Integrand size = 86, antiderivative size = 23 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=\left (4+e^{\frac {2 e^2}{3 x}}\right ) \left (1+x+x^2\right )^2 \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.37 (sec) , antiderivative size = 297, normalized size of antiderivative = 12.91, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {12, 14, 6874, 2237, 2241, 2240, 2245, 2250} \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=-\frac {8}{243} e^6 \left (9-e^2\right ) \operatorname {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \operatorname {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{27} e^4 \left (9-2 e^2\right ) \operatorname {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )+\frac {4}{3} e^2 \operatorname {ExpIntegralEi}\left (\frac {2 e^2}{3 x}\right )+4 x^4+\frac {2}{9} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}} x^3+8 x^3+\frac {2}{27} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}+2} x^2+\frac {1}{3} \left (9-2 e^2\right ) e^{\frac {2 e^2}{3 x}} x^2+12 x^2+\frac {4}{81} \left (9-e^2\right ) e^{\frac {2 e^2}{3 x}+4} x+2 \left (1-e^2\right ) e^{\frac {2 e^2}{3 x}} x+\frac {2}{9} \left (9-2 e^2\right ) e^{\frac {2 e^2}{3 x}+2} x+8 x+e^{\frac {2 e^2}{3 x}}+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right ) \]
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Rule 12
Rule 14
Rule 2237
Rule 2240
Rule 2241
Rule 2245
Rule 2250
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{x^2} \, dx \\ & = \frac {1}{3} \int \left (24 \left (1+3 x+3 x^2+2 x^3\right )+\frac {2 e^{\frac {2 e^2}{3 x}} \left (1+x+x^2\right ) \left (-e^2-e^2 x+\left (3-e^2\right ) x^2+6 x^3\right )}{x^2}\right ) \, dx \\ & = \frac {2}{3} \int \frac {e^{\frac {2 e^2}{3 x}} \left (1+x+x^2\right ) \left (-e^2-e^2 x+\left (3-e^2\right ) x^2+6 x^3\right )}{x^2} \, dx+8 \int \left (1+3 x+3 x^2+2 x^3\right ) \, dx \\ & = 8 x+12 x^2+8 x^3+4 x^4+\frac {2}{3} \int \left (-3 e^{\frac {2 e^2}{3 x}} \left (-1+e^2\right )-\frac {e^{2+\frac {2 e^2}{3 x}}}{x^2}-\frac {2 e^{2+\frac {2 e^2}{3 x}}}{x}-e^{\frac {2 e^2}{3 x}} \left (-9+2 e^2\right ) x-e^{\frac {2 e^2}{3 x}} \left (-9+e^2\right ) x^2+6 e^{\frac {2 e^2}{3 x}} x^3\right ) \, dx \\ & = 8 x+12 x^2+8 x^3+4 x^4-\frac {2}{3} \int \frac {e^{2+\frac {2 e^2}{3 x}}}{x^2} \, dx-\frac {4}{3} \int \frac {e^{2+\frac {2 e^2}{3 x}}}{x} \, dx+4 \int e^{\frac {2 e^2}{3 x}} x^3 \, dx+\frac {1}{3} \left (2 \left (9-2 e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} x \, dx+\left (2 \left (1-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} \, dx+\frac {1}{3} \left (2 \left (9-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} x^2 \, dx \\ & = e^{\frac {2 e^2}{3 x}}+8 x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )+\frac {1}{9} \left (2 e^2 \left (9-2 e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} \, dx+\frac {1}{3} \left (4 e^2 \left (1-e^2\right )\right ) \int \frac {e^{\frac {2 e^2}{3 x}}}{x} \, dx+\frac {1}{27} \left (4 e^2 \left (9-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} x \, dx \\ & = e^{\frac {2 e^2}{3 x}}+8 x+\frac {2}{9} e^{2+\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+\frac {2}{27} e^{2+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )+\frac {1}{27} \left (4 e^4 \left (9-2 e^2\right )\right ) \int \frac {e^{\frac {2 e^2}{3 x}}}{x} \, dx+\frac {1}{81} \left (4 e^4 \left (9-e^2\right )\right ) \int e^{\frac {2 e^2}{3 x}} \, dx \\ & = e^{\frac {2 e^2}{3 x}}+8 x+\frac {2}{9} e^{2+\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+\frac {4}{81} e^{4+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+\frac {2}{27} e^{2+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{27} e^4 \left (9-2 e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right )+\frac {1}{243} \left (8 e^6 \left (9-e^2\right )\right ) \int \frac {e^{\frac {2 e^2}{3 x}}}{x} \, dx \\ & = e^{\frac {2 e^2}{3 x}}+8 x+\frac {2}{9} e^{2+\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x+2 e^{\frac {2 e^2}{3 x}} \left (1-e^2\right ) x+\frac {4}{81} e^{4+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x+12 x^2+\frac {1}{3} e^{\frac {2 e^2}{3 x}} \left (9-2 e^2\right ) x^2+\frac {2}{27} e^{2+\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^2+8 x^3+\frac {2}{9} e^{\frac {2 e^2}{3 x}} \left (9-e^2\right ) x^3+4 x^4+\frac {4}{3} e^2 \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{27} e^4 \left (9-2 e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {4}{3} e^2 \left (1-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )-\frac {8}{243} e^6 \left (9-e^2\right ) \text {Ei}\left (\frac {2 e^2}{3 x}\right )+\frac {64}{81} e^8 \Gamma \left (-4,-\frac {2 e^2}{3 x}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=\left (4+e^{\frac {2 e^2}{3 x}}\right ) \left (1+x+x^2\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(19)=38\).
Time = 0.47 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22
method | result | size |
risch | \(4 x^{4}+8 x^{3}+12 x^{2}+8 x +4+\frac {\left (3 x^{4}+6 x^{3}+9 x^{2}+6 x +3\right ) {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{3}\) | \(51\) |
parallelrisch | \({\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{4}+4 x^{4}+2 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{3}+8 x^{3}+3 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{2}+12 x^{2}+2 x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}+8 x +{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}\) | \(77\) |
norman | \(\frac {x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}+{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{5}+8 x^{2}+12 x^{3}+8 x^{4}+4 x^{5}+2 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{2}+3 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{3}+2 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}} x^{4}}{x}\) | \(87\) |
derivativedivides | \(4 x^{4}+8 x^{3}+12 x^{2}+8 x -\frac {{\mathrm e}^{-2} \left (-162 \,{\mathrm e}^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}+128 \,{\mathrm e}^{10} \left (-\frac {81 x^{4} {\mathrm e}^{-8} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{64}-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{16}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{24}\right )+288 \,{\mathrm e}^{8} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )-32 \,{\mathrm e}^{10} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )+432 \,{\mathrm e}^{6} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )-96 \,{\mathrm e}^{8} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )+216 \,{\mathrm e}^{4} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )-216 \,{\mathrm e}^{6} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )+216 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )}{162}\) | \(438\) |
default | \(4 x^{4}+8 x^{3}+12 x^{2}+8 x -\frac {{\mathrm e}^{-2} \left (-162 \,{\mathrm e}^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}+128 \,{\mathrm e}^{10} \left (-\frac {81 x^{4} {\mathrm e}^{-8} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{64}-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{16}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{24}\right )+288 \,{\mathrm e}^{8} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )-32 \,{\mathrm e}^{10} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )+432 \,{\mathrm e}^{6} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )-96 \,{\mathrm e}^{8} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )+216 \,{\mathrm e}^{4} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )-216 \,{\mathrm e}^{6} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )+216 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )}{162}\) | \(438\) |
parts | \(4 x^{4}+8 x^{3}+12 x^{2}+8 x +\frac {{\mathrm e}^{-2} \left (81 \,{\mathrm e}^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}-64 \,{\mathrm e}^{10} \left (-\frac {81 x^{4} {\mathrm e}^{-8} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{64}-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{32}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{16}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{24}\right )-144 \,{\mathrm e}^{8} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )+16 \,{\mathrm e}^{10} \left (-\frac {9 x^{3} {\mathrm e}^{-6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{6}\right )-216 \,{\mathrm e}^{6} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )+48 \,{\mathrm e}^{8} \left (-\frac {9 x^{2} {\mathrm e}^{-4} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{8}-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{4}-\frac {\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )}{2}\right )-108 \,{\mathrm e}^{4} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )-108 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )+108 \,{\mathrm e}^{6} \left (-\frac {3 x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{2}}{3 x}}}{2}-\operatorname {Ei}_{1}\left (-\frac {2 \,{\mathrm e}^{2}}{3 x}\right )\right )\right )}{81}\) | \(438\) |
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=4 \, x^{4} + 8 \, x^{3} + 12 \, x^{2} + {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )} e^{\left (\frac {2 \, e^{2}}{3 \, x}\right )} + 8 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=4 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + \left (x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1\right ) e^{\frac {2 e^{2}}{3 x}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 5.65 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=4 \, x^{4} + 8 \, x^{3} + 12 \, x^{2} + \frac {4}{3} \, {\rm Ei}\left (\frac {2 \, e^{2}}{3 \, x}\right ) e^{2} + \frac {4}{3} \, e^{4} \Gamma \left (-1, -\frac {2 \, e^{2}}{3 \, x}\right ) - \frac {4}{3} \, e^{2} \Gamma \left (-1, -\frac {2 \, e^{2}}{3 \, x}\right ) - \frac {16}{27} \, e^{6} \Gamma \left (-2, -\frac {2 \, e^{2}}{3 \, x}\right ) + \frac {8}{3} \, e^{4} \Gamma \left (-2, -\frac {2 \, e^{2}}{3 \, x}\right ) + \frac {16}{81} \, e^{8} \Gamma \left (-3, -\frac {2 \, e^{2}}{3 \, x}\right ) - \frac {16}{9} \, e^{6} \Gamma \left (-3, -\frac {2 \, e^{2}}{3 \, x}\right ) + \frac {64}{81} \, e^{8} \Gamma \left (-4, -\frac {2 \, e^{2}}{3 \, x}\right ) + 8 \, x + e^{\left (\frac {2 \, e^{2}}{3 \, x}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.39 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx=x^{4} {\left (\frac {8 \, e^{10}}{x} + \frac {2 \, e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x} + \frac {12 \, e^{10}}{x^{2}} + \frac {3 \, e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x^{2}} + \frac {8 \, e^{10}}{x^{3}} + \frac {2 \, e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x^{3}} + \frac {e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}}{x^{4}} + 4 \, e^{10} + e^{\left (\frac {2 \, e^{2}}{3 \, x} + 10\right )}\right )} e^{\left (-10\right )} \]
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Time = 11.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.00 \[ \int \frac {24 x^2+72 x^3+72 x^4+48 x^5+e^{\frac {2 e^2}{3 x}} \left (6 x^2+18 x^3+18 x^4+12 x^5+e^2 \left (-2-4 x-6 x^2-4 x^3-2 x^4\right )\right )}{3 x^2} \, dx={\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+x\,\left (2\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+8\right )+x^4\,\left ({\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+4\right )+x^3\,\left (2\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+8\right )+x^2\,\left (3\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^2}{3\,x}}+12\right ) \]
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