3.914 \(\int \frac{1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{5 \sqrt{2-e x}}{256 \sqrt{3} e (e x+2)}-\frac{5 \sqrt{2-e x}}{96 \sqrt{3} e (e x+2)^2}+\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (e x+2)^2}-\frac{5 \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{512 \sqrt{3} e} \]

[Out]

1/(6*Sqrt[3]*e*Sqrt[2 - e*x]*(2 + e*x)^2) - (5*Sqrt[2 - e*x])/(96*Sqrt[3]*e*(2 +
e*x)^2) - (5*Sqrt[2 - e*x])/(256*Sqrt[3]*e*(2 + e*x)) - (5*ArcTanh[Sqrt[2 - e*x
]/2])/(512*Sqrt[3]*e)

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Rubi [A]  time = 0.162117, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{5 \sqrt{2-e x}}{256 \sqrt{3} e (e x+2)}-\frac{5 \sqrt{2-e x}}{96 \sqrt{3} e (e x+2)^2}+\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (e x+2)^2}-\frac{5 \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{512 \sqrt{3} e} \]

Antiderivative was successfully verified.

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

[Out]

1/(6*Sqrt[3]*e*Sqrt[2 - e*x]*(2 + e*x)^2) - (5*Sqrt[2 - e*x])/(96*Sqrt[3]*e*(2 +
e*x)^2) - (5*Sqrt[2 - e*x])/(256*Sqrt[3]*e*(2 + e*x)) - (5*ArcTanh[Sqrt[2 - e*x
]/2])/(512*Sqrt[3]*e)

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Rubi in Sympy [A]  time = 17.9652, size = 92, normalized size = 0.8 \[ - \frac{5 \sqrt{- 3 e x + 6}}{768 e \left (e x + 2\right )} - \frac{5 \sqrt{- 3 e x + 6}}{288 e \left (e x + 2\right )^{2}} - \frac{5 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{1536 e} + \frac{1}{6 e \sqrt{- 3 e x + 6} \left (e x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(-3*e*x + 6)/(768*e*(e*x + 2)) - 5*sqrt(-3*e*x + 6)/(288*e*(e*x + 2)**2)
- 5*sqrt(3)*atanh(sqrt(3)*sqrt(-3*e*x + 6)/6)/(1536*e) + 1/(6*e*sqrt(-3*e*x + 6)
*(e*x + 2)**2)

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Mathematica [A]  time = 0.1032, size = 76, normalized size = 0.66 \[ \frac{30 e^2 x^2+80 e x+15 \sqrt{e x-2} (e x+2)^2 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )-24}{1536 e (e x+2)^{3/2} \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-24 + 80*e*x + 30*e^2*x^2 + 15*Sqrt[-2 + e*x]*(2 + e*x)^2*ArcTan[Sqrt[-2 + e*x]
/2])/(1536*e*(2 + e*x)^(3/2)*Sqrt[12 - 3*e^2*x^2])

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Maple [A]  time = 0.028, size = 135, normalized size = 1.2 \[{\frac{1}{ \left ( 4608\,ex-9216 \right ) e}\sqrt{-3\,{e}^{2}{x}^{2}+12} \left ( 5\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}\sqrt{-3\,ex+6}{x}^{2}{e}^{2}+20\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}\sqrt{-3\,ex+6}xe-30\,{e}^{2}{x}^{2}+20\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{-3\,ex+6}-80\,ex+24 \right ) \left ( ex+2 \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4608/(e*x+2)^(5/2)*(-3*e^2*x^2+12)^(1/2)*(5*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/
2))*3^(1/2)*(-3*e*x+6)^(1/2)*x^2*e^2+20*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^
(1/2)*(-3*e*x+6)^(1/2)*x*e-30*e^2*x^2+20*3^(1/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6)^
(1/2))*(-3*e*x+6)^(1/2)-80*e*x+24)/(e*x-2)/e

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Maxima [A]  time = 0.847618, size = 107, normalized size = 0.93 \[ \frac{-15 i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - \frac{2 \,{\left (15 i \, \sqrt{3}{\left (e x - 2\right )}^{2} + 100 i \, \sqrt{3}{\left (e x - 2\right )} + 128 i \, \sqrt{3}\right )}}{{\left (e x - 2\right )}^{\frac{5}{2}} + 8 \,{\left (e x - 2\right )}^{\frac{3}{2}} + 16 \, \sqrt{e x - 2}}}{4608 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(3/2)*(e*x + 2)^(3/2)),x, algorithm="maxima")

[Out]

1/4608*(-15*I*sqrt(3)*arctan(1/2*sqrt(e*x - 2)) - 2*(15*I*sqrt(3)*(e*x - 2)^2 +
100*I*sqrt(3)*(e*x - 2) + 128*I*sqrt(3))/((e*x - 2)^(5/2) + 8*(e*x - 2)^(3/2) +
16*sqrt(e*x - 2)))/e

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Fricas [A]  time = 0.215857, size = 204, normalized size = 1.77 \[ -\frac{\sqrt{3}{\left (4 \, \sqrt{3}{\left (15 \, e^{2} x^{2} + 40 \, e x - 12\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 45 \,{\left (e^{4} x^{4} + 4 \, e^{3} x^{3} - 16 \, e x - 16\right )} \log \left (-\frac{\sqrt{3}{\left (e^{2} x^{2} - 4 \, e x - 12\right )} + 4 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{e^{2} x^{2} + 4 \, e x + 4}\right )\right )}}{27648 \,{\left (e^{5} x^{4} + 4 \, e^{4} x^{3} - 16 \, e^{2} x - 16 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(3/2)*(e*x + 2)^(3/2)),x, algorithm="fricas")

[Out]

-1/27648*sqrt(3)*(4*sqrt(3)*(15*e^2*x^2 + 40*e*x - 12)*sqrt(-3*e^2*x^2 + 12)*sqr
t(e*x + 2) - 45*(e^4*x^4 + 4*e^3*x^3 - 16*e*x - 16)*log(-(sqrt(3)*(e^2*x^2 - 4*e
*x - 12) + 4*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2))/(e^2*x^2 + 4*e*x + 4)))/(e^5*x
^4 + 4*e^4*x^3 - 16*e^2*x - 16*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(1/(e*x+2)**(3/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, -2\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(3/2)*(e*x + 2)^(3/2)),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, -2]