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

Optimal. Leaf size=55 \[ \frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{2 e}-\frac{\sqrt{3} \sqrt{2-e x}}{e (e x+2)} \]

[Out]

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

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Rubi [A]  time = 0.0944513, antiderivative size = 55, 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 \[ \frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{2 e}-\frac{\sqrt{3} \sqrt{2-e x}}{e (e x+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.1173, size = 42, normalized size = 0.76 \[ - \frac{\sqrt{- 3 e x + 6}}{e \left (e x + 2\right )} + \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{2 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0618818, size = 71, normalized size = 1.29 \[ \frac{\sqrt{12-3 e^2 x^2} \left ((e x+2) \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )-2 \sqrt{e x-2}\right )}{2 e \sqrt{e x-2} (e x+2)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.027, size = 88, normalized size = 1.6 \[{\frac{\sqrt{3}}{2\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ({\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ) \sqrt{3}xe+2\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -2\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{3}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(-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 = 0.812804, size = 50, normalized size = 0.91 \[ -\frac{i \, \sqrt{3}{\left (\frac{2 \, \sqrt{e x - 2}}{e x + 2} - \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right )\right )}}{2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218794, size = 157, normalized size = 2.85 \[ \frac{\sqrt{3}{\left (e^{2} x^{2} + 4 \, e x + 4\right )} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x - 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{4 \,{\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(sqrt(3)*(e^2*x^2 + 4*e*x + 4)*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*sqrt(-3*e^2*x^2 + 1
2)*sqrt(e*x + 2))/(e^3*x^2 + 4*e^2*x + 4*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((-3*e**2*x**2+12)**(1/2)/(e*x+2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-3 \, e^{2} x^{2} + 12}}{{\left (e x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(5/2), x)