3.896 \(\int \frac{\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{7/2}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{3 \sqrt{3} (2-e x)^{3/2}}{e (e x+2)}-\frac{9 \sqrt{3} \sqrt{2-e x}}{e}+\frac{18 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]

[Out]

(-9*Sqrt[3]*Sqrt[2 - e*x])/e - (3*Sqrt[3]*(2 - e*x)^(3/2))/(e*(2 + e*x)) + (18*S
qrt[3]*ArcTanh[Sqrt[2 - e*x]/2])/e

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Rubi [A]  time = 0.121022, antiderivative size = 73, 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 \[ -\frac{3 \sqrt{3} (2-e x)^{3/2}}{e (e x+2)}-\frac{9 \sqrt{3} \sqrt{2-e x}}{e}+\frac{18 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

(-9*Sqrt[3]*Sqrt[2 - e*x])/e - (3*Sqrt[3]*(2 - e*x)^(3/2))/(e*(2 + e*x)) + (18*S
qrt[3]*ArcTanh[Sqrt[2 - e*x]/2])/e

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Rubi in Sympy [A]  time = 14.3675, size = 56, normalized size = 0.77 \[ - \frac{\left (- 3 e x + 6\right )^{\frac{3}{2}}}{e \left (e x + 2\right )} - \frac{9 \sqrt{- 3 e x + 6}}{e} + \frac{18 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-3*e*x + 6)**(3/2)/(e*(e*x + 2)) - 9*sqrt(-3*e*x + 6)/e + 18*sqrt(3)*atanh(sqr
t(3)*sqrt(-3*e*x + 6)/6)/e

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Mathematica [A]  time = 0.0776781, size = 74, normalized size = 1.01 \[ -\frac{6 \sqrt{12-3 e^2 x^2} \left (\sqrt{e x-2} (e x+4)-3 (e x+2) \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )\right )}{e \sqrt{e x-2} (e x+2)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[12 - 3*e^2*x^2]*(Sqrt[-2 + e*x]*(4 + e*x) - 3*(2 + e*x)*ArcTan[Sqrt[-2
+ e*x]/2]))/(e*Sqrt[-2 + e*x]*(2 + e*x)^(3/2))

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Maple [A]  time = 0.028, size = 101, normalized size = 1.4 \[ 6\,{\frac{\sqrt{-{e}^{2}{x}^{2}+4} \left ( 3\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}xe-xe\sqrt{-3\,ex+6}+6\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -4\,\sqrt{-3\,ex+6} \right ) \sqrt{3}}{\sqrt{ \left ( ex+2 \right ) ^{3}}\sqrt{-3\,ex+6}e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*(-e^2*x^2+4)^(1/2)*(3*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x*e-x*e*(-
3*e*x+6)^(1/2)+6*3^(1/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))-4*(-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 = 0.856059, size = 70, normalized size = 0.96 \[ -\frac{2 \,{\left (-9 i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) + 3 i \, \sqrt{3} \sqrt{e x - 2} + \frac{6 i \, \sqrt{3} \sqrt{e x - 2}}{e x + 2}\right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-9*I*sqrt(3)*arctan(1/2*sqrt(e*x - 2)) + 3*I*sqrt(3)*sqrt(e*x - 2) + 6*I*sqr
t(3)*sqrt(e*x - 2)/(e*x + 2))/e

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Fricas [A]  time = 0.220111, size = 158, normalized size = 2.16 \[ \frac{9 \,{\left (2 \, e^{2} x^{2} + \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x - 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) + 4 \, e x - 16\right )}}{\sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*(2*e^2*x^2 + 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 - 16)/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)*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)**(3/2)/(e*x+2)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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