result | size |
risch | \(\frac {906250 x -12890625}{x^{2}-50 x +625}+25 \,{\mathrm e}^{2 x \left (1+4 x \right )}-\frac {50 \left (29 x -100\right ) {\mathrm e}^{x \left (1+4 x \right )}}{x -25}\) | \(49\) |
parts | \(25 \,{\mathrm e}^{8 x^{2}+2 x}+\frac {9765625}{\left (x -25\right )^{2}}+\frac {906250}{x -25}+\frac {-1450 \,{\mathrm e}^{4 x^{2}+x} x +5000 \,{\mathrm e}^{4 x^{2}+x}}{x -25}\) | \(56\) |
norman | \(\frac {906250 x +15625 \,{\mathrm e}^{8 x^{2}+2 x}+41250 \,{\mathrm e}^{4 x^{2}+x} x -1450 \,{\mathrm e}^{4 x^{2}+x} x^{2}-1250 \,{\mathrm e}^{8 x^{2}+2 x} x +25 \,{\mathrm e}^{8 x^{2}+2 x} x^{2}-125000 \,{\mathrm e}^{4 x^{2}+x}-12890625}{\left (x -25\right )^{2}}\) | \(86\) |
parallelrisch | \(\frac {906250 x +15625 \,{\mathrm e}^{8 x^{2}+2 x}+41250 \,{\mathrm e}^{4 x^{2}+x} x -1450 \,{\mathrm e}^{4 x^{2}+x} x^{2}-1250 \,{\mathrm e}^{8 x^{2}+2 x} x +25 \,{\mathrm e}^{8 x^{2}+2 x} x^{2}-125000 \,{\mathrm e}^{4 x^{2}+x}-12890625}{x^{2}-50 x +625}\) | \(91\) |
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=\frac {25 \, {\left ({\left (x^{2} - 50 \, x + 625\right )} e^{\left (8 \, x^{2} + 2 \, x\right )} - 2 \, {\left (29 \, x^{2} - 825 \, x + 2500\right )} e^{\left (4 \, x^{2} + x\right )} + 36250 \, x - 515625\right )}}{x^{2} - 50 \, x + 625} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=- \frac {12890625 - 906250 x}{x^{2} - 50 x + 625} + \frac {\left (5000 - 1450 x\right ) e^{4 x^{2} + x} + \left (25 x - 625\right ) e^{8 x^{2} + 2 x}}{x - 25} \]
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=\frac {453125 \, {\left (2 \, x - 25\right )}}{x^{2} - 50 \, x + 625} + \frac {25 \, {\left ({\left (x - 25\right )} e^{\left (8 \, x^{2} + 2 \, x\right )} - 2 \, {\left (29 \, x - 100\right )} e^{\left (4 \, x^{2} + x\right )}\right )}}{x - 25} - \frac {1562500}{x^{2} - 50 \, x + 625} \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.60 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=\frac {25 \, {\left (x^{2} e^{\left (8 \, x^{2} + 2 \, x\right )} - 58 \, x^{2} e^{\left (4 \, x^{2} + x\right )} - 50 \, x e^{\left (8 \, x^{2} + 2 \, x\right )} + 1650 \, x e^{\left (4 \, x^{2} + x\right )} + 36250 \, x + 625 \, e^{\left (8 \, x^{2} + 2 \, x\right )} - 5000 \, e^{\left (4 \, x^{2} + x\right )} - 515625\right )}}{x^{2} - 50 \, x + 625} \]
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Time = 13.54 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {3125000-906250 x+e^{x+4 x^2} \left (2343750+23875000 x-9172500 x^2+618550 x^3-11600 x^4\right )+e^{2 x+8 x^2} \left (-781250-6156250 x+746250 x^2-29950 x^3+400 x^4\right )}{-15625+1875 x-75 x^2+x^3} \, dx=25\,{\mathrm {e}}^{8\,x^2+2\,x}-1450\,{\mathrm {e}}^{4\,x^2+x}-\frac {x\,\left (31250\,{\mathrm {e}}^{4\,x^2+x}-906250\right )-781250\,{\mathrm {e}}^{4\,x^2+x}+12890625}{{\left (x-25\right )}^2} \]
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