Integrand size = 42, antiderivative size = 24 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {e^x x}{4 \left (5+\frac {4 (4-x)^2}{x^2}\right )} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.82 (sec) , antiderivative size = 544, normalized size of antiderivative = 22.67, number of steps used = 30, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1608, 6820, 12, 6874, 2225, 2207, 2208, 2209, 2302} \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {\left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )}{3645}-\frac {176 \left (2-i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )}{3645}+\frac {7 i e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )}{81 \sqrt {5}}+\frac {8}{405} e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )+\frac {\left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )}{3645}-\frac {176 \left (2+i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )}{3645}-\frac {7 i e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )}{81 \sqrt {5}}+\frac {8}{405} e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )+\frac {e^x x}{36}+\frac {8 e^x}{81}-\frac {176 \left (2-i \sqrt {5}\right ) e^x}{405 \left (-9 x+8 \left (2-i \sqrt {5}\right )\right )}+\frac {8 e^x}{45 \left (-9 x+8 \left (2-i \sqrt {5}\right )\right )}-\frac {176 \left (2+i \sqrt {5}\right ) e^x}{405 \left (-9 x+8 \left (2+i \sqrt {5}\right )\right )}+\frac {8 e^x}{45 \left (-9 x+8 \left (2+i \sqrt {5}\right )\right )} \]
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Rule 12
Rule 1608
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2302
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x x^2 \left (192-23 x^2+9 x^3\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx \\ & = \int \frac {e^x x^2 \left (192-23 x^2+9 x^3\right )}{4 \left (64-32 x+9 x^2\right )^2} \, dx \\ & = \frac {1}{4} \int \frac {e^x x^2 \left (192-23 x^2+9 x^3\right )}{\left (64-32 x+9 x^2\right )^2} \, dx \\ & = \frac {1}{4} \int \left (\frac {41 e^x}{81}+\frac {e^x x}{9}+\frac {2048 e^x (-4+11 x)}{81 \left (64-32 x+9 x^2\right )^2}+\frac {64 e^x (-39+7 x)}{81 \left (64-32 x+9 x^2\right )}\right ) \, dx \\ & = \frac {1}{36} \int e^x x \, dx+\frac {41 \int e^x \, dx}{324}+\frac {16}{81} \int \frac {e^x (-39+7 x)}{64-32 x+9 x^2} \, dx+\frac {512}{81} \int \frac {e^x (-4+11 x)}{\left (64-32 x+9 x^2\right )^2} \, dx \\ & = \frac {41 e^x}{324}+\frac {e^x x}{36}-\frac {\int e^x \, dx}{36}+\frac {16}{81} \int \left (\frac {\left (7+\frac {239 i}{8 \sqrt {5}}\right ) e^x}{-32-16 i \sqrt {5}+18 x}+\frac {\left (7-\frac {239 i}{8 \sqrt {5}}\right ) e^x}{-32+16 i \sqrt {5}+18 x}\right ) \, dx+\frac {512}{81} \int \left (-\frac {4 e^x}{\left (64-32 x+9 x^2\right )^2}+\frac {11 e^x x}{\left (64-32 x+9 x^2\right )^2}\right ) \, dx \\ & = \frac {8 e^x}{81}+\frac {e^x x}{36}-\frac {2048}{81} \int \frac {e^x}{\left (64-32 x+9 x^2\right )^2} \, dx+\frac {5632}{81} \int \frac {e^x x}{\left (64-32 x+9 x^2\right )^2} \, dx+\frac {1}{405} \left (2 \left (280-239 i \sqrt {5}\right )\right ) \int \frac {e^x}{-32+16 i \sqrt {5}+18 x} \, dx+\frac {1}{405} \left (2 \left (280+239 i \sqrt {5}\right )\right ) \int \frac {e^x}{-32-16 i \sqrt {5}+18 x} \, dx \\ & = \frac {8 e^x}{81}+\frac {e^x x}{36}+\frac {\left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {\left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{3645}-\frac {2048}{81} \int \left (-\frac {81 e^x}{320 \left (32+16 i \sqrt {5}-18 x\right )^2}+\frac {81 i e^x}{5120 \sqrt {5} \left (32+16 i \sqrt {5}-18 x\right )}-\frac {81 e^x}{320 \left (-32+16 i \sqrt {5}+18 x\right )^2}+\frac {81 i e^x}{5120 \sqrt {5} \left (-32+16 i \sqrt {5}+18 x\right )}\right ) \, dx+\frac {5632}{81} \int \left (-\frac {9 \left (32+16 i \sqrt {5}\right ) e^x}{640 \left (32+16 i \sqrt {5}-18 x\right )^2}+\frac {9 i e^x}{320 \sqrt {5} \left (32+16 i \sqrt {5}-18 x\right )}-\frac {9 \left (32-16 i \sqrt {5}\right ) e^x}{640 \left (-32+16 i \sqrt {5}+18 x\right )^2}+\frac {9 i e^x}{320 \sqrt {5} \left (-32+16 i \sqrt {5}+18 x\right )}\right ) \, dx \\ & = \frac {8 e^x}{81}+\frac {e^x x}{36}+\frac {\left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {\left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {32}{5} \int \frac {e^x}{\left (32+16 i \sqrt {5}-18 x\right )^2} \, dx+\frac {32}{5} \int \frac {e^x}{\left (-32+16 i \sqrt {5}+18 x\right )^2} \, dx-\frac {(2 i) \int \frac {e^x}{32+16 i \sqrt {5}-18 x} \, dx}{5 \sqrt {5}}-\frac {(2 i) \int \frac {e^x}{-32+16 i \sqrt {5}+18 x} \, dx}{5 \sqrt {5}}+\frac {(88 i) \int \frac {e^x}{32+16 i \sqrt {5}-18 x} \, dx}{45 \sqrt {5}}+\frac {(88 i) \int \frac {e^x}{-32+16 i \sqrt {5}+18 x} \, dx}{45 \sqrt {5}}-\frac {1}{45} \left (704 \left (2-i \sqrt {5}\right )\right ) \int \frac {e^x}{\left (-32+16 i \sqrt {5}+18 x\right )^2} \, dx-\frac {1}{45} \left (704 \left (2+i \sqrt {5}\right )\right ) \int \frac {e^x}{\left (32+16 i \sqrt {5}-18 x\right )^2} \, dx \\ & = \frac {8 e^x}{81}+\frac {8 e^x}{45 \left (8 \left (2-i \sqrt {5}\right )-9 x\right )}-\frac {176 \left (2-i \sqrt {5}\right ) e^x}{405 \left (8 \left (2-i \sqrt {5}\right )-9 x\right )}+\frac {8 e^x}{45 \left (8 \left (2+i \sqrt {5}\right )-9 x\right )}-\frac {176 \left (2+i \sqrt {5}\right ) e^x}{405 \left (8 \left (2+i \sqrt {5}\right )-9 x\right )}+\frac {e^x x}{36}+\frac {7 i e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{81 \sqrt {5}}+\frac {\left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{3645}-\frac {7 i e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{81 \sqrt {5}}+\frac {\left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{3645}-\frac {16}{45} \int \frac {e^x}{32+16 i \sqrt {5}-18 x} \, dx+\frac {16}{45} \int \frac {e^x}{-32+16 i \sqrt {5}+18 x} \, dx-\frac {1}{405} \left (352 \left (2-i \sqrt {5}\right )\right ) \int \frac {e^x}{-32+16 i \sqrt {5}+18 x} \, dx+\frac {1}{405} \left (352 \left (2+i \sqrt {5}\right )\right ) \int \frac {e^x}{32+16 i \sqrt {5}-18 x} \, dx \\ & = \frac {8 e^x}{81}+\frac {8 e^x}{45 \left (8 \left (2-i \sqrt {5}\right )-9 x\right )}-\frac {176 \left (2-i \sqrt {5}\right ) e^x}{405 \left (8 \left (2-i \sqrt {5}\right )-9 x\right )}+\frac {8 e^x}{45 \left (8 \left (2+i \sqrt {5}\right )-9 x\right )}-\frac {176 \left (2+i \sqrt {5}\right ) e^x}{405 \left (8 \left (2+i \sqrt {5}\right )-9 x\right )}+\frac {e^x x}{36}+\frac {8}{405} e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )+\frac {7 i e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{81 \sqrt {5}}-\frac {176 \left (2-i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {\left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {8}{405} e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )-\frac {7 i e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{81 \sqrt {5}}-\frac {176 \left (2+i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {\left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{3645} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {e^x x^3}{4 \left (64-32 x+9 x^2\right )} \]
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Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\) | \(20\) |
norman | \(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\) | \(20\) |
risch | \(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\) | \(20\) |
parallelrisch | \(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\) | \(20\) |
default | \(-\frac {8 \,{\mathrm e}^{x} \left (x +16\right )}{15 \left (9 x^{2}-32 x +64\right )}-\frac {23 \,{\mathrm e}^{x}}{324}+\frac {184 \,{\mathrm e}^{x} \left (79 x -176\right )}{3645 \left (9 x^{2}-32 x +64\right )}+\frac {\left (9 x +55\right ) {\mathrm e}^{x}}{324}-\frac {128 \,{\mathrm e}^{x} \left (59 x -316\right )}{3645 \left (9 x^{2}-32 x +64\right )}\) | \(76\) |
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Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {x^{3} e^{x}}{4 \, {\left (9 \, x^{2} - 32 \, x + 64\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {x^{3} e^{x}}{36 x^{2} - 128 x + 256} \]
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {x^{3} e^{x}}{4 \, {\left (9 \, x^{2} - 32 \, x + 64\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {x^{3} e^{x}}{4 \, {\left (9 \, x^{2} - 32 \, x + 64\right )}} \]
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Time = 13.53 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {x^3\,{\mathrm {e}}^x}{4\,\left (9\,x^2-32\,x+64\right )} \]
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