3.898 \(\int \frac{\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{11/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{\sqrt{3} (2-e x)^{3/2}}{e (e x+2)^3}-\frac{3 \sqrt{3} \sqrt{2-e x}}{32 e (e x+2)}+\frac{3 \sqrt{3} \sqrt{2-e x}}{4 e (e x+2)^2}-\frac{3 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{64 e} \]

[Out]

-((Sqrt[3]*(2 - e*x)^(3/2))/(e*(2 + e*x)^3)) + (3*Sqrt[3]*Sqrt[2 - e*x])/(4*e*(2
+ e*x)^2) - (3*Sqrt[3]*Sqrt[2 - e*x])/(32*e*(2 + e*x)) - (3*Sqrt[3]*ArcTanh[Sqr
t[2 - e*x]/2])/(64*e)

_______________________________________________________________________________________

Rubi [A]  time = 0.156911, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{3} (2-e x)^{3/2}}{e (e x+2)^3}-\frac{3 \sqrt{3} \sqrt{2-e x}}{32 e (e x+2)}+\frac{3 \sqrt{3} \sqrt{2-e x}}{4 e (e x+2)^2}-\frac{3 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{64 e} \]

Antiderivative was successfully verified.

[In]  Int[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(11/2),x]

[Out]

-((Sqrt[3]*(2 - e*x)^(3/2))/(e*(2 + e*x)^3)) + (3*Sqrt[3]*Sqrt[2 - e*x])/(4*e*(2
+ e*x)^2) - (3*Sqrt[3]*Sqrt[2 - e*x])/(32*e*(2 + e*x)) - (3*Sqrt[3]*ArcTanh[Sqr
t[2 - e*x]/2])/(64*e)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.9008, size = 90, normalized size = 0.8 \[ - \frac{\left (- 3 e x + 6\right )^{\frac{3}{2}}}{3 e \left (e x + 2\right )^{3}} - \frac{3 \sqrt{- 3 e x + 6}}{32 e \left (e x + 2\right )} + \frac{3 \sqrt{- 3 e x + 6}}{4 e \left (e x + 2\right )^{2}} - \frac{3 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{64 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3*e**2*x**2+12)**(3/2)/(e*x+2)**(11/2),x)

[Out]

-(-3*e*x + 6)**(3/2)/(3*e*(e*x + 2)**3) - 3*sqrt(-3*e*x + 6)/(32*e*(e*x + 2)) +
3*sqrt(-3*e*x + 6)/(4*e*(e*x + 2)**2) - 3*sqrt(3)*atanh(sqrt(3)*sqrt(-3*e*x + 6)
/6)/(64*e)

_______________________________________________________________________________________

Mathematica [A]  time = 0.106018, size = 88, normalized size = 0.78 \[ -\frac{\sqrt{12-3 e^2 x^2} \left (2 \sqrt{e x-2} \left (3 e^2 x^2-44 e x+28\right )+3 (e x+2)^3 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )\right )}{64 e \sqrt{e x-2} (e x+2)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(11/2),x]

[Out]

-(Sqrt[12 - 3*e^2*x^2]*(2*Sqrt[-2 + e*x]*(28 - 44*e*x + 3*e^2*x^2) + 3*(2 + e*x)
^3*ArcTan[Sqrt[-2 + e*x]/2]))/(64*e*Sqrt[-2 + e*x]*(2 + e*x)^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.032, size = 167, normalized size = 1.5 \[ -{\frac{\sqrt{3}}{64\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ( 3\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}{x}^{3}{e}^{3}+18\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}{x}^{2}{e}^{2}+6\,{x}^{2}{e}^{2}\sqrt{-3\,ex+6}+36\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}xe-88\,xe\sqrt{-3\,ex+6}+24\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) +56\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{7}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*e^2*x^2+12)^(3/2)/(e*x+2)^(11/2),x)

[Out]

-1/64*(-e^2*x^2+4)^(1/2)*(3*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x^3*e^
3+18*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x^2*e^2+6*x^2*e^2*(-3*e*x+6)^
(1/2)+36*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x*e-88*x*e*(-3*e*x+6)^(1/
2)+24*3^(1/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))+56*(-3*e*x+6)^(1/2))*3^(1/2)
/((e*x+2)^7)^(1/2)/(-3*e*x+6)^(1/2)/e

_______________________________________________________________________________________

Maxima [A]  time = 0.868658, size = 113, normalized size = 1. \[ \frac{-3 i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - \frac{2 \,{\left (3 i \, \sqrt{3}{\left (e x - 2\right )}^{\frac{5}{2}} - 32 i \, \sqrt{3}{\left (e x - 2\right )}^{\frac{3}{2}} - 48 i \, \sqrt{3} \sqrt{e x - 2}\right )}}{{\left (e x - 2\right )}^{3} + 12 \,{\left (e x - 2\right )}^{2} + 48 \, e x - 32}}{64 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(11/2),x, algorithm="maxima")

[Out]

1/64*(-3*I*sqrt(3)*arctan(1/2*sqrt(e*x - 2)) - 2*(3*I*sqrt(3)*(e*x - 2)^(5/2) -
32*I*sqrt(3)*(e*x - 2)^(3/2) - 48*I*sqrt(3)*sqrt(e*x - 2))/((e*x - 2)^3 + 12*(e*
x - 2)^2 + 48*e*x - 32))/e

_______________________________________________________________________________________

Fricas [A]  time = 0.223542, size = 220, normalized size = 1.95 \[ \frac{3 \, \sqrt{3}{\left (e^{4} x^{4} + 8 \, e^{3} x^{3} + 24 \, e^{2} x^{2} + 32 \, e x + 16\right )} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \,{\left (3 \, e^{2} x^{2} - 44 \, e x + 28\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{128 \,{\left (e^{5} x^{4} + 8 \, e^{4} x^{3} + 24 \, e^{3} x^{2} + 32 \, e^{2} x + 16 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(11/2),x, algorithm="fricas")

[Out]

1/128*(3*sqrt(3)*(e^4*x^4 + 8*e^3*x^3 + 24*e^2*x^2 + 32*e*x + 16)*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*(3*e^2*x^2 - 44*e*x + 28)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2))/(e^5*
x^4 + 8*e^4*x^3 + 24*e^3*x^2 + 32*e^2*x + 16*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)**(11/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}}{{\left (e x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(11/2),x, algorithm="giac")

[Out]

integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(11/2), x)