Integrand size = 17, antiderivative size = 37 \[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\frac {1}{6} e^{-2 x} \left (-3+e^{7 x}\right )^{5/3} \operatorname {Hypergeometric2F1}\left (1,\frac {29}{21},\frac {5}{7},\frac {e^{7 x}}{3}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2281, 342, 372, 371} \[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=-\frac {3^{2/3} e^{-2 x} \left (e^{7 x}-3\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{7},\frac {5}{7},\frac {e^{7 x}}{3}\right )}{2 \left (3-e^{7 x}\right )^{2/3}} \]
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Rule 342
Rule 371
Rule 372
Rule 2281
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \left (-3+\frac {1}{x^7}\right )^{2/3} x \, dx,x,e^{-x}\right ) \\ & = \text {Subst}\left (\int \frac {\left (-3+x^7\right )^{2/3}}{x^3} \, dx,x,e^x\right ) \\ & = \frac {\left (-3+e^{7 x}\right )^{2/3} \text {Subst}\left (\int \frac {\left (1-\frac {x^7}{3}\right )^{2/3}}{x^3} \, dx,x,e^x\right )}{\left (1-\frac {e^{7 x}}{3}\right )^{2/3}} \\ & = -\frac {3^{2/3} e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{7},\frac {5}{7},\frac {e^{7 x}}{3}\right )}{2 \left (3-e^{7 x}\right )^{2/3}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.46 \[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=-\frac {e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{7},\frac {5}{7},\frac {e^{7 x}}{3}\right )}{2 \left (1-\frac {e^{7 x}}{3}\right )^{2/3}} \]
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\[\int \left (-3+{\mathrm e}^{7 x}\right )^{\frac {2}{3}} {\mathrm e}^{-2 x}d x\]
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\[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\int { {\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )} \,d x } \]
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\[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\int \left (e^{7 x} - 3\right )^{\frac {2}{3}} e^{- 2 x}\, dx \]
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\[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\int { {\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )} \,d x } \]
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\[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\int { {\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )} \,d x } \]
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Timed out. \[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\int {\mathrm {e}}^{-2\,x}\,{\left ({\mathrm {e}}^{7\,x}-3\right )}^{2/3} \,d x \]
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