3.711 \(\int \frac{e^{6 x}}{(9-e^x)^{5/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2}{7} \left (9-e^x\right )^{7/2}-18 \left (9-e^x\right )^{5/2}+540 \left (9-e^x\right )^{3/2}-14580 \sqrt{9-e^x}-\frac{65610}{\sqrt{9-e^x}}+\frac{39366}{\left (9-e^x\right )^{3/2}} \]

[Out]

39366/(9 - E^x)^(3/2) - 65610/Sqrt[9 - E^x] - 14580*Sqrt[9 - E^x] + 540*(9 - E^x)^(3/2) - 18*(9 - E^x)^(5/2) +
(2*(9 - E^x)^(7/2))/7

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Rubi [A]  time = 0.0451012, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2248, 43} \[ \frac{2}{7} \left (9-e^x\right )^{7/2}-18 \left (9-e^x\right )^{5/2}+540 \left (9-e^x\right )^{3/2}-14580 \sqrt{9-e^x}-\frac{65610}{\sqrt{9-e^x}}+\frac{39366}{\left (9-e^x\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(6*x)/(9 - E^x)^(5/2),x]

[Out]

39366/(9 - E^x)^(3/2) - 65610/Sqrt[9 - E^x] - 14580*Sqrt[9 - E^x] + 540*(9 - E^x)^(3/2) - 18*(9 - E^x)^(5/2) +
(2*(9 - E^x)^(7/2))/7

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^{6 x}}{\left (9-e^x\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{x^5}{(9-x)^{5/2}} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{59049}{(9-x)^{5/2}}-\frac{32805}{(9-x)^{3/2}}+\frac{7290}{\sqrt{9-x}}-810 \sqrt{9-x}+45 (9-x)^{3/2}-(9-x)^{5/2}\right ) \, dx,x,e^x\right )\\ &=\frac{39366}{\left (9-e^x\right )^{3/2}}-\frac{65610}{\sqrt{9-e^x}}-14580 \sqrt{9-e^x}+540 \left (9-e^x\right )^{3/2}-18 \left (9-e^x\right )^{5/2}+\frac{2}{7} \left (9-e^x\right )^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0268025, size = 48, normalized size = 0.59 \[ -\frac{2 \left (-839808 e^x+23328 e^{2 x}+432 e^{3 x}+18 e^{4 x}+e^{5 x}+5038848\right )}{7 \left (9-e^x\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(6*x)/(9 - E^x)^(5/2),x]

[Out]

(-2*(5038848 - 839808*E^x + 23328*E^(2*x) + 432*E^(3*x) + 18*E^(4*x) + E^(5*x)))/(7*(9 - E^x)^(3/2))

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Maple [A]  time = 0.057, size = 62, normalized size = 0.8 \begin{align*} 39366\, \left ( 9-{{\rm e}^{x}} \right ) ^{-3/2}+540\, \left ( 9-{{\rm e}^{x}} \right ) ^{3/2}-18\, \left ( 9-{{\rm e}^{x}} \right ) ^{5/2}+{\frac{2}{7} \left ( 9-{{\rm e}^{x}} \right ) ^{{\frac{7}{2}}}}-65610\,{\frac{1}{\sqrt{9-{{\rm e}^{x}}}}}-14580\,\sqrt{9-{{\rm e}^{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(6*x)/(9-exp(x))^(5/2),x)

[Out]

39366/(9-exp(x))^(3/2)+540*(9-exp(x))^(3/2)-18*(9-exp(x))^(5/2)+2/7*(9-exp(x))^(7/2)-65610/(9-exp(x))^(1/2)-14
580*(9-exp(x))^(1/2)

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Maxima [A]  time = 0.983339, size = 82, normalized size = 1.01 \begin{align*} \frac{2}{7} \,{\left (-e^{x} + 9\right )}^{\frac{7}{2}} - 18 \,{\left (-e^{x} + 9\right )}^{\frac{5}{2}} + 540 \,{\left (-e^{x} + 9\right )}^{\frac{3}{2}} - 14580 \, \sqrt{-e^{x} + 9} - \frac{65610}{\sqrt{-e^{x} + 9}} + \frac{39366}{{\left (-e^{x} + 9\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)/(9-exp(x))^(5/2),x, algorithm="maxima")

[Out]

2/7*(-e^x + 9)^(7/2) - 18*(-e^x + 9)^(5/2) + 540*(-e^x + 9)^(3/2) - 14580*sqrt(-e^x + 9) - 65610/sqrt(-e^x + 9
) + 39366/(-e^x + 9)^(3/2)

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Fricas [A]  time = 0.863059, size = 163, normalized size = 2.01 \begin{align*} -\frac{2 \,{\left (e^{\left (5 \, x\right )} + 18 \, e^{\left (4 \, x\right )} + 432 \, e^{\left (3 \, x\right )} + 23328 \, e^{\left (2 \, x\right )} - 839808 \, e^{x} + 5038848\right )} \sqrt{-e^{x} + 9}}{7 \,{\left (e^{\left (2 \, x\right )} - 18 \, e^{x} + 81\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)/(9-exp(x))^(5/2),x, algorithm="fricas")

[Out]

-2/7*(e^(5*x) + 18*e^(4*x) + 432*e^(3*x) + 23328*e^(2*x) - 839808*e^x + 5038848)*sqrt(-e^x + 9)/(e^(2*x) - 18*
e^x + 81)

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Sympy [A]  time = 60.0249, size = 61, normalized size = 0.75 \begin{align*} \frac{2 \left (9 - e^{x}\right )^{\frac{7}{2}}}{7} - 18 \left (9 - e^{x}\right )^{\frac{5}{2}} + 540 \left (9 - e^{x}\right )^{\frac{3}{2}} - 14580 \sqrt{9 - e^{x}} - \frac{65610}{\sqrt{9 - e^{x}}} + \frac{39366}{\left (9 - e^{x}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)/(9-exp(x))**(5/2),x)

[Out]

2*(9 - exp(x))**(7/2)/7 - 18*(9 - exp(x))**(5/2) + 540*(9 - exp(x))**(3/2) - 14580*sqrt(9 - exp(x)) - 65610/sq
rt(9 - exp(x)) + 39366/(9 - exp(x))**(3/2)

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Giac [A]  time = 1.28495, size = 101, normalized size = 1.25 \begin{align*} -\frac{2}{7} \,{\left (e^{x} - 9\right )}^{3} \sqrt{-e^{x} + 9} - 18 \,{\left (e^{x} - 9\right )}^{2} \sqrt{-e^{x} + 9} + 540 \,{\left (-e^{x} + 9\right )}^{\frac{3}{2}} - 14580 \, \sqrt{-e^{x} + 9} - \frac{13122 \,{\left (5 \, e^{x} - 42\right )}}{{\left (e^{x} - 9\right )} \sqrt{-e^{x} + 9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(6*x)/(9-exp(x))^(5/2),x, algorithm="giac")

[Out]

-2/7*(e^x - 9)^3*sqrt(-e^x + 9) - 18*(e^x - 9)^2*sqrt(-e^x + 9) + 540*(-e^x + 9)^(3/2) - 14580*sqrt(-e^x + 9)
- 13122*(5*e^x - 42)/((e^x - 9)*sqrt(-e^x + 9))