Integrand size = 78, antiderivative size = 18 \[ \int \frac {10240000+e^{2+x} \left (-73728000 x-36864000 x^2\right )+e^{4+2 x} \left (12441600 x^3+6220800 x^4\right )+e^{6+3 x} \left (-699840 x^5-349920 x^6\right )+e^{8+4 x} \left (13122 x^7+6561 x^8\right )}{10240000} \, dx=x+\left (2-\frac {9}{80} e^{2+x} x^2\right )^4 \]
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Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(18)=36\).
Time = 0.42 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.11, number of steps used = 58, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {12, 1607, 2227, 2207, 2225} \[ \int \frac {10240000+e^{2+x} \left (-73728000 x-36864000 x^2\right )+e^{4+2 x} \left (12441600 x^3+6220800 x^4\right )+e^{6+3 x} \left (-699840 x^5-349920 x^6\right )+e^{8+4 x} \left (13122 x^7+6561 x^8\right )}{10240000} \, dx=\frac {6561 e^{4 x+8} x^8}{40960000}-\frac {729 e^{3 x+6} x^6}{64000}+\frac {243}{800} e^{2 x+4} x^4-\frac {18}{5} e^{x+2} x^2+x \]
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Rule 12
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (10240000+e^{2+x} \left (-73728000 x-36864000 x^2\right )+e^{4+2 x} \left (12441600 x^3+6220800 x^4\right )+e^{6+3 x} \left (-699840 x^5-349920 x^6\right )+e^{8+4 x} \left (13122 x^7+6561 x^8\right )\right ) \, dx}{10240000} \\ & = x+\frac {\int e^{2+x} \left (-73728000 x-36864000 x^2\right ) \, dx}{10240000}+\frac {\int e^{4+2 x} \left (12441600 x^3+6220800 x^4\right ) \, dx}{10240000}+\frac {\int e^{6+3 x} \left (-699840 x^5-349920 x^6\right ) \, dx}{10240000}+\frac {\int e^{8+4 x} \left (13122 x^7+6561 x^8\right ) \, dx}{10240000} \\ & = x+\frac {\int e^{2+x} (-73728000-36864000 x) x \, dx}{10240000}+\frac {\int e^{6+3 x} (-699840-349920 x) x^5 \, dx}{10240000}+\frac {\int e^{8+4 x} x^7 (13122+6561 x) \, dx}{10240000}+\frac {\int e^{4+2 x} x^3 (12441600+6220800 x) \, dx}{10240000} \\ & = x+\frac {\int \left (-73728000 e^{2+x} x-36864000 e^{2+x} x^2\right ) \, dx}{10240000}+\frac {\int \left (12441600 e^{4+2 x} x^3+6220800 e^{4+2 x} x^4\right ) \, dx}{10240000}+\frac {\int \left (-699840 e^{6+3 x} x^5-349920 e^{6+3 x} x^6\right ) \, dx}{10240000}+\frac {\int \left (13122 e^{8+4 x} x^7+6561 e^{8+4 x} x^8\right ) \, dx}{10240000} \\ & = x+\frac {6561 \int e^{8+4 x} x^8 \, dx}{10240000}+\frac {6561 \int e^{8+4 x} x^7 \, dx}{5120000}-\frac {2187 \int e^{6+3 x} x^6 \, dx}{64000}-\frac {2187 \int e^{6+3 x} x^5 \, dx}{32000}+\frac {243}{400} \int e^{4+2 x} x^4 \, dx+\frac {243}{200} \int e^{4+2 x} x^3 \, dx-\frac {18}{5} \int e^{2+x} x^2 \, dx-\frac {36}{5} \int e^{2+x} x \, dx \\ & = x-\frac {36}{5} e^{2+x} x-\frac {18}{5} e^{2+x} x^2+\frac {243}{400} e^{4+2 x} x^3+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^5}{32000}-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^7}{20480000}+\frac {6561 e^{8+4 x} x^8}{40960000}-\frac {6561 \int e^{8+4 x} x^7 \, dx}{5120000}-\frac {45927 \int e^{8+4 x} x^6 \, dx}{20480000}+\frac {2187 \int e^{6+3 x} x^5 \, dx}{32000}+\frac {729 \int e^{6+3 x} x^4 \, dx}{6400}-\frac {243}{200} \int e^{4+2 x} x^3 \, dx-\frac {729}{400} \int e^{4+2 x} x^2 \, dx+\frac {36}{5} \int e^{2+x} \, dx+\frac {36}{5} \int e^{2+x} x \, dx \\ & = \frac {36 e^{2+x}}{5}+x-\frac {18}{5} e^{2+x} x^2-\frac {729}{800} e^{4+2 x} x^2+\frac {243}{800} e^{4+2 x} x^4+\frac {243 e^{6+3 x} x^4}{6400}-\frac {729 e^{6+3 x} x^6}{64000}-\frac {45927 e^{8+4 x} x^6}{81920000}+\frac {6561 e^{8+4 x} x^8}{40960000}+\frac {45927 \int e^{8+4 x} x^6 \, dx}{20480000}+\frac {137781 \int e^{8+4 x} x^5 \, dx}{40960000}-\frac {729 \int e^{6+3 x} x^4 \, dx}{6400}-\frac {243 \int e^{6+3 x} x^3 \, dx}{1600}+\frac {729}{400} \int e^{4+2 x} x \, dx+\frac {729}{400} \int e^{4+2 x} x^2 \, dx-\frac {36}{5} \int e^{2+x} \, dx \\ & = x+\frac {729}{800} e^{4+2 x} x-\frac {18}{5} e^{2+x} x^2-\frac {81 e^{6+3 x} x^3}{1600}+\frac {243}{800} e^{4+2 x} x^4+\frac {137781 e^{8+4 x} x^5}{163840000}-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}-\frac {137781 \int e^{8+4 x} x^5 \, dx}{40960000}-\frac {137781 \int e^{8+4 x} x^4 \, dx}{32768000}+\frac {243 \int e^{6+3 x} x^2 \, dx}{1600}+\frac {243 \int e^{6+3 x} x^3 \, dx}{1600}-\frac {729}{800} \int e^{4+2 x} \, dx-\frac {729}{400} \int e^{4+2 x} x \, dx \\ & = -\frac {729 e^{4+2 x}}{1600}+x-\frac {18}{5} e^{2+x} x^2+\frac {81 e^{6+3 x} x^2}{1600}+\frac {243}{800} e^{4+2 x} x^4-\frac {137781 e^{8+4 x} x^4}{131072000}-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}+\frac {137781 \int e^{8+4 x} x^3 \, dx}{32768000}+\frac {137781 \int e^{8+4 x} x^4 \, dx}{32768000}-\frac {81}{800} \int e^{6+3 x} x \, dx-\frac {243 \int e^{6+3 x} x^2 \, dx}{1600}+\frac {729}{800} \int e^{4+2 x} \, dx \\ & = x-\frac {27}{800} e^{6+3 x} x-\frac {18}{5} e^{2+x} x^2+\frac {137781 e^{8+4 x} x^3}{131072000}+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}-\frac {413343 \int e^{8+4 x} x^2 \, dx}{131072000}-\frac {137781 \int e^{8+4 x} x^3 \, dx}{32768000}+\frac {27}{800} \int e^{6+3 x} \, dx+\frac {81}{800} \int e^{6+3 x} x \, dx \\ & = \frac {9}{800} e^{6+3 x}+x-\frac {18}{5} e^{2+x} x^2-\frac {413343 e^{8+4 x} x^2}{524288000}+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}+\frac {413343 \int e^{8+4 x} x \, dx}{262144000}+\frac {413343 \int e^{8+4 x} x^2 \, dx}{131072000}-\frac {27}{800} \int e^{6+3 x} \, dx \\ & = x+\frac {413343 e^{8+4 x} x}{1048576000}-\frac {18}{5} e^{2+x} x^2+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}-\frac {413343 \int e^{8+4 x} \, dx}{1048576000}-\frac {413343 \int e^{8+4 x} x \, dx}{262144000} \\ & = -\frac {413343 e^{8+4 x}}{4194304000}+x-\frac {18}{5} e^{2+x} x^2+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}+\frac {413343 \int e^{8+4 x} \, dx}{1048576000} \\ & = x-\frac {18}{5} e^{2+x} x^2+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(18)=36\).
Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.11 \[ \int \frac {10240000+e^{2+x} \left (-73728000 x-36864000 x^2\right )+e^{4+2 x} \left (12441600 x^3+6220800 x^4\right )+e^{6+3 x} \left (-699840 x^5-349920 x^6\right )+e^{8+4 x} \left (13122 x^7+6561 x^8\right )}{10240000} \, dx=x-\frac {18}{5} e^{2+x} x^2+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs. \(2(15)=30\).
Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.50
method | result | size |
risch | \(x -\frac {18 x^{2} {\mathrm e}^{2+x}}{5}+\frac {243 \,{\mathrm e}^{4+2 x} x^{4}}{800}-\frac {729 x^{6} {\mathrm e}^{6+3 x}}{64000}+\frac {6561 x^{8} {\mathrm e}^{4 x +8}}{40960000}\) | \(45\) |
default | \(x -\frac {18 x^{2} {\mathrm e}^{2} {\mathrm e}^{x}}{5}+\frac {243 \,{\mathrm e}^{4} {\mathrm e}^{2 x} x^{4}}{800}-\frac {729 \,{\mathrm e}^{6} {\mathrm e}^{3 x} x^{6}}{64000}+\frac {6561 \,{\mathrm e}^{8} {\mathrm e}^{4 x} x^{8}}{40960000}\) | \(51\) |
parallelrisch | \(x -\frac {18 x^{2} {\mathrm e}^{2} {\mathrm e}^{x}}{5}+\frac {243 \,{\mathrm e}^{4} {\mathrm e}^{2 x} x^{4}}{800}-\frac {729 \,{\mathrm e}^{6} {\mathrm e}^{3 x} x^{6}}{64000}+\frac {6561 \,{\mathrm e}^{8} {\mathrm e}^{4 x} x^{8}}{40960000}\) | \(51\) |
parts | \(x -\frac {18 x^{2} {\mathrm e}^{2} {\mathrm e}^{x}}{5}+\frac {243 \,{\mathrm e}^{4} {\mathrm e}^{2 x} x^{4}}{800}-\frac {729 \,{\mathrm e}^{6} {\mathrm e}^{3 x} x^{6}}{64000}+\frac {6561 \,{\mathrm e}^{8} {\mathrm e}^{4 x} x^{8}}{40960000}\) | \(51\) |
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (17) = 34\).
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.44 \[ \int \frac {10240000+e^{2+x} \left (-73728000 x-36864000 x^2\right )+e^{4+2 x} \left (12441600 x^3+6220800 x^4\right )+e^{6+3 x} \left (-699840 x^5-349920 x^6\right )+e^{8+4 x} \left (13122 x^7+6561 x^8\right )}{10240000} \, dx=\frac {6561}{40960000} \, x^{8} e^{\left (4 \, x + 8\right )} - \frac {729}{64000} \, x^{6} e^{\left (3 \, x + 6\right )} + \frac {243}{800} \, x^{4} e^{\left (2 \, x + 4\right )} - \frac {18}{5} \, x^{2} e^{\left (x + 2\right )} + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (17) = 34\).
Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.33 \[ \int \frac {10240000+e^{2+x} \left (-73728000 x-36864000 x^2\right )+e^{4+2 x} \left (12441600 x^3+6220800 x^4\right )+e^{6+3 x} \left (-699840 x^5-349920 x^6\right )+e^{8+4 x} \left (13122 x^7+6561 x^8\right )}{10240000} \, dx=\frac {6561 x^{8} e^{8} e^{4 x}}{40960000} - \frac {729 x^{6} e^{6} e^{3 x}}{64000} + \frac {243 x^{4} e^{4} e^{2 x}}{800} - \frac {18 x^{2} e^{2} e^{x}}{5} + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (17) = 34\).
Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.78 \[ \int \frac {10240000+e^{2+x} \left (-73728000 x-36864000 x^2\right )+e^{4+2 x} \left (12441600 x^3+6220800 x^4\right )+e^{6+3 x} \left (-699840 x^5-349920 x^6\right )+e^{8+4 x} \left (13122 x^7+6561 x^8\right )}{10240000} \, dx=\frac {6561}{40960000} \, x^{8} e^{\left (4 \, x + 8\right )} - \frac {729}{64000} \, x^{6} e^{\left (3 \, x + 6\right )} + \frac {243}{800} \, x^{4} e^{\left (2 \, x + 4\right )} - \frac {18}{5} \, {\left (x^{2} e^{2} - 2 \, x e^{2} + 2 \, e^{2}\right )} e^{x} - \frac {36}{5} \, {\left (x e^{2} - e^{2}\right )} e^{x} + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (17) = 34\).
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.44 \[ \int \frac {10240000+e^{2+x} \left (-73728000 x-36864000 x^2\right )+e^{4+2 x} \left (12441600 x^3+6220800 x^4\right )+e^{6+3 x} \left (-699840 x^5-349920 x^6\right )+e^{8+4 x} \left (13122 x^7+6561 x^8\right )}{10240000} \, dx=\frac {6561}{40960000} \, x^{8} e^{\left (4 \, x + 8\right )} - \frac {729}{64000} \, x^{6} e^{\left (3 \, x + 6\right )} + \frac {243}{800} \, x^{4} e^{\left (2 \, x + 4\right )} - \frac {18}{5} \, x^{2} e^{\left (x + 2\right )} + x \]
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Time = 14.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.44 \[ \int \frac {10240000+e^{2+x} \left (-73728000 x-36864000 x^2\right )+e^{4+2 x} \left (12441600 x^3+6220800 x^4\right )+e^{6+3 x} \left (-699840 x^5-349920 x^6\right )+e^{8+4 x} \left (13122 x^7+6561 x^8\right )}{10240000} \, dx=x-\frac {18\,x^2\,{\mathrm {e}}^{x+2}}{5}+\frac {243\,x^4\,{\mathrm {e}}^{2\,x+4}}{800}-\frac {729\,x^6\,{\mathrm {e}}^{3\,x+6}}{64000}+\frac {6561\,x^8\,{\mathrm {e}}^{4\,x+8}}{40960000} \]
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