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

Optimal. Leaf size=22 \[ -\frac{6 \sqrt{3} (2-e x)^{5/2}}{5 e} \]

[Out]

(-6*Sqrt[3]*(2 - e*x)^(5/2))/(5*e)

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Rubi [A]  time = 0.0406682, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{6 \sqrt{3} (2-e x)^{5/2}}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[3]*(2 - e*x)^(5/2))/(5*e)

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Rubi in Sympy [A]  time = 4.01232, size = 15, normalized size = 0.68 \[ - \frac{2 \left (- 3 e x + 6\right )^{\frac{5}{2}}}{15 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-3*e*x + 6)**(5/2)/(15*e)

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Mathematica [A]  time = 0.0346433, size = 37, normalized size = 1.68 \[ -\frac{6 (e x-2)^2 \sqrt{12-3 e^2 x^2}}{5 e \sqrt{e x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2])/(5*e*Sqrt[2 + e*x])

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Maple [A]  time = 0.005, size = 30, normalized size = 1.4 \[{\frac{2\,ex-4}{5\,e} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{{\frac{3}{2}}} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.83685, size = 49, normalized size = 2.23 \[ -\frac{{\left (6 i \, \sqrt{3} e^{2} x^{2} - 24 i \, \sqrt{3} e x + 24 i \, \sqrt{3}\right )} \sqrt{e x - 2}}{5 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(6*I*sqrt(3)*e^2*x^2 - 24*I*sqrt(3)*e*x + 24*I*sqrt(3))*sqrt(e*x - 2)/e

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Fricas [A]  time = 0.220176, size = 61, normalized size = 2.77 \[ \frac{18 \,{\left (e^{4} x^{4} - 4 \, e^{3} x^{3} + 16 \, e x - 16\right )}}{5 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18/5*(e^4*x^4 - 4*e^3*x^3 + 16*e*x - 16)/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)*e)

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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((-3*e**2*x**2+12)**(3/2)/(e*x+2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}}{{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(3/2), x)