3.8.11 \(\int \frac {e^{6 x}}{(9-e^x)^{5/2}} \, dx\) [711]

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

[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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Rubi [A]
time = 0.03, antiderivative size = 81, normalized size of antiderivative = 1.00, 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 = {2280, 45} \begin {gather*} \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}} \end {gather*}

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 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*} \int \frac {e^{6 x}}{\left (9-e^x\right )^{5/2}} \, dx &=\text {Subst}\left (\int \frac {x^5}{(9-x)^{5/2}} \, dx,x,e^x\right )\\ &=\text {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*}

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Mathematica [A]
time = 0.04, size = 48, normalized size = 0.59 \begin {gather*} -\frac {2 \left (5038848-839808 e^x+23328 e^{2 x}+432 e^{3 x}+18 e^{4 x}+e^{5 x}\right )}{7 \left (9-e^x\right )^{3/2}} \end {gather*}

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 = 0.02, size = 62, normalized size = 0.77

method	result	size
default	\(\frac {39366}{\left (9-{\mathrm e}^{x}\right )^{\frac {3}{2}}}+540 \left (9-{\mathrm e}^{x}\right )^{\frac {3}{2}}-18 \left (9-{\mathrm e}^{x}\right )^{\frac {5}{2}}+\frac {2 \left (9-{\mathrm e}^{x}\right )^{\frac {7}{2}}}{7}-\frac {65610}{\sqrt {9-{\mathrm e}^{x}}}-14580 \sqrt {9-{\mathrm e}^{x}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 61, normalized size = 0.75 \begin {gather*} \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 {gather*}

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 = 0.40, size = 50, normalized size = 0.62 \begin {gather*} -\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 {gather*}

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 = 13.01, size = 61, normalized size = 0.75 \begin {gather*} \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 {gather*}

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 = 4.48, size = 75, normalized size = 0.93 \begin {gather*} -\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 {gather*}

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))

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Mupad [B]
time = 0.19, size = 38, normalized size = 0.47 \begin {gather*} -\frac {2\,\left (23328\,{\mathrm {e}}^{2\,x}+432\,{\mathrm {e}}^{3\,x}+18\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}-839808\,{\mathrm {e}}^x+5038848\right )}{7\,{\left (9-{\mathrm {e}}^x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(6*x)/(9 - exp(x))^(5/2),x)

[Out]

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

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