Optimal. Leaf size=18 \[ \frac {81 x^4}{25 \left (-4+\frac {5 e^x}{4}\right )^4} \]
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Rubi [A] time = 5.90, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 226, number of rules used = 16, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6688, 12, 6742, 2185, 2184, 2190, 2531, 6609, 2282, 6589, 2191, 2279, 2391, 36, 29, 31} \begin {gather*} \frac {20736 x^4}{25 \left (16-5 e^x\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 6589
Rule 6609
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {82944 \left (16+5 e^x (-1+x)\right ) x^3}{25 \left (16-5 e^x\right )^5} \, dx\\ &=\frac {82944}{25} \int \frac {\left (16+5 e^x (-1+x)\right ) x^3}{\left (16-5 e^x\right )^5} \, dx\\ &=\frac {82944}{25} \int \left (-\frac {(-1+x) x^3}{\left (-16+5 e^x\right )^4}-\frac {16 x^4}{\left (-16+5 e^x\right )^5}\right ) \, dx\\ &=-\left (\frac {82944}{25} \int \frac {(-1+x) x^3}{\left (-16+5 e^x\right )^4} \, dx\right )-\frac {1327104}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^5} \, dx\\ &=\frac {82944}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^4} \, dx-\frac {82944}{25} \int \left (-\frac {x^3}{\left (-16+5 e^x\right )^4}+\frac {x^4}{\left (-16+5 e^x\right )^4}\right ) \, dx-\frac {82944}{5} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^5} \, dx\\ &=\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}-\frac {5184}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^3} \, dx+\frac {5184}{5} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^4} \, dx-\frac {82944}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^4} \, dx\\ &=\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {1728 x^4}{25 \left (16-5 e^x\right )^3}+\frac {324}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^2} \, dx-\frac {324}{5} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^3} \, dx+\frac {5184}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^3} \, dx+\frac {6912}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^3} \, dx-\frac {5184}{5} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^4} \, dx\\ &=\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {162 x^4}{25 \left (16-5 e^x\right )^2}-\frac {81}{100} \int \frac {x^4}{-16+5 e^x} \, dx+\frac {81}{20} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^2} \, dx-\frac {324}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^2} \, dx-\frac {432}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^2} \, dx-\frac {648}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^2} \, dx+\frac {324}{5} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^3} \, dx+\frac {432}{5} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^3} \, dx-\frac {6912}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^3} \, dx\\ &=-\frac {216 x^3}{25 \left (16-5 e^x\right )^2}+\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {81 x^4}{100 \left (16-5 e^x\right )}+\frac {81 x^5}{8000}-\frac {81}{320} \int \frac {e^x x^4}{-16+5 e^x} \, dx+\frac {81}{100} \int \frac {x^4}{-16+5 e^x} \, dx+\frac {27}{25} \int \frac {x^3}{-16+5 e^x} \, dx+\frac {81}{50} \int \frac {x^3}{-16+5 e^x} \, dx+\frac {81}{25} \int \frac {x^3}{-16+5 e^x} \, dx-\frac {81}{20} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^2} \, dx-\frac {27}{5} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^2} \, dx-\frac {81}{10} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^2} \, dx+\frac {432}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^2} \, dx+\frac {648}{25} \int \frac {x^2}{\left (-16+5 e^x\right )^2} \, dx+\frac {648}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^2} \, dx-\frac {432}{5} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^3} \, dx\\ &=-\frac {27 x^3}{10 \left (16-5 e^x\right )}-\frac {297 x^4}{3200}+\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}-\frac {81 x^4 \log \left (1-\frac {5 e^x}{16}\right )}{1600}+\frac {81}{400} \int x^3 \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {81}{320} \int \frac {e^x x^4}{-16+5 e^x} \, dx+\frac {27}{80} \int \frac {e^x x^3}{-16+5 e^x} \, dx+\frac {81}{160} \int \frac {e^x x^3}{-16+5 e^x} \, dx+\frac {81}{80} \int \frac {e^x x^3}{-16+5 e^x} \, dx-\frac {27}{25} \int \frac {x^3}{-16+5 e^x} \, dx-\frac {81}{50} \int \frac {x^2}{-16+5 e^x} \, dx-\frac {81}{50} \int \frac {x^3}{-16+5 e^x} \, dx-\frac {81}{25} \int \frac {x^2}{-16+5 e^x} \, dx-\frac {81}{25} \int \frac {x^3}{-16+5 e^x} \, dx-\frac {243}{50} \int \frac {x^2}{-16+5 e^x} \, dx+\frac {27}{5} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^2} \, dx+\frac {81}{10} \int \frac {e^x x^2}{\left (-16+5 e^x\right )^2} \, dx+\frac {81}{10} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^2} \, dx-\frac {648}{25} \int \frac {x^2}{\left (-16+5 e^x\right )^2} \, dx\\ &=\frac {81 x^2}{50 \left (16-5 e^x\right )}+\frac {81 x^3}{400}+\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {297}{800} x^3 \log \left (1-\frac {5 e^x}{16}\right )-\frac {81}{400} x^3 \text {Li}_2\left (\frac {5 e^x}{16}\right )-\frac {81}{400} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {81}{400} \int x^3 \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {243}{800} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {27}{80} \int \frac {e^x x^3}{-16+5 e^x} \, dx-\frac {81}{160} \int \frac {e^x x^2}{-16+5 e^x} \, dx-\frac {81}{160} \int \frac {e^x x^3}{-16+5 e^x} \, dx-\frac {243}{400} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \int x^2 \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx-\frac {81}{80} \int \frac {e^x x^2}{-16+5 e^x} \, dx-\frac {81}{80} \int \frac {e^x x^3}{-16+5 e^x} \, dx-\frac {243}{160} \int \frac {e^x x^2}{-16+5 e^x} \, dx+\frac {81}{50} \int \frac {x^2}{-16+5 e^x} \, dx+\frac {81}{25} \int \frac {x}{-16+5 e^x} \, dx+\frac {81}{25} \int \frac {x^2}{-16+5 e^x} \, dx+\frac {243}{50} \int \frac {x^2}{-16+5 e^x} \, dx-\frac {81}{10} \int \frac {e^x x^2}{\left (-16+5 e^x\right )^2} \, dx\\ &=-\frac {81 x^2}{800}+\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}-\frac {243}{400} x^2 \log \left (1-\frac {5 e^x}{16}\right )+\frac {891}{800} x^2 \text {Li}_2\left (\frac {5 e^x}{16}\right )+\frac {243}{400} x^2 \text {Li}_3\left (\frac {5 e^x}{16}\right )+\frac {81}{400} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {81}{400} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {243}{800} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {81}{200} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {81}{200} \int x \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx+\frac {81}{160} \int \frac {e^x x^2}{-16+5 e^x} \, dx+\frac {243}{400} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {243}{400} \int x \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx-\frac {243}{400} \int x^2 \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx+\frac {81}{80} \int \frac {e^x x}{-16+5 e^x} \, dx+\frac {81}{80} \int \frac {e^x x^2}{-16+5 e^x} \, dx-\frac {243}{200} \int x \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx-\frac {243}{200} \int x \text {Li}_3\left (\frac {5 e^x}{16}\right ) \, dx+\frac {243}{160} \int \frac {e^x x^2}{-16+5 e^x} \, dx-\frac {81}{25} \int \frac {x}{-16+5 e^x} \, dx\\ &=\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {81}{400} x \log \left (1-\frac {5 e^x}{16}\right )-\frac {243}{200} x \text {Li}_2\left (\frac {5 e^x}{16}\right )-\frac {891}{400} x \text {Li}_3\left (\frac {5 e^x}{16}\right )-\frac {243}{200} x \text {Li}_4\left (\frac {5 e^x}{16}\right )-\frac {81}{400} \int \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {81}{400} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {81}{400} \int \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx-\frac {81}{200} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {81}{200} \int \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx+\frac {81}{200} \int x \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx+\frac {81}{200} \int \text {Li}_3\left (\frac {5 e^x}{16}\right ) \, dx-\frac {243}{400} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \int \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \int x \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \int \text {Li}_3\left (\frac {5 e^x}{16}\right ) \, dx-\frac {81}{80} \int \frac {e^x x}{-16+5 e^x} \, dx+\frac {243}{200} \int x \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx+\frac {243}{200} \int \text {Li}_3\left (\frac {5 e^x}{16}\right ) \, dx+\frac {243}{200} \int x \text {Li}_3\left (\frac {5 e^x}{16}\right ) \, dx+\frac {243}{200} \int \text {Li}_4\left (\frac {5 e^x}{16}\right ) \, dx\\ &=\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {81}{400} \int \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {81}{400} \int \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx-\frac {81}{400} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )+\frac {81}{400} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {81}{200} \int \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx-\frac {81}{200} \int \text {Li}_3\left (\frac {5 e^x}{16}\right ) \, dx+\frac {81}{200} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )+\frac {81}{200} \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{400} \int \text {Li}_2\left (\frac {5 e^x}{16}\right ) \, dx-\frac {243}{400} \int \text {Li}_3\left (\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )+\frac {243}{400} \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{200} \int \text {Li}_3\left (\frac {5 e^x}{16}\right ) \, dx-\frac {243}{200} \int \text {Li}_4\left (\frac {5 e^x}{16}\right ) \, dx+\frac {243}{200} \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )+\frac {243}{200} \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )\\ &=\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {81}{400} \text {Li}_2\left (\frac {5 e^x}{16}\right )+\frac {243}{200} \text {Li}_3\left (\frac {5 e^x}{16}\right )+\frac {891}{400} \text {Li}_4\left (\frac {5 e^x}{16}\right )+\frac {243}{200} \text {Li}_5\left (\frac {5 e^x}{16}\right )+\frac {81}{400} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {81}{400} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {81}{200} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {81}{200} \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{400} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{400} \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{200} \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{200} \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )\\ &=\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 16, normalized size = 0.89 \begin {gather*} \frac {20736 x^4}{25 \left (-16+5 e^x\right )^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 47, normalized size = 2.61 \begin {gather*} \frac {81 \, x^{4}}{25 \, {\left (e^{\left (4 \, x + 4 \, \log \left (\frac {5}{4}\right )\right )} - 16 \, e^{\left (3 \, x + 3 \, \log \left (\frac {5}{4}\right )\right )} + 96 \, e^{\left (2 \, x + 2 \, \log \left (\frac {5}{4}\right )\right )} - 256 \, e^{\left (x + \log \left (\frac {5}{4}\right )\right )} + 256\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 31, normalized size = 1.72 \begin {gather*} \frac {20736 \, x^{4}}{25 \, {\left (625 \, e^{\left (4 \, x\right )} - 8000 \, e^{\left (3 \, x\right )} + 38400 \, e^{\left (2 \, x\right )} - 81920 \, e^{x} + 65536\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 14, normalized size = 0.78
method | result | size |
risch | \(\frac {81 x^{4}}{25 \left (\frac {5 \,{\mathrm e}^{x}}{4}-4\right )^{4}}\) | \(14\) |
norman | \(\frac {81 x^{4}}{25 \left ({\mathrm e}^{\ln \left (\frac {5}{4}\right )+x}-4\right )^{4}}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 31, normalized size = 1.72 \begin {gather*} \frac {20736 \, x^{4}}{25 \, {\left (625 \, e^{\left (4 \, x\right )} - 8000 \, e^{\left (3 \, x\right )} + 38400 \, e^{\left (2 \, x\right )} - 81920 \, e^{x} + 65536\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 31, normalized size = 1.72 \begin {gather*} \frac {81\,x^4}{25\,\left (150\,{\mathrm {e}}^{2\,x}-\frac {125\,{\mathrm {e}}^{3\,x}}{4}+\frac {625\,{\mathrm {e}}^{4\,x}}{256}-320\,{\mathrm {e}}^x+256\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 31, normalized size = 1.72 \begin {gather*} \frac {20736 x^{4}}{15625 e^{4 x} - 200000 e^{3 x} + 960000 e^{2 x} - 2048000 e^{x} + 1638400} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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