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

Optimal. Leaf size=50 \[ \frac{1}{6 \sqrt{3} e \sqrt{2-e x}}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{12 \sqrt{3} e} \]

[Out]

1/(6*Sqrt[3]*e*Sqrt[2 - e*x]) - ArcTanh[Sqrt[2 - e*x]/2]/(12*Sqrt[3]*e)

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Rubi [A]  time = 0.0902509, antiderivative size = 50, 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 \[ \frac{1}{6 \sqrt{3} e \sqrt{2-e x}}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{12 \sqrt{3} e} \]

Antiderivative was successfully verified.

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

[Out]

1/(6*Sqrt[3]*e*Sqrt[2 - e*x]) - ArcTanh[Sqrt[2 - e*x]/2]/(12*Sqrt[3]*e)

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Rubi in Sympy [A]  time = 10.5316, size = 41, normalized size = 0.82 \[ - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{36 e} + \frac{1}{6 e \sqrt{- 3 e x + 6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*atanh(sqrt(3)*sqrt(-3*e*x + 6)/6)/(36*e) + 1/(6*e*sqrt(-3*e*x + 6))

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Mathematica [A]  time = 0.0437858, size = 56, normalized size = 1.12 \[ \frac{\sqrt{e x+2} \left (\sqrt{e x-2} \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )+2\right )}{12 e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 60, normalized size = 1.2 \[{\frac{1}{ \left ( 108\,ex-216 \right ) e}\sqrt{-3\,{e}^{2}{x}^{2}+12} \left ( \sqrt{3}{\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ) \sqrt{-3\,ex+6}-6 \right ){\frac{1}{\sqrt{ex+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/108/(e*x+2)^(1/2)*(-3*e^2*x^2+12)^(1/2)*(3^(1/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6
)^(1/2))*(-3*e*x+6)^(1/2)-6)/(e*x-2)/e

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Maxima [A]  time = 0.820391, size = 45, normalized size = 0.9 \[ \frac{-i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - \frac{2 i \, \sqrt{3}}{\sqrt{e x - 2}}}{36 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36*(-I*sqrt(3)*arctan(1/2*sqrt(e*x - 2)) - 2*I*sqrt(3)/sqrt(e*x - 2))/e

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Fricas [A]  time = 0.220725, size = 150, normalized size = 3. \[ \frac{\sqrt{3}{\left (3 \,{\left (e^{2} x^{2} - 4\right )} \log \left (-\frac{\sqrt{3}{\left (e^{2} x^{2} - 4 \, e x - 12\right )} + 4 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}\right )}}{216 \,{\left (e^{3} x^{2} - 4 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/216*sqrt(3)*(3*(e^2*x^2 - 4)*log(-(sqrt(3)*(e^2*x^2 - 4*e*x - 12) + 4*sqrt(-3*
e^2*x^2 + 12)*sqrt(e*x + 2))/(e^2*x^2 + 4*e*x + 4)) - 4*sqrt(3)*sqrt(-3*e^2*x^2
+ 12)*sqrt(e*x + 2))/(e^3*x^2 - 4*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{3} \int \frac{\sqrt{e x + 2}}{- e^{2} x^{2} \sqrt{- e^{2} x^{2} + 4} + 4 \sqrt{- e^{2} x^{2} + 4}}\, dx}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*Integral(sqrt(e*x + 2)/(-e**2*x**2*sqrt(-e**2*x**2 + 4) + 4*sqrt(-e**2*x
**2 + 4)), x)/9

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GIAC/XCAS [A]  time = 0.625924, size = 4, normalized size = 0.08 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x