Optimal. Leaf size=37 \[ \frac {1}{6} e^{-2 x} \left (e^{7 x}-3\right )^{5/3} \, _2F_1\left (1,\frac {29}{21};\frac {5}{7};\frac {e^{7 x}}{3}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.54, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2249, 335, 365, 364} \[ -\frac {3^{2/3} e^{-2 x} \left (e^{7 x}-3\right )^{2/3} \text {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{7},\frac {5}{7},\frac {e^{7 x}}{3}\right )}{2 \left (3-e^{7 x}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 335
Rule 364
Rule 365
Rule 2249
Rubi steps
\begin {align*} \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx &=-\operatorname {Subst}\left (\int \left (-3+\frac {1}{x^7}\right )^{2/3} x \, dx,x,e^{-x}\right )\\ &=\operatorname {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} \operatorname {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} \, _2F_1\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*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 54, normalized size = 1.46 \[ -\frac {e^{-2 x} \left (e^{7 x}-3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {2}{7};\frac {5}{7};\frac {e^{7 x}}{3}\right )}{2 \left (1-\frac {e^{7 x}}{3}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \left (-3+{\mathrm e}^{7 x}\right )^{\frac {2}{3}} {\mathrm e}^{-2 x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\mathrm {e}}^{-2\,x}\,{\left ({\mathrm {e}}^{7\,x}-3\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e^{7 x} - 3\right )^{\frac {2}{3}} e^{- 2 x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________