3.913 \(\int \frac{1}{\sqrt{2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{\sqrt{2-e x}}{16 \sqrt{3} e (e x+2)}+\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (e x+2)}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{32 \sqrt{3} e} \]

[Out]

1/(6*Sqrt[3]*e*Sqrt[2 - e*x]*(2 + e*x)) - Sqrt[2 - e*x]/(16*Sqrt[3]*e*(2 + e*x))
- ArcTanh[Sqrt[2 - e*x]/2]/(32*Sqrt[3]*e)

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Rubi [A]  time = 0.124963, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\sqrt{2-e x}}{16 \sqrt{3} e (e x+2)}+\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (e x+2)}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{32 \sqrt{3} e} \]

Antiderivative was successfully verified.

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

[Out]

1/(6*Sqrt[3]*e*Sqrt[2 - e*x]*(2 + e*x)) - Sqrt[2 - e*x]/(16*Sqrt[3]*e*(2 + e*x))
- ArcTanh[Sqrt[2 - e*x]/2]/(32*Sqrt[3]*e)

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Rubi in Sympy [A]  time = 14.1645, size = 65, normalized size = 0.76 \[ - \frac{\sqrt{- 3 e x + 6}}{48 e \left (e x + 2\right )} - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{96 e} + \frac{1}{6 e \sqrt{- 3 e x + 6} \left (e x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-3*e*x + 6)/(48*e*(e*x + 2)) - sqrt(3)*atanh(sqrt(3)*sqrt(-3*e*x + 6)/6)/(
96*e) + 1/(6*e*sqrt(-3*e*x + 6)*(e*x + 2))

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Mathematica [A]  time = 0.0806527, size = 67, normalized size = 0.78 \[ \frac{6 e x+3 \sqrt{e x-2} (e x+2) \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )+4}{96 e \sqrt{3 e x+6} \sqrt{4-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(4 + 6*e*x + 3*Sqrt[-2 + e*x]*(2 + e*x)*ArcTan[Sqrt[-2 + e*x]/2])/(96*e*Sqrt[6 +
3*e*x]*Sqrt[4 - e^2*x^2])

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Maple [A]  time = 0.026, size = 93, normalized size = 1.1 \[{\frac{1}{ \left ( 288\,ex-576 \right ) e}\sqrt{-3\,{e}^{2}{x}^{2}+12} \left ({\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ) \sqrt{3}\sqrt{-3\,ex+6}xe+2\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{-3\,ex+6}-6\,ex-4 \right ) \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/288/(e*x+2)^(3/2)*(-3*e^2*x^2+12)^(1/2)*(arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))
*3^(1/2)*(-3*e*x+6)^(1/2)*x*e+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-4)/(e*x-2)/e

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Maxima [A]  time = 0.833139, size = 76, normalized size = 0.88 \[ \frac{-i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) + \frac{192 \,{\left (3 \, e x + 2\right )}}{96 i \, \sqrt{3}{\left (e x - 2\right )}^{\frac{3}{2}} + 384 i \, \sqrt{3} \sqrt{e x - 2}}}{96 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(3/2)*sqrt(e*x + 2)),x, algorithm="maxima")

[Out]

1/96*(-I*sqrt(3)*arctan(1/2*sqrt(e*x - 2)) + 192*(3*e*x + 2)/(96*I*sqrt(3)*(e*x
- 2)^(3/2) + 384*I*sqrt(3)*sqrt(e*x - 2)))/e

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Fricas [A]  time = 0.216419, size = 193, normalized size = 2.24 \[ -\frac{\sqrt{3}{\left (4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12}{\left (3 \, e x + 2\right )} \sqrt{e x + 2} - 9 \,{\left (e^{3} x^{3} + 2 \, e^{2} x^{2} - 4 \, e x - 8\right )} \log \left (-\frac{\sqrt{3}{\left (e^{2} x^{2} - 4 \, e x - 12\right )} + 4 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{e^{2} x^{2} + 4 \, e x + 4}\right )\right )}}{1728 \,{\left (e^{4} x^{3} + 2 \, e^{3} x^{2} - 4 \, e^{2} x - 8 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(3/2)*sqrt(e*x + 2)),x, algorithm="fricas")

[Out]

-1/1728*sqrt(3)*(4*sqrt(3)*sqrt(-3*e^2*x^2 + 12)*(3*e*x + 2)*sqrt(e*x + 2) - 9*(
e^3*x^3 + 2*e^2*x^2 - 4*e*x - 8)*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)))/(e^4*x^3 + 2*e^3*x^2 - 4*
e^2*x - 8*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{3} \int \frac{1}{- e^{2} x^{2} \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4} + 4 \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}}\, dx}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*Integral(1/(-e**2*x**2*sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4) + 4*sqrt(e*x +
2)*sqrt(-e**2*x**2 + 4)), x)/9

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GIAC/XCAS [A]  time = 0.695498, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(3/2)*sqrt(e*x + 2)),x, algorithm="giac")

[Out]

sage0*x