3.899 \(\int \frac{\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{13/2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{3 \sqrt{3} (2-e x)^{3/2}}{4 e (e x+2)^4}-\frac{9 \sqrt{3} \sqrt{2-e x}}{1024 e (e x+2)}-\frac{3 \sqrt{3} \sqrt{2-e x}}{128 e (e x+2)^2}+\frac{3 \sqrt{3} \sqrt{2-e x}}{8 e (e x+2)^3}-\frac{9 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{2048 e} \]

[Out]

(-3*Sqrt[3]*(2 - e*x)^(3/2))/(4*e*(2 + e*x)^4) + (3*Sqrt[3]*Sqrt[2 - e*x])/(8*e*
(2 + e*x)^3) - (3*Sqrt[3]*Sqrt[2 - e*x])/(128*e*(2 + e*x)^2) - (9*Sqrt[3]*Sqrt[2
- e*x])/(1024*e*(2 + e*x)) - (9*Sqrt[3]*ArcTanh[Sqrt[2 - e*x]/2])/(2048*e)

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Rubi [A]  time = 0.199858, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{3 \sqrt{3} (2-e x)^{3/2}}{4 e (e x+2)^4}-\frac{9 \sqrt{3} \sqrt{2-e x}}{1024 e (e x+2)}-\frac{3 \sqrt{3} \sqrt{2-e x}}{128 e (e x+2)^2}+\frac{3 \sqrt{3} \sqrt{2-e x}}{8 e (e x+2)^3}-\frac{9 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{2048 e} \]

Antiderivative was successfully verified.

[In]  Int[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(13/2),x]

[Out]

(-3*Sqrt[3]*(2 - e*x)^(3/2))/(4*e*(2 + e*x)^4) + (3*Sqrt[3]*Sqrt[2 - e*x])/(8*e*
(2 + e*x)^3) - (3*Sqrt[3]*Sqrt[2 - e*x])/(128*e*(2 + e*x)^2) - (9*Sqrt[3]*Sqrt[2
- e*x])/(1024*e*(2 + e*x)) - (9*Sqrt[3]*ArcTanh[Sqrt[2 - e*x]/2])/(2048*e)

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Rubi in Sympy [A]  time = 23.0459, size = 112, normalized size = 0.78 \[ - \frac{\left (- 3 e x + 6\right )^{\frac{3}{2}}}{4 e \left (e x + 2\right )^{4}} - \frac{9 \sqrt{- 3 e x + 6}}{1024 e \left (e x + 2\right )} - \frac{3 \sqrt{- 3 e x + 6}}{128 e \left (e x + 2\right )^{2}} + \frac{3 \sqrt{- 3 e x + 6}}{8 e \left (e x + 2\right )^{3}} - \frac{9 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{2048 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3*e**2*x**2+12)**(3/2)/(e*x+2)**(13/2),x)

[Out]

-(-3*e*x + 6)**(3/2)/(4*e*(e*x + 2)**4) - 9*sqrt(-3*e*x + 6)/(1024*e*(e*x + 2))
- 3*sqrt(-3*e*x + 6)/(128*e*(e*x + 2)**2) + 3*sqrt(-3*e*x + 6)/(8*e*(e*x + 2)**3
) - 9*sqrt(3)*atanh(sqrt(3)*sqrt(-3*e*x + 6)/6)/(2048*e)

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Mathematica [A]  time = 0.128793, size = 96, normalized size = 0.67 \[ -\frac{3 \sqrt{12-3 e^2 x^2} \left (2 \sqrt{e x-2} \left (3 e^3 x^3+26 e^2 x^2-316 e x+312\right )+3 (e x+2)^4 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )\right )}{2048 e \sqrt{e x-2} (e x+2)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(13/2),x]

[Out]

(-3*Sqrt[12 - 3*e^2*x^2]*(2*Sqrt[-2 + e*x]*(312 - 316*e*x + 26*e^2*x^2 + 3*e^3*x
^3) + 3*(2 + e*x)^4*ArcTan[Sqrt[-2 + e*x]/2]))/(2048*e*Sqrt[-2 + e*x]*(2 + e*x)^
(9/2))

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Maple [A]  time = 0.032, size = 208, normalized size = 1.4 \[ -{\frac{3\,\sqrt{3}}{2048\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ( 3\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}{x}^{4}{e}^{4}+24\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}{x}^{3}{e}^{3}+6\,{x}^{3}{e}^{3}\sqrt{-3\,ex+6}+72\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}{x}^{2}{e}^{2}+52\,{x}^{2}{e}^{2}\sqrt{-3\,ex+6}+96\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}xe-632\,xe\sqrt{-3\,ex+6}+48\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) +624\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{9}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*e^2*x^2+12)^(3/2)/(e*x+2)^(13/2),x)

[Out]

-3/2048*(-e^2*x^2+4)^(1/2)*(3*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x^4*
e^4+24*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x^3*e^3+6*x^3*e^3*(-3*e*x+6
)^(1/2)+72*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x^2*e^2+52*x^2*e^2*(-3*
e*x+6)^(1/2)+96*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))*3^(1/2)*x*e-632*x*e*(-3*e*
x+6)^(1/2)+48*3^(1/2)*arctanh(1/6*3^(1/2)*(-3*e*x+6)^(1/2))+624*(-3*e*x+6)^(1/2)
)*3^(1/2)/((e*x+2)^9)^(1/2)/(-3*e*x+6)^(1/2)/e

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Maxima [A]  time = 0.865674, size = 142, normalized size = 0.99 \[ \frac{-9 i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - \frac{2 \,{\left (9 i \, \sqrt{3}{\left (e x - 2\right )}^{\frac{7}{2}} + 132 i \, \sqrt{3}{\left (e x - 2\right )}^{\frac{5}{2}} - 528 i \, \sqrt{3}{\left (e x - 2\right )}^{\frac{3}{2}} - 576 i \, \sqrt{3} \sqrt{e x - 2}\right )}}{{\left (e x - 2\right )}^{4} + 16 \,{\left (e x - 2\right )}^{3} + 96 \,{\left (e x - 2\right )}^{2} + 256 \, e x - 256}}{2048 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(13/2),x, algorithm="maxima")

[Out]

1/2048*(-9*I*sqrt(3)*arctan(1/2*sqrt(e*x - 2)) - 2*(9*I*sqrt(3)*(e*x - 2)^(7/2)
+ 132*I*sqrt(3)*(e*x - 2)^(5/2) - 528*I*sqrt(3)*(e*x - 2)^(3/2) - 576*I*sqrt(3)*
sqrt(e*x - 2))/((e*x - 2)^4 + 16*(e*x - 2)^3 + 96*(e*x - 2)^2 + 256*e*x - 256))/
e

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Fricas [A]  time = 0.232016, size = 252, normalized size = 1.75 \[ \frac{3 \,{\left (3 \, \sqrt{3}{\left (e^{5} x^{5} + 10 \, e^{4} x^{4} + 40 \, e^{3} x^{3} + 80 \, e^{2} x^{2} + 80 \, e x + 32\right )} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \,{\left (3 \, e^{3} x^{3} + 26 \, e^{2} x^{2} - 316 \, e x + 312\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}\right )}}{4096 \,{\left (e^{6} x^{5} + 10 \, e^{5} x^{4} + 40 \, e^{4} x^{3} + 80 \, e^{3} x^{2} + 80 \, e^{2} x + 32 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(13/2),x, algorithm="fricas")

[Out]

3/4096*(3*sqrt(3)*(e^5*x^5 + 10*e^4*x^4 + 40*e^3*x^3 + 80*e^2*x^2 + 80*e*x + 32)
*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^3*x^3 + 26*e^2*x^2 - 316*e*x + 312)*sqrt(-3*e^2*
x^2 + 12)*sqrt(e*x + 2))/(e^6*x^5 + 10*e^5*x^4 + 40*e^4*x^3 + 80*e^3*x^2 + 80*e^
2*x + 32*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)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}}{{\left (e x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(13/2),x, algorithm="giac")

[Out]

integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(13/2), x)