Integrand size = 21, antiderivative size = 73 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=-216 \sqrt [4]{3-e^{x/2}}+\frac {216}{5} \left (3-e^{x/2}\right )^{5/4}-8 \left (3-e^{x/2}\right )^{9/4}+\frac {8}{13} \left (3-e^{x/2}\right )^{13/4} \]
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Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2280, 45} \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=\frac {8}{13} \left (3-e^{x/2}\right )^{13/4}-8 \left (3-e^{x/2}\right )^{9/4}+\frac {216}{5} \left (3-e^{x/2}\right )^{5/4}-216 \sqrt [4]{3-e^{x/2}} \]
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Rule 45
Rule 2280
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^3}{(3-x)^{3/4}} \, dx,x,e^{x/2}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {27}{(3-x)^{3/4}}-27 \sqrt [4]{3-x}+9 (3-x)^{5/4}-(3-x)^{9/4}\right ) \, dx,x,e^{x/2}\right ) \\ & = -216 \sqrt [4]{3-e^{x/2}}+\frac {216}{5} \left (3-e^{x/2}\right )^{5/4}-8 \left (3-e^{x/2}\right )^{9/4}+\frac {8}{13} \left (3-e^{x/2}\right )^{13/4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.60 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=-\frac {8}{65} \sqrt [4]{3-e^{x/2}} \left (1152+96 e^{x/2}+20 e^x+5 e^{3 x/2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.51
method | result | size |
risch | \(\frac {8 \left (5 \,{\mathrm e}^{\frac {3 x}{2}}+20 \,{\mathrm e}^{x}+96 \,{\mathrm e}^{\frac {x}{2}}+1152\right ) \left (-3+{\mathrm e}^{\frac {x}{2}}\right )}{65 \left (3-{\mathrm e}^{\frac {x}{2}}\right )^{\frac {3}{4}}}\) | \(37\) |
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.41 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=-\frac {8}{65} \, {\left (5 \, e^{\left (\frac {3}{2} \, x\right )} + 96 \, e^{\left (\frac {1}{2} \, x\right )} + 20 \, e^{x} + 1152\right )} {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} \]
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Time = 0.92 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=\frac {8 \left (3 - e^{\frac {x}{2}}\right )^{\frac {13}{4}}}{13} - 8 \left (3 - e^{\frac {x}{2}}\right )^{\frac {9}{4}} + \frac {216 \left (3 - e^{\frac {x}{2}}\right )^{\frac {5}{4}}}{5} - 216 \sqrt [4]{3 - e^{\frac {x}{2}}} \]
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Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=\frac {8}{13} \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {13}{4}} - 8 \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {9}{4}} + \frac {216}{5} \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {5}{4}} - 216 \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} \]
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Time = 0.32 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=-\frac {8}{13} \, {\left (e^{\left (\frac {1}{2} \, x\right )} - 3\right )}^{3} {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} - 8 \, {\left (e^{\left (\frac {1}{2} \, x\right )} - 3\right )}^{2} {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} + \frac {216}{5} \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {5}{4}} - 216 \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} \]
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Time = 0.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.41 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=-{\left (3-{\mathrm {e}}^{x/2}\right )}^{1/4}\,\left (\frac {768\,{\mathrm {e}}^{x/2}}{65}+\frac {8\,{\mathrm {e}}^{\frac {3\,x}{2}}}{13}+\frac {32\,{\mathrm {e}}^x}{13}+\frac {9216}{65}\right ) \]
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