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

Optimal. Leaf size=57 \[ -\frac{\sqrt{2-e x}}{4 \sqrt{3} e (e x+2)}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{8 \sqrt{3} e} \]

[Out]

-Sqrt[2 - e*x]/(4*Sqrt[3]*e*(2 + e*x)) - ArcTanh[Sqrt[2 - e*x]/2]/(8*Sqrt[3]*e)

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Rubi [A]  time = 0.0945073, antiderivative size = 57, 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 \[ -\frac{\sqrt{2-e x}}{4 \sqrt{3} e (e x+2)}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{8 \sqrt{3} e} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[2 - e*x]/(4*Sqrt[3]*e*(2 + e*x)) - ArcTanh[Sqrt[2 - e*x]/2]/(8*Sqrt[3]*e)

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Rubi in Sympy [A]  time = 10.7422, size = 46, normalized size = 0.81 \[ - \frac{\sqrt{- 3 e x + 6}}{12 e \left (e x + 2\right )} - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{24 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-3*e*x + 6)/(12*e*(e*x + 2)) - sqrt(3)*atanh(sqrt(3)*sqrt(-3*e*x + 6)/6)/(
24*e)

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Mathematica [A]  time = 0.0710026, size = 66, normalized size = 1.16 \[ \frac{2 e x+\sqrt{e x-2} (e x+2) \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )-4}{8 e \sqrt{3 e x+6} \sqrt{4-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4 + 2*e*x + Sqrt[-2 + e*x]*(2 + e*x)*ArcTan[Sqrt[-2 + e*x]/2])/(8*e*Sqrt[6 + 3
*e*x]*Sqrt[4 - e^2*x^2])

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Maple [B]  time = 0.023, size = 88, normalized size = 1.5 \[ -{\frac{\sqrt{3}}{24\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ({\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ) \sqrt{3}xe+2\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) +2\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{3}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(-e^2*x^2+4)^(1/2)*(arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x*e+2*3^
(1/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))+2*(-3*e*x+6)^(1/2))/((e*x+2)^3)^(1/2
)*3^(1/2)/(-3*e*x+6)^(1/2)/e

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Maxima [A]  time = 0.834697, size = 65, normalized size = 1.14 \[ -\frac{i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - \frac{48 \, \sqrt{e x - 2}}{8 i \, \sqrt{3}{\left (e x - 2\right )} + 32 i \, \sqrt{3}}}{24 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(I*sqrt(3)*arctan(1/2*sqrt(e*x - 2)) - 48*sqrt(e*x - 2)/(8*I*sqrt(3)*(e*x
- 2) + 32*I*sqrt(3)))/e

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Fricas [A]  time = 0.223308, size = 163, normalized size = 2.86 \[ \frac{\sqrt{3}{\left (3 \,{\left (e^{2} x^{2} + 4 \, e x + 4\right )} \log \left (-\frac{\sqrt{3}{\left (e^{2} x^{2} - 4 \, e x - 12\right )} + 4 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}\right )}}{144 \,{\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/144*sqrt(3)*(3*(e^2*x^2 + 4*e*x + 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^2*x + 4*e)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-3 \, e^{2} x^{2} + 12}{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(3/2)), x)