Integrand size = 69, antiderivative size = 26 \[ \int \frac {5832+440 e^x+8 e^{2 x}+e^{2 e^{5 e^x}+5 e^x} \left (91125 e^x+6750 e^{2 x}+125 e^{3 x}\right )}{5832+432 e^x+8 e^{2 x}} \, dx=\frac {25}{16} e^{2 e^{5 e^x}}-\frac {1}{27+e^x}+x \]
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Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.072, Rules used = {2320, 12, 6874, 2225, 907} \[ \int \frac {5832+440 e^x+8 e^{2 x}+e^{2 e^{5 e^x}+5 e^x} \left (91125 e^x+6750 e^{2 x}+125 e^{3 x}\right )}{5832+432 e^x+8 e^{2 x}} \, dx=x+\frac {25}{16} e^{2 e^{5 e^x}}-\frac {1}{e^x+27} \]
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Rule 12
Rule 907
Rule 2225
Rule 2320
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {5832+440 x+8 x^2+125 e^{2 e^{5 x}+5 x} x (27+x)^2}{8 x (27+x)^2} \, dx,x,e^x\right ) \\ & = \frac {1}{8} \text {Subst}\left (\int \frac {5832+440 x+8 x^2+125 e^{2 e^{5 x}+5 x} x (27+x)^2}{x (27+x)^2} \, dx,x,e^x\right ) \\ & = \frac {1}{8} \text {Subst}\left (\int \left (125 e^{2 e^{5 x}+5 x}+\frac {8 \left (729+55 x+x^2\right )}{x (27+x)^2}\right ) \, dx,x,e^x\right ) \\ & = \frac {125}{8} \text {Subst}\left (\int e^{2 e^{5 x}+5 x} \, dx,x,e^x\right )+\text {Subst}\left (\int \frac {729+55 x+x^2}{x (27+x)^2} \, dx,x,e^x\right ) \\ & = \frac {25}{8} \text {Subst}\left (\int e^{2 x} \, dx,x,e^{5 e^x}\right )+\text {Subst}\left (\int \left (\frac {1}{x}+\frac {1}{(27+x)^2}\right ) \, dx,x,e^x\right ) \\ & = \frac {25}{16} e^{2 e^{5 e^x}}-\frac {1}{27+e^x}+x \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {5832+440 e^x+8 e^{2 x}+e^{2 e^{5 e^x}+5 e^x} \left (91125 e^x+6750 e^{2 x}+125 e^{3 x}\right )}{5832+432 e^x+8 e^{2 x}} \, dx=\frac {1}{8} \left (\frac {25}{2} e^{2 e^{5 e^x}}-\frac {8}{27+e^x}+8 x\right ) \]
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Time = 0.39 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81
method | result | size |
risch | \(x -\frac {1}{27+{\mathrm e}^{x}}+\frac {25 \,{\mathrm e}^{2 \,{\mathrm e}^{5 \,{\mathrm e}^{x}}}}{16}\) | \(21\) |
parallelrisch | \(\frac {25 \,{\mathrm e}^{2 \,{\mathrm e}^{5 \,{\mathrm e}^{x}}} {\mathrm e}^{x}+16 \,{\mathrm e}^{x} x +675 \,{\mathrm e}^{2 \,{\mathrm e}^{5 \,{\mathrm e}^{x}}}+432 x -16}{432+16 \,{\mathrm e}^{x}}\) | \(41\) |
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {5832+440 e^x+8 e^{2 x}+e^{2 e^{5 e^x}+5 e^x} \left (91125 e^x+6750 e^{2 x}+125 e^{3 x}\right )}{5832+432 e^x+8 e^{2 x}} \, dx=\frac {{\left (16 \, {\left (x e^{x} + 27 \, x - 1\right )} e^{\left (5 \, e^{x}\right )} + 25 \, {\left (e^{x} + 27\right )} e^{\left (5 \, e^{x} + 2 \, e^{\left (5 \, e^{x}\right )}\right )}\right )} e^{\left (-5 \, e^{x}\right )}}{16 \, {\left (e^{x} + 27\right )}} \]
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Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {5832+440 e^x+8 e^{2 x}+e^{2 e^{5 e^x}+5 e^x} \left (91125 e^x+6750 e^{2 x}+125 e^{3 x}\right )}{5832+432 e^x+8 e^{2 x}} \, dx=x + \frac {25 e^{2 e^{5 e^{x}}}}{16} - \frac {1}{e^{x} + 27} \]
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none
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {5832+440 e^x+8 e^{2 x}+e^{2 e^{5 e^x}+5 e^x} \left (91125 e^x+6750 e^{2 x}+125 e^{3 x}\right )}{5832+432 e^x+8 e^{2 x}} \, dx=x - \frac {1}{e^{x} + 27} + \frac {25}{16} \, e^{\left (2 \, e^{\left (5 \, e^{x}\right )}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92 \[ \int \frac {5832+440 e^x+8 e^{2 x}+e^{2 e^{5 e^x}+5 e^x} \left (91125 e^x+6750 e^{2 x}+125 e^{3 x}\right )}{5832+432 e^x+8 e^{2 x}} \, dx=\frac {16 \, x e^{\left (x + 5 \, e^{x}\right )} + 432 \, x e^{\left (5 \, e^{x}\right )} + 25 \, e^{\left (x + 5 \, e^{x} + 2 \, e^{\left (5 \, e^{x}\right )}\right )} - 16 \, e^{\left (5 \, e^{x}\right )} + 675 \, e^{\left (5 \, e^{x} + 2 \, e^{\left (5 \, e^{x}\right )}\right )}}{16 \, {\left (e^{\left (x + 5 \, e^{x}\right )} + 27 \, e^{\left (5 \, e^{x}\right )}\right )}} \]
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Time = 12.98 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {5832+440 e^x+8 e^{2 x}+e^{2 e^{5 e^x}+5 e^x} \left (91125 e^x+6750 e^{2 x}+125 e^{3 x}\right )}{5832+432 e^x+8 e^{2 x}} \, dx=x+\frac {25\,{\mathrm {e}}^{2\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}}}{16}-\frac {1}{{\mathrm {e}}^x+27} \]
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