3.889 \(\int \frac{\sqrt{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.124003, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\sqrt{3} \sqrt{2-e x}}{16 e (e x+2)}-\frac{\sqrt{3} \sqrt{2-e x}}{2 e (e x+2)^2}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{32 e} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.6124, size = 65, normalized size = 0.76 \[ \frac{\sqrt{- 3 e x + 6}}{16 e \left (e x + 2\right )} - \frac{\sqrt{- 3 e x + 6}}{2 e \left (e x + 2\right )^{2}} + \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{32 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3*e**2*x**2+12)**(1/2)/(e*x+2)**(7/2),x)

[Out]

sqrt(-3*e*x + 6)/(16*e*(e*x + 2)) - sqrt(-3*e*x + 6)/(2*e*(e*x + 2)**2) + sqrt(3
)*atanh(sqrt(3)*sqrt(-3*e*x + 6)/6)/(32*e)

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Mathematica [A]  time = 0.0814277, size = 78, normalized size = 0.91 \[ \frac{\sqrt{12-3 e^2 x^2} \left (2 (e x-6) \sqrt{e x-2}+(e x+2)^2 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )\right )}{32 e \sqrt{e x-2} (e x+2)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(7/2),x]

[Out]

(Sqrt[12 - 3*e^2*x^2]*(2*(-6 + e*x)*Sqrt[-2 + e*x] + (2 + e*x)^2*ArcTan[Sqrt[-2
+ e*x]/2]))/(32*e*Sqrt[-2 + e*x]*(2 + e*x)^(5/2))

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Maple [A]  time = 0.028, size = 125, normalized size = 1.5 \[{\frac{\sqrt{3}}{32\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ({\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ) \sqrt{3}{x}^{2}{e}^{2}+4\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}xe+2\,xe\sqrt{-3\,ex+6}+4\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -12\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{5}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(7/2),x)

[Out]

1/32*(-e^2*x^2+4)^(1/2)*(arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x^2*e^2+4
*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x*e+2*x*e*(-3*e*x+6)^(1/2)+4*3^(1
/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))-12*(-3*e*x+6)^(1/2))*3^(1/2)/((e*x+2)^
5)^(1/2)/(-3*e*x+6)^(1/2)/e

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Maxima [A]  time = 0.815255, size = 70, normalized size = 0.81 \[ \frac{i \, \sqrt{3}{\left (\frac{2 \,{\left ({\left (e x - 2\right )}^{\frac{3}{2}} - 4 \, \sqrt{e x - 2}\right )}}{{\left (e x - 2\right )}^{2} + 8 \, e x} + \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right )\right )}}{32 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(7/2),x, algorithm="maxima")

[Out]

1/32*I*sqrt(3)*(2*((e*x - 2)^(3/2) - 4*sqrt(e*x - 2))/((e*x - 2)^2 + 8*e*x) + ar
ctan(1/2*sqrt(e*x - 2)))/e

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Fricas [A]  time = 0.21587, size = 185, normalized size = 2.15 \[ \frac{\sqrt{3}{\left (e^{3} x^{3} + 6 \, e^{2} x^{2} + 12 \, e x + 8\right )} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x - 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) + 4 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}{\left (e x - 6\right )}}{64 \,{\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(7/2),x, algorithm="fricas")

[Out]

1/64*(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*sq
rt(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 + 12)*sqrt(e*x + 2)*(e*x - 6))/(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((-3*e**2*x**2+12)**(1/2)/(e*x+2)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-3 \, e^{2} x^{2} + 12}}{{\left (e x + 2\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(7/2),x, algorithm="giac")

[Out]

integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(7/2), x)