3.897 \(\int \frac{\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{9/2}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{3 \sqrt{3} (2-e x)^{3/2}}{2 e (e x+2)^2}+\frac{9 \sqrt{3} \sqrt{2-e x}}{4 e (e x+2)}-\frac{9 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{8 e} \]

[Out]

(-3*Sqrt[3]*(2 - e*x)^(3/2))/(2*e*(2 + e*x)^2) + (9*Sqrt[3]*Sqrt[2 - e*x])/(4*e*
(2 + e*x)) - (9*Sqrt[3]*ArcTanh[Sqrt[2 - e*x]/2])/(8*e)

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Rubi [A]  time = 0.124431, antiderivative size = 86, 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 \[ -\frac{3 \sqrt{3} (2-e x)^{3/2}}{2 e (e x+2)^2}+\frac{9 \sqrt{3} \sqrt{2-e x}}{4 e (e x+2)}-\frac{9 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{8 e} \]

Antiderivative was successfully verified.

[In]  Int[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(9/2),x]

[Out]

(-3*Sqrt[3]*(2 - e*x)^(3/2))/(2*e*(2 + e*x)^2) + (9*Sqrt[3]*Sqrt[2 - e*x])/(4*e*
(2 + e*x)) - (9*Sqrt[3]*ArcTanh[Sqrt[2 - e*x]/2])/(8*e)

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Rubi in Sympy [A]  time = 15.1612, size = 68, normalized size = 0.79 \[ - \frac{\left (- 3 e x + 6\right )^{\frac{3}{2}}}{2 e \left (e x + 2\right )^{2}} + \frac{9 \sqrt{- 3 e x + 6}}{4 e \left (e x + 2\right )} - \frac{9 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{8 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3*e**2*x**2+12)**(3/2)/(e*x+2)**(9/2),x)

[Out]

-(-3*e*x + 6)**(3/2)/(2*e*(e*x + 2)**2) + 9*sqrt(-3*e*x + 6)/(4*e*(e*x + 2)) - 9
*sqrt(3)*atanh(sqrt(3)*sqrt(-3*e*x + 6)/6)/(8*e)

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Mathematica [A]  time = 0.103321, size = 80, normalized size = 0.93 \[ -\frac{3 \sqrt{12-3 e^2 x^2} \left (3 (e x+2)^2 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )-2 \sqrt{e x-2} (5 e x+2)\right )}{8 e \sqrt{e x-2} (e x+2)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(9/2),x]

[Out]

(-3*Sqrt[12 - 3*e^2*x^2]*(-2*Sqrt[-2 + e*x]*(2 + 5*e*x) + 3*(2 + e*x)^2*ArcTan[S
qrt[-2 + e*x]/2]))/(8*e*Sqrt[-2 + e*x]*(2 + e*x)^(5/2))

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Maple [A]  time = 0.028, size = 126, normalized size = 1.5 \[ -{\frac{3\,\sqrt{3}}{8\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ( 3\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}{x}^{2}{e}^{2}+12\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}xe-10\,xe\sqrt{-3\,ex+6}+12\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -4\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{5}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*e^2*x^2+12)^(3/2)/(e*x+2)^(9/2),x)

[Out]

-3/8*(-e^2*x^2+4)^(1/2)*(3*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x^2*e^2
+12*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x*e-10*x*e*(-3*e*x+6)^(1/2)+12
*3^(1/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))-4*(-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 = 0.859386, size = 84, normalized size = 0.98 \[ \frac{-9 i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - \frac{2 \,{\left (-15 i \, \sqrt{3}{\left (e x - 2\right )}^{\frac{3}{2}} - 36 i \, \sqrt{3} \sqrt{e x - 2}\right )}}{{\left (e x - 2\right )}^{2} + 8 \, e x}}{8 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(9/2),x, algorithm="maxima")

[Out]

1/8*(-9*I*sqrt(3)*arctan(1/2*sqrt(e*x - 2)) - 2*(-15*I*sqrt(3)*(e*x - 2)^(3/2) -
36*I*sqrt(3)*sqrt(e*x - 2))/((e*x - 2)^2 + 8*e*x))/e

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Fricas [A]  time = 0.222431, size = 188, normalized size = 2.19 \[ \frac{3 \,{\left (3 \, \sqrt{3}{\left (e^{3} x^{3} + 6 \, e^{2} x^{2} + 12 \, e x + 8\right )} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) + 4 \, \sqrt{-3 \, e^{2} x^{2} + 12}{\left (5 \, e x + 2\right )} \sqrt{e x + 2}\right )}}{16 \,{\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(9/2),x, algorithm="fricas")

[Out]

3/16*(3*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*
sqrt(3)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2) - 36)/(e^2*x^2 + 4*e*x + 4)) + 4*sqr
t(-3*e^2*x^2 + 12)*(5*e*x + 2)*sqrt(e*x + 2))/(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)**(3/2)/(e*x+2)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}}{{\left (e x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(9/2),x, algorithm="giac")

[Out]

integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(9/2), x)