3.906 \(\int \frac{1}{(2+e x)^{5/2} \sqrt{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{\sqrt{3} \sqrt{2-e x}}{64 e (e x+2)}-\frac{\sqrt{2-e x}}{8 \sqrt{3} e (e x+2)^2}-\frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{128 e} \]

[Out]

-Sqrt[2 - e*x]/(8*Sqrt[3]*e*(2 + e*x)^2) - (Sqrt[3]*Sqrt[2 - e*x])/(64*e*(2 + e*
x)) - (Sqrt[3]*ArcTanh[Sqrt[2 - e*x]/2])/(128*e)

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Rubi [A]  time = 0.125842, 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 \[ -\frac{\sqrt{3} \sqrt{2-e x}}{64 e (e x+2)}-\frac{\sqrt{2-e x}}{8 \sqrt{3} e (e x+2)^2}-\frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{128 e} \]

Antiderivative was successfully verified.

[In]  Int[1/((2 + e*x)^(5/2)*Sqrt[12 - 3*e^2*x^2]),x]

[Out]

-Sqrt[2 - e*x]/(8*Sqrt[3]*e*(2 + e*x)^2) - (Sqrt[3]*Sqrt[2 - e*x])/(64*e*(2 + e*
x)) - (Sqrt[3]*ArcTanh[Sqrt[2 - e*x]/2])/(128*e)

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Rubi in Sympy [A]  time = 14.3813, size = 66, normalized size = 0.77 \[ - \frac{\sqrt{- 3 e x + 6}}{64 e \left (e x + 2\right )} - \frac{\sqrt{- 3 e x + 6}}{24 e \left (e x + 2\right )^{2}} - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{128 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+2)**(5/2)/(-3*e**2*x**2+12)**(1/2),x)

[Out]

-sqrt(-3*e*x + 6)/(64*e*(e*x + 2)) - sqrt(-3*e*x + 6)/(24*e*(e*x + 2)**2) - sqrt
(3)*atanh(sqrt(3)*sqrt(-3*e*x + 6)/6)/(128*e)

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Mathematica [A]  time = 0.0941284, size = 76, normalized size = 0.88 \[ \frac{6 e^2 x^2+16 e x+3 \sqrt{e x-2} (e x+2)^2 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )-56}{128 e (e x+2)^{3/2} \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((2 + e*x)^(5/2)*Sqrt[12 - 3*e^2*x^2]),x]

[Out]

(-56 + 16*e*x + 6*e^2*x^2 + 3*Sqrt[-2 + e*x]*(2 + e*x)^2*ArcTan[Sqrt[-2 + e*x]/2
])/(128*e*(2 + e*x)^(3/2)*Sqrt[12 - 3*e^2*x^2])

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Maple [A]  time = 0.021, size = 126, normalized size = 1.5 \[ -{\frac{\sqrt{3}}{384\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ( 3\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}{x}^{2}{e}^{2}+12\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}xe+6\,xe\sqrt{-3\,ex+6}+12\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) +28\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{5}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+2)^(5/2)/(-3*e^2*x^2+12)^(1/2),x)

[Out]

-1/384*(-e^2*x^2+4)^(1/2)*(3*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x^2*e
^2+12*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x*e+6*x*e*(-3*e*x+6)^(1/2)+1
2*3^(1/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))+28*(-3*e*x+6)^(1/2))/((e*x+2)^5)
^(1/2)*3^(1/2)/(-3*e*x+6)^(1/2)/e

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Maxima [A]  time = 0.839566, size = 97, normalized size = 1.13 \[ -\frac{i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - \frac{256 \,{\left (3 \,{\left (e x - 2\right )}^{\frac{3}{2}} + 20 \, \sqrt{e x - 2}\right )}}{128 i \, \sqrt{3}{\left (e x - 2\right )}^{2} + 1024 i \, \sqrt{3}{\left (e x - 2\right )} + 2048 i \, \sqrt{3}}}{128 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2)),x, algorithm="maxima")

[Out]

-1/128*(I*sqrt(3)*arctan(1/2*sqrt(e*x - 2)) - 256*(3*(e*x - 2)^(3/2) + 20*sqrt(e
*x - 2))/(128*I*sqrt(3)*(e*x - 2)^2 + 1024*I*sqrt(3)*(e*x - 2) + 2048*I*sqrt(3))
)/e

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Fricas [A]  time = 0.224518, size = 188, normalized size = 2.19 \[ \frac{3 \, \sqrt{3}{\left (e^{3} x^{3} + 6 \, e^{2} x^{2} + 12 \, e x + 8\right )} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt{-3 \, e^{2} x^{2} + 12}{\left (3 \, e x + 14\right )} \sqrt{e x + 2}}{768 \,{\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2)),x, algorithm="fricas")

[Out]

1/768*(3*sqrt(3)*(e^3*x^3 + 6*e^2*x^2 + 12*e*x + 8)*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)) - 4*sq
rt(-3*e^2*x^2 + 12)*(3*e*x + 14)*sqrt(e*x + 2))/(e^4*x^3 + 6*e^3*x^2 + 12*e^2*x
+ 8*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(1/(e*x+2)**(5/2)/(-3*e**2*x**2+12)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.617204, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2)),x, algorithm="giac")

[Out]

sage0*x