Integrand size = 166, antiderivative size = 29 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=\left (5-\left (21-\frac {5}{\log \left (2+e^{5-\frac {2 x}{5+x}}\right )}\right )^2\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(29)=58\).
Time = 1.89 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6820, 12, 6874, 6818} \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=\frac {625}{\log ^4\left (e^{\frac {3 x+25}{x+5}}+2\right )}-\frac {10500}{\log ^3\left (e^{\frac {3 x+25}{x+5}}+2\right )}+\frac {65900}{\log ^2\left (e^{\frac {3 x+25}{x+5}}+2\right )}-\frac {183120}{\log \left (e^{\frac {3 x+25}{x+5}}+2\right )} \]
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Rule 12
Rule 6818
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {200 e^{\frac {25+3 x}{5+x}} \left (125-1575 \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+6590 \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-9156 \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )\right )}{\left (2+e^{\frac {25+3 x}{5+x}}\right ) (5+x)^2 \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx \\ & = 200 \int \frac {e^{\frac {25+3 x}{5+x}} \left (125-1575 \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+6590 \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-9156 \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )\right )}{\left (2+e^{\frac {25+3 x}{5+x}}\right ) (5+x)^2 \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx \\ & = 200 \int \left (\frac {125 e^{\frac {25+3 x}{5+x}}}{\left (2+e^{\frac {25}{5+x}+\frac {3 x}{5+x}}\right ) (5+x)^2 \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )}-\frac {1575 e^{\frac {25+3 x}{5+x}}}{\left (2+e^{\frac {25}{5+x}+\frac {3 x}{5+x}}\right ) (5+x)^2 \log ^4\left (2+e^{\frac {25+3 x}{5+x}}\right )}+\frac {6590 e^{\frac {25+3 x}{5+x}}}{\left (2+e^{\frac {25}{5+x}+\frac {3 x}{5+x}}\right ) (5+x)^2 \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}-\frac {9156 e^{\frac {25+3 x}{5+x}}}{\left (2+e^{\frac {25}{5+x}+\frac {3 x}{5+x}}\right ) (5+x)^2 \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )}\right ) \, dx \\ & = 25000 \int \frac {e^{\frac {25+3 x}{5+x}}}{\left (2+e^{\frac {25}{5+x}+\frac {3 x}{5+x}}\right ) (5+x)^2 \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx-315000 \int \frac {e^{\frac {25+3 x}{5+x}}}{\left (2+e^{\frac {25}{5+x}+\frac {3 x}{5+x}}\right ) (5+x)^2 \log ^4\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx+1318000 \int \frac {e^{\frac {25+3 x}{5+x}}}{\left (2+e^{\frac {25}{5+x}+\frac {3 x}{5+x}}\right ) (5+x)^2 \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx-1831200 \int \frac {e^{\frac {25+3 x}{5+x}}}{\left (2+e^{\frac {25}{5+x}+\frac {3 x}{5+x}}\right ) (5+x)^2 \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx \\ & = \frac {625}{\log ^4\left (2+e^{\frac {25+3 x}{5+x}}\right )}-\frac {10500}{\log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}+\frac {65900}{\log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )}-\frac {183120}{\log \left (2+e^{\frac {25+3 x}{5+x}}\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(29)=58\).
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.14 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=-200 \left (-\frac {25}{8 \log ^4\left (2+e^{\frac {25+3 x}{5+x}}\right )}+\frac {105}{2 \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}-\frac {659}{2 \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )}+\frac {4578}{5 \log \left (2+e^{\frac {25+3 x}{5+x}}\right )}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs. \(2(28)=56\).
Time = 0.62 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.66
method | result | size |
risch | \(-\frac {5 \left (36624 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{3}-13180 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{2}+2100 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )-125\right )}{\ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{4}}\) | \(77\) |
parallelrisch | \(-\frac {-50000+14649600 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{3}-5272000 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{2}+840000 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )}{80 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{4}}\) | \(77\) |
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.62 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=-\frac {5 \, {\left (36624 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{3} - 13180 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{2} + 2100 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right ) - 125\right )}}{\log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.24 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=\frac {- 183120 \log {\left (e^{\frac {3 x + 25}{x + 5}} + 2 \right )}^{3} + 65900 \log {\left (e^{\frac {3 x + 25}{x + 5}} + 2 \right )}^{2} - 10500 \log {\left (e^{\frac {3 x + 25}{x + 5}} + 2 \right )} + 625}{\log {\left (e^{\frac {3 x + 25}{x + 5}} + 2 \right )}^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 297, normalized size of antiderivative = 10.24 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=-\frac {45780 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )^{3}}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{4}} - \frac {45780 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )^{2}}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{3}} - \frac {45780 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{2}} - \frac {45780}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )} + \frac {32950 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )^{2}}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{4}} + \frac {65900 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )}{3 \, \log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{3}} + \frac {32950}{3 \, \log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{2}} - \frac {7875 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{4}} - \frac {2625}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{3}} + \frac {625}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.62 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=-\frac {5 \, {\left (36624 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{3} - 13180 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{2} + 2100 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right ) - 125\right )}}{\log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{4}} \]
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Time = 9.50 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.48 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=\frac {65900}{{\ln \left ({\mathrm {e}}^{\frac {3\,x}{x+5}}\,{\mathrm {e}}^{\frac {25}{x+5}}+2\right )}^2}-\frac {183120}{\ln \left ({\mathrm {e}}^{\frac {3\,x}{x+5}}\,{\mathrm {e}}^{\frac {25}{x+5}}+2\right )}-\frac {10500}{{\ln \left ({\mathrm {e}}^{\frac {3\,x}{x+5}}\,{\mathrm {e}}^{\frac {25}{x+5}}+2\right )}^3}+\frac {625}{{\ln \left ({\mathrm {e}}^{\frac {3\,x}{x+5}}\,{\mathrm {e}}^{\frac {25}{x+5}}+2\right )}^4} \]
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