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

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 = 0.0819464, 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 \[ \frac{2}{7} \left (9-e^x\right )^{7/2}-18 \left (9-e^x\right )^{5/2}+540 \left (9-e^x\right )^{3/2}-14580 \sqrt{9-e^x}-\frac{65610}{\sqrt{9-e^x}}+\frac{39366}{\left (9-e^x\right )^{3/2}} \]

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

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Rubi in Sympy [A]  time = 11.1197, size = 61, normalized size = 0.75 \[ \frac{2 \left (- e^{x} + 9\right )^{\frac{7}{2}}}{7} - 18 \left (- e^{x} + 9\right )^{\frac{5}{2}} + 540 \left (- e^{x} + 9\right )^{\frac{3}{2}} - 14580 \sqrt{- e^{x} + 9} - \frac{65610}{\sqrt{- e^{x} + 9}} + \frac{39366}{\left (- e^{x} + 9\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(6*x)/(9-exp(x))**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0439276, 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 = 0.014, size = 62, normalized size = 0.8 \[ 39366\, \left ( 9-{{\rm e}^{x}} \right ) ^{-3/2}+540\, \left ( 9-{{\rm e}^{x}} \right ) ^{3/2}-18\, \left ( 9-{{\rm e}^{x}} \right ) ^{5/2}+{\frac{2}{7} \left ( 9-{{\rm e}^{x}} \right ) ^{{\frac{7}{2}}}}-65610\,{\frac{1}{\sqrt{9-{{\rm e}^{x}}}}}-14580\,\sqrt{9-{{\rm e}^{x}}} \]

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)-14580*(9-exp(x))^(1/2)

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Maxima [A]  time = 0.786708, size = 82, normalized size = 1.01 \[ \frac{2}{7} \,{\left (-e^{x} + 9\right )}^{\frac{7}{2}} - 18 \,{\left (-e^{x} + 9\right )}^{\frac{5}{2}} + 540 \,{\left (-e^{x} + 9\right )}^{\frac{3}{2}} - 14580 \, \sqrt{-e^{x} + 9} - \frac{65610}{\sqrt{-e^{x} + 9}} + \frac{39366}{{\left (-e^{x} + 9\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(6*x)/(-e^x + 9)^(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 = 0.23467, size = 59, normalized size = 0.73 \[ \frac{2 \,{\left (e^{\left (5 \, x\right )} + 18 \, e^{\left (4 \, x\right )} + 432 \, e^{\left (3 \, x\right )} + 23328 \, e^{\left (2 \, x\right )} - 839808 \, e^{x} + 5038848\right )}}{7 \,{\left (e^{x} - 9\right )} \sqrt{-e^{x} + 9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(6*x)/(-e^x + 9)^(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)/
((e^x - 9)*sqrt(-e^x + 9))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{6 x}}{\left (- e^{x} + 9\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232066, size = 101, normalized size = 1.25 \[ -\frac{2}{7} \,{\left (e^{x} - 9\right )}^{3} \sqrt{-e^{x} + 9} - 18 \,{\left (e^{x} - 9\right )}^{2} \sqrt{-e^{x} + 9} + 540 \,{\left (-e^{x} + 9\right )}^{\frac{3}{2}} - 14580 \, \sqrt{-e^{x} + 9} - \frac{13122 \,{\left (5 \, e^{x} - 42\right )}}{{\left (e^{x} - 9\right )} \sqrt{-e^{x} + 9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(6*x)/(-e^x + 9)^(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))