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

Optimal. Leaf size=73 \[ \frac{8}{13} \left (3-e^{x/2}\right )^{13/4}-8 \left (3-e^{x/2}\right )^{9/4}+\frac{216}{5} \left (3-e^{x/2}\right )^{5/4}-216 \sqrt [4]{3-e^{x/2}} \]

[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

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Rubi [A]  time = 0.0474549, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2248, 43} \[ \frac{8}{13} \left (3-e^{x/2}\right )^{13/4}-8 \left (3-e^{x/2}\right )^{9/4}+\frac{216}{5} \left (3-e^{x/2}\right )^{5/4}-216 \sqrt [4]{3-e^{x/2}} \]

Antiderivative was successfully verified.

[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 2248

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]

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{e^{2 x}}{\left (3-e^{x/2}\right )^{3/4}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{(3-x)^{3/4}} \, dx,x,e^{x/2}\right )\\ &=2 \operatorname{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]  time = 0.0195386, size = 44, normalized size = 0.6 \[ -\frac{8}{65} \sqrt [4]{3-e^{x/2}} \left (96 e^{x/2}+20 e^x+5 e^{3 x/2}+1152\right ) \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.019, size = 37, normalized size = 0.5 \begin{align*}{\frac{8}{65} \left ( 5\,{{\rm e}^{3/2\,x}}+20\,{{\rm e}^{x}}+96\,{{\rm e}^{x/2}}+1152 \right ) \left ( -3+{{\rm e}^{{\frac{x}{2}}}} \right ) \left ( 3-{{\rm e}^{{\frac{x}{2}}}} \right ) ^{-{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

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Maxima [A]  time = 0.931849, size = 66, normalized size = 0.9 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.83639, size = 101, normalized size = 1.38 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{2 x}}{\left (3 - e^{\frac{x}{2}}\right )^{\frac{3}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(2*x)/(3 - exp(x/2))**(3/4), x)

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Giac [A]  time = 1.14824, size = 88, normalized size = 1.21 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)