3.887 \(\int \frac{\sqrt{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx\)

Optimal. Leaf size=46 \[ \frac{2 \sqrt{3} \sqrt{2-e x}}{e}-\frac{4 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]

[Out]

(2*Sqrt[3]*Sqrt[2 - e*x])/e - (4*Sqrt[3]*ArcTanh[Sqrt[2 - e*x]/2])/e

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Rubi [A]  time = 0.0894362, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 \sqrt{3} \sqrt{2-e x}}{e}-\frac{4 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3]*Sqrt[2 - e*x])/e - (4*Sqrt[3]*ArcTanh[Sqrt[2 - e*x]/2])/e

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Rubi in Sympy [A]  time = 10.328, size = 39, normalized size = 0.85 \[ \frac{2 \sqrt{- 3 e x + 6}}{e} - \frac{4 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3*e**2*x**2+12)**(1/2)/(e*x+2)**(3/2),x)

[Out]

2*sqrt(-3*e*x + 6)/e - 4*sqrt(3)*atanh(sqrt(3)*sqrt(-3*e*x + 6)/6)/e

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Mathematica [A]  time = 0.0536746, size = 63, normalized size = 1.37 \[ \frac{2 \sqrt{12-3 e^2 x^2} \left (\sqrt{e x-2}-2 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )\right )}{e \sqrt{e x-2} \sqrt{e x+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 66, normalized size = 1.4 \[ -2\,{\frac{\sqrt{-{e}^{2}{x}^{2}+4} \left ( 2\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -\sqrt{-3\,ex+6} \right ) \sqrt{3}}{\sqrt{ex+2}\sqrt{-3\,ex+6}e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(3/2),x)

[Out]

-2*(-e^2*x^2+4)^(1/2)*(2*3^(1/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))-(-3*e*x+6
)^(1/2))/(e*x+2)^(1/2)*3^(1/2)/(-3*e*x+6)^(1/2)/e

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Maxima [A]  time = 0.802196, size = 38, normalized size = 0.83 \[ \frac{2 i \, \sqrt{3}{\left (\sqrt{e x - 2} - 2 \, \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right )\right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*I*sqrt(3)*(sqrt(e*x - 2) - 2*arctan(1/2*sqrt(e*x - 2)))/e

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Fricas [A]  time = 0.228745, size = 154, normalized size = 3.35 \[ -\frac{2 \,{\left (3 \, e^{2} x^{2} - \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 12\right )}}{\sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(3*e^2*x^2 - sqrt(3)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)*log(-(3*e^2*x^2 - 12
*e*x + 4*sqrt(3)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2) - 36)/(e^2*x^2 + 4*e*x + 4)
) - 12)/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \sqrt{3} \int \frac{\sqrt{- e^{2} x^{2} + 4}}{e x \sqrt{e x + 2} + 2 \sqrt{e x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e**2*x**2+12)**(1/2)/(e*x+2)**(3/2),x)

[Out]

sqrt(3)*Integral(sqrt(-e**2*x**2 + 4)/(e*x*sqrt(e*x + 2) + 2*sqrt(e*x + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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