\(\int \frac {e^{2 x}}{(3-e^{x/2})^{3/4}} \, dx\) [534]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 21, antiderivative size = 73 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=-216 \sqrt [4]{3-e^{x/2}}+\frac {216}{5} \left (3-e^{x/2}\right )^{5/4}-8 \left (3-e^{x/2}\right )^{9/4}+\frac {8}{13} \left (3-e^{x/2}\right )^{13/4} \]

[Out]

-216*(3-exp(1/2*x))^(1/4)+216/5*(3-exp(1/2*x))^(5/4)-8*(3-exp(1/2*x))^(9/4)+8/13*(3-exp(1/2*x))^(13/4)

Rubi [A] (verified)

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} \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=\frac {8}{13} \left (3-e^{x/2}\right )^{13/4}-8 \left (3-e^{x/2}\right )^{9/4}+\frac {216}{5} \left (3-e^{x/2}\right )^{5/4}-216 \sqrt [4]{3-e^{x/2}} \]

[In]

Int[E^(2*x)/(3 - E^(x/2))^(3/4),x]

[Out]

-216*(3 - E^(x/2))^(1/4) + (216*(3 - E^(x/2))^(5/4))/5 - 8*(3 - E^(x/2))^(9/4) + (8*(3 - E^(x/2))^(13/4))/13

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2280

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[g*h*(Log[G]/(d*e*Log[F]))]}, Dist[Denominator[m]*(G^(f*h - c*g*(h/d))/(d*e*Log[F])), Subst
[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^(e*((c + d*x)/Denominator[m]))], x] /; LeQ[m, -
1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

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*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.60 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=-\frac {8}{65} \sqrt [4]{3-e^{x/2}} \left (1152+96 e^{x/2}+20 e^x+5 e^{3 x/2}\right ) \]

[In]

Integrate[E^(2*x)/(3 - E^(x/2))^(3/4),x]

[Out]

(-8*(3 - E^(x/2))^(1/4)*(1152 + 96*E^(x/2) + 20*E^x + 5*E^((3*x)/2)))/65

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.51

method result size
risch \(\frac {8 \left (5 \,{\mathrm e}^{\frac {3 x}{2}}+20 \,{\mathrm e}^{x}+96 \,{\mathrm e}^{\frac {x}{2}}+1152\right ) \left (-3+{\mathrm e}^{\frac {x}{2}}\right )}{65 \left (3-{\mathrm e}^{\frac {x}{2}}\right )^{\frac {3}{4}}}\) \(37\)

[In]

int(exp(2*x)/(3-exp(1/2*x))^(3/4),x,method=_RETURNVERBOSE)

[Out]

8/65/(3-exp(1/2*x))^(3/4)*(5*exp(3/2*x)+20*exp(x)+96*exp(1/2*x)+1152)*(-3+exp(1/2*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.41 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=-\frac {8}{65} \, {\left (5 \, e^{\left (\frac {3}{2} \, x\right )} + 96 \, e^{\left (\frac {1}{2} \, x\right )} + 20 \, e^{x} + 1152\right )} {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} \]

[In]

integrate(exp(2*x)/(3-exp(1/2*x))^(3/4),x, algorithm="fricas")

[Out]

-8/65*(5*e^(3/2*x) + 96*e^(1/2*x) + 20*e^x + 1152)*(-e^(1/2*x) + 3)^(1/4)

Sympy [A] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=\frac {8 \left (3 - e^{\frac {x}{2}}\right )^{\frac {13}{4}}}{13} - 8 \left (3 - e^{\frac {x}{2}}\right )^{\frac {9}{4}} + \frac {216 \left (3 - e^{\frac {x}{2}}\right )^{\frac {5}{4}}}{5} - 216 \sqrt [4]{3 - e^{\frac {x}{2}}} \]

[In]

integrate(exp(2*x)/(3-exp(1/2*x))**(3/4),x)

[Out]

8*(3 - exp(x/2))**(13/4)/13 - 8*(3 - exp(x/2))**(9/4) + 216*(3 - exp(x/2))**(5/4)/5 - 216*(3 - exp(x/2))**(1/4
)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=\frac {8}{13} \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {13}{4}} - 8 \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {9}{4}} + \frac {216}{5} \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {5}{4}} - 216 \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} \]

[In]

integrate(exp(2*x)/(3-exp(1/2*x))^(3/4),x, algorithm="maxima")

[Out]

8/13*(-e^(1/2*x) + 3)^(13/4) - 8*(-e^(1/2*x) + 3)^(9/4) + 216/5*(-e^(1/2*x) + 3)^(5/4) - 216*(-e^(1/2*x) + 3)^
(1/4)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=-\frac {8}{13} \, {\left (e^{\left (\frac {1}{2} \, x\right )} - 3\right )}^{3} {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} - 8 \, {\left (e^{\left (\frac {1}{2} \, x\right )} - 3\right )}^{2} {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} + \frac {216}{5} \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {5}{4}} - 216 \, {\left (-e^{\left (\frac {1}{2} \, x\right )} + 3\right )}^{\frac {1}{4}} \]

[In]

integrate(exp(2*x)/(3-exp(1/2*x))^(3/4),x, algorithm="giac")

[Out]

-8/13*(e^(1/2*x) - 3)^3*(-e^(1/2*x) + 3)^(1/4) - 8*(e^(1/2*x) - 3)^2*(-e^(1/2*x) + 3)^(1/4) + 216/5*(-e^(1/2*x
) + 3)^(5/4) - 216*(-e^(1/2*x) + 3)^(1/4)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.41 \[ \int \frac {e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx=-{\left (3-{\mathrm {e}}^{x/2}\right )}^{1/4}\,\left (\frac {768\,{\mathrm {e}}^{x/2}}{65}+\frac {8\,{\mathrm {e}}^{\frac {3\,x}{2}}}{13}+\frac {32\,{\mathrm {e}}^x}{13}+\frac {9216}{65}\right ) \]

[In]

int(exp(2*x)/(3 - exp(x/2))^(3/4),x)

[Out]

-(3 - exp(x/2))^(1/4)*((768*exp(x/2))/65 + (8*exp((3*x)/2))/13 + (32*exp(x))/13 + 9216/65)