3.895 \(\int \frac{\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{5/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{2 \sqrt{3} (2-e x)^{3/2}}{e}+\frac{24 \sqrt{3} \sqrt{2-e x}}{e}-\frac{48 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]

[Out]

(24*Sqrt[3]*Sqrt[2 - e*x])/e + (2*Sqrt[3]*(2 - e*x)^(3/2))/e - (48*Sqrt[3]*ArcTa
nh[Sqrt[2 - e*x]/2])/e

_______________________________________________________________________________________

Rubi [A]  time = 0.117541, antiderivative size = 66, 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 \[ \frac{2 \sqrt{3} (2-e x)^{3/2}}{e}+\frac{24 \sqrt{3} \sqrt{2-e x}}{e}-\frac{48 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

(24*Sqrt[3]*Sqrt[2 - e*x])/e + (2*Sqrt[3]*(2 - e*x)^(3/2))/e - (48*Sqrt[3]*ArcTa
nh[Sqrt[2 - e*x]/2])/e

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.7975, size = 54, normalized size = 0.82 \[ \frac{2 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{3 e} + \frac{24 \sqrt{- 3 e x + 6}}{e} - \frac{48 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-3*e*x + 6)**(3/2)/(3*e) + 24*sqrt(-3*e*x + 6)/e - 48*sqrt(3)*atanh(sqrt(3)*s
qrt(-3*e*x + 6)/6)/e

_______________________________________________________________________________________

Mathematica [A]  time = 0.0678738, size = 69, normalized size = 1.05 \[ -\frac{2 \sqrt{12-3 e^2 x^2} \left (\sqrt{e x-2} (e x-14)+24 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )\right )}{e \sqrt{e x-2} \sqrt{e x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[12 - 3*e^2*x^2]*((-14 + e*x)*Sqrt[-2 + e*x] + 24*ArcTan[Sqrt[-2 + e*x]/
2]))/(e*Sqrt[-2 + e*x]*Sqrt[2 + e*x])

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 77, normalized size = 1.2 \[ -2\,{\frac{\sqrt{-{e}^{2}{x}^{2}+4} \left ( xe\sqrt{-3\,ex+6}+24\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -14\,\sqrt{-3\,ex+6} \right ) \sqrt{3}}{\sqrt{ex+2}\sqrt{-3\,ex+6}e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-e^2*x^2+4)^(1/2)*(x*e*(-3*e*x+6)^(1/2)+24*3^(1/2)*arctanh(1/6*3^(1/2)*(-3*e
*x+6)^(1/2))-14*(-3*e*x+6)^(1/2))*3^(1/2)/(e*x+2)^(1/2)/(-3*e*x+6)^(1/2)/e

_______________________________________________________________________________________

Maxima [A]  time = 0.878036, size = 61, normalized size = 0.92 \[ -\frac{2 \,{\left (i \, \sqrt{3}{\left (e x - 2\right )}^{\frac{3}{2}} + 24 i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - 12 i \, \sqrt{3} \sqrt{e x - 2}\right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(I*sqrt(3)*(e*x - 2)^(3/2) + 24*I*sqrt(3)*arctan(1/2*sqrt(e*x - 2)) - 12*I*sq
rt(3)*sqrt(e*x - 2))/e

_______________________________________________________________________________________

Fricas [A]  time = 0.217694, size = 169, normalized size = 2.56 \[ \frac{6 \,{\left (e^{3} x^{3} - 14 \, e^{2} x^{2} + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, e x + 56\right )}}{\sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*(e^3*x^3 - 14*e^2*x^2 + 4*sqrt(3)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)*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*e*x + 56)/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)*e)

_______________________________________________________________________________________

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)**(3/2)/(e*x+2)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: AttributeError