3.908 \(\int \frac{(2+e x)^{9/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=87 \[ \frac{2 (2-e x)^{5/2}}{15 \sqrt{3} e}-\frac{8 (2-e x)^{3/2}}{3 \sqrt{3} e}+\frac{32 \sqrt{2-e x}}{\sqrt{3} e}+\frac{128}{3 \sqrt{3} e \sqrt{2-e x}} \]

[Out]

128/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (32*Sqrt[2 - e*x])/(Sqrt[3]*e) - (8*(2 - e*x)^
(3/2))/(3*Sqrt[3]*e) + (2*(2 - e*x)^(5/2))/(15*Sqrt[3]*e)

_______________________________________________________________________________________

Rubi [A]  time = 0.0977314, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 (2-e x)^{5/2}}{15 \sqrt{3} e}-\frac{8 (2-e x)^{3/2}}{3 \sqrt{3} e}+\frac{32 \sqrt{2-e x}}{\sqrt{3} e}+\frac{128}{3 \sqrt{3} e \sqrt{2-e x}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + e*x)^(9/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

128/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (32*Sqrt[2 - e*x])/(Sqrt[3]*e) - (8*(2 - e*x)^
(3/2))/(3*Sqrt[3]*e) + (2*(2 - e*x)^(5/2))/(15*Sqrt[3]*e)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.2638, size = 70, normalized size = 0.8 \[ - \frac{8 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{27 e} + \frac{2 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{45 e} + \frac{32 \sqrt{3} \sqrt{- e x + 2}}{3 e} + \frac{128 \sqrt{3}}{9 e \sqrt{- e x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+2)**(9/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

-8*(-3*e*x + 6)**(3/2)/(27*e) + 2*sqrt(3)*(-e*x + 2)**(5/2)/(45*e) + 32*sqrt(3)*
sqrt(-e*x + 2)/(3*e) + 128*sqrt(3)/(9*e*sqrt(-e*x + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0494287, size = 51, normalized size = 0.59 \[ -\frac{2 \sqrt{e x+2} \left (e^3 x^3+14 e^2 x^2+172 e x-728\right )}{15 e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + e*x)^(9/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[2 + e*x]*(-728 + 172*e*x + 14*e^2*x^2 + e^3*x^3))/(15*e*Sqrt[12 - 3*e^2
*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 51, normalized size = 0.6 \[{\frac{ \left ( 2\,ex-4 \right ) \left ({e}^{3}{x}^{3}+14\,{e}^{2}{x}^{2}+172\,ex-728 \right ) }{5\,e} \left ( ex+2 \right ) ^{{\frac{3}{2}}} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+2)^(9/2)/(-3*e^2*x^2+12)^(3/2),x)

[Out]

2/5*(e*x-2)*(e^3*x^3+14*e^2*x^2+172*e*x-728)*(e*x+2)^(3/2)/e/(-3*e^2*x^2+12)^(3/
2)

_______________________________________________________________________________________

Maxima [A]  time = 0.794662, size = 49, normalized size = 0.56 \[ \frac{2 i \, \sqrt{3}{\left (e^{3} x^{3} + 14 \, e^{2} x^{2} + 172 \, e x - 728\right )}}{45 \, \sqrt{e x - 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(9/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="maxima")

[Out]

2/45*I*sqrt(3)*(e^3*x^3 + 14*e^2*x^2 + 172*e*x - 728)/(sqrt(e*x - 2)*e)

_______________________________________________________________________________________

Fricas [A]  time = 0.21823, size = 72, normalized size = 0.83 \[ -\frac{2 \,{\left (e^{4} x^{4} + 16 \, e^{3} x^{3} + 200 \, e^{2} x^{2} - 384 \, e x - 1456\right )}}{15 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(9/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="fricas")

[Out]

-2/15*(e^4*x^4 + 16*e^3*x^3 + 200*e^2*x^2 - 384*e*x - 1456)/(sqrt(-3*e^2*x^2 + 1
2)*sqrt(e*x + 2)*e)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+2)**(9/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.577285, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(9/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="giac")

[Out]

sage0*x