Integrand size = 17, antiderivative size = 37 \[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\frac {1}{6} e^{-2 x} \left (-3+e^{7 x}\right )^{5/3} \operatorname {Hypergeometric2F1}\left (1,\frac {29}{21},\frac {5}{7},\frac {e^{7 x}}{3}\right ) \] Output:
1/6*(-3+exp(7*x))^(5/3)*hypergeom([1, 29/21],[5/7],1/3*exp(7*x))/exp(2*x)
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.46 \[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=-\frac {e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{7},\frac {5}{7},\frac {e^{7 x}}{3}\right )}{2 \left (1-\frac {e^{7 x}}{3}\right )^{2/3}} \] Input:
Integrate[(-3 + E^(7*x))^(2/3)/E^(2*x),x]
Output:
-1/2*((-3 + E^(7*x))^(2/3)*Hypergeometric2F1[-2/3, -2/7, 5/7, E^(7*x)/3])/ (E^(2*x)*(1 - E^(7*x)/3)^(2/3))
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2679, 858, 889, 27, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^{-2 x} \left (e^{7 x}-3\right )^{2/3} \, dx\) |
\(\Big \downarrow \) 2679 |
\(\displaystyle -\int e^{-x} \left (-3+e^{7 x}\right )^{2/3}de^{-x}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \int e^{3 x} \left (e^{-7 x}-3\right )^{2/3}de^x\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {3^{2/3} \left (e^{-7 x}-3\right )^{2/3} \int \frac {e^{3 x} \left (3-e^{-7 x}\right )^{2/3}}{3^{2/3}}de^x}{\left (3-e^{-7 x}\right )^{2/3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (e^{-7 x}-3\right )^{2/3} \int e^{3 x} \left (3-e^{-7 x}\right )^{2/3}de^x}{\left (3-e^{-7 x}\right )^{2/3}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle -\frac {3^{2/3} e^{2 x} \left (e^{-7 x}-3\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{7},\frac {5}{7},\frac {e^{-7 x}}{3}\right )}{2 \left (3-e^{-7 x}\right )^{2/3}}\) |
Input:
Int[(-3 + E^(7*x))^(2/3)/E^(2*x),x]
Output:
-1/2*(3^(2/3)*E^(2*x)*(-3 + E^(-7*x))^(2/3)*Hypergeometric2F1[-2/3, -2/7, 5/7, 1/(3*E^(7*x))])/(3 - E^(-7*x))^(2/3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_ .) + (g_.)*(x_))), x_Symbol] :> With[{m = FullSimplify[d*e*(Log[F]/(g*h*Log [G]))]}, Simp[Denominator[m]/(g*h*Log[G]) Subst[Int[x^(Denominator[m] - 1 )*(a + b*F^(c*e - d*e*(f/g))*x^Numerator[m])^p, x], x, G^(h*((f + g*x)/Deno minator[m]))], x] /; LtQ[m, -1] || GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e , f, g, h, p}, x]
\[\int \left (-3+{\mathrm e}^{7 x}\right )^{\frac {2}{3}} {\mathrm e}^{-2 x}d x\]
Input:
int((-3+exp(7*x))^(2/3)/exp(2*x),x)
Output:
int((-3+exp(7*x))^(2/3)/exp(2*x),x)
\[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\int { {\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )} \,d x } \] Input:
integrate((-3+exp(7*x))^(2/3)/exp(2*x),x, algorithm="fricas")
Output:
integral((e^(7*x) - 3)^(2/3)*e^(-2*x), x)
\[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\int \left (e^{7 x} - 3\right )^{\frac {2}{3}} e^{- 2 x}\, dx \] Input:
integrate((-3+exp(7*x))**(2/3)/exp(2*x),x)
Output:
Integral((exp(7*x) - 3)**(2/3)*exp(-2*x), x)
\[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\int { {\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )} \,d x } \] Input:
integrate((-3+exp(7*x))^(2/3)/exp(2*x),x, algorithm="maxima")
Output:
integrate((e^(7*x) - 3)^(2/3)*e^(-2*x), x)
\[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\int { {\left (e^{\left (7 \, x\right )} - 3\right )}^{\frac {2}{3}} e^{\left (-2 \, x\right )} \,d x } \] Input:
integrate((-3+exp(7*x))^(2/3)/exp(2*x),x, algorithm="giac")
Output:
integrate((e^(7*x) - 3)^(2/3)*e^(-2*x), x)
Timed out. \[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\int {\mathrm {e}}^{-2\,x}\,{\left ({\mathrm {e}}^{7\,x}-3\right )}^{2/3} \,d x \] Input:
int(exp(-2*x)*(exp(7*x) - 3)^(2/3),x)
Output:
int(exp(-2*x)*(exp(7*x) - 3)^(2/3), x)
\[ \int e^{-2 x} \left (-3+e^{7 x}\right )^{2/3} \, dx=\frac {\frac {3 \left (e^{7 x}-3\right )^{\frac {2}{3}}}{8}-\frac {21 e^{2 x} \left (\int \frac {\left (e^{7 x}-3\right )^{\frac {2}{3}}}{e^{9 x}-3 e^{2 x}}d x \right )}{4}}{e^{2 x}} \] Input:
int((-3+exp(7*x))^(2/3)/exp(2*x),x)
Output:
(3*((e**(7*x) - 3)**(2/3) - 14*e**(2*x)*int((e**(7*x) - 3)**(2/3)/(e**(9*x ) - 3*e**(2*x)),x)))/(8*e**(2*x))