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

Optimal. Leaf size=22 \[ \frac{2}{3 \sqrt{3} e \sqrt{2-e x}} \]

[Out]

2/(3*Sqrt[3]*e*Sqrt[2 - e*x])

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Rubi [A]  time = 0.0406753, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2}{3 \sqrt{3} e \sqrt{2-e x}} \]

Antiderivative was successfully verified.

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

[Out]

2/(3*Sqrt[3]*e*Sqrt[2 - e*x])

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Rubi in Sympy [A]  time = 3.94468, size = 14, normalized size = 0.64 \[ \frac{2}{3 e \sqrt{- 3 e x + 6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(3*e*sqrt(-3*e*x + 6))

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Mathematica [A]  time = 0.0224465, size = 30, normalized size = 1.36 \[ \frac{2 \sqrt{e x+2}}{3 e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 30, normalized size = 1.4 \[ -2\,{\frac{ \left ( ex-2 \right ) \left ( ex+2 \right ) ^{3/2}}{e \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.787998, size = 20, normalized size = 0.91 \[ -\frac{2 i \, \sqrt{3}}{9 \, \sqrt{e x - 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*I*sqrt(3)/(sqrt(e*x - 2)*e)

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Fricas [A]  time = 0.219496, size = 46, normalized size = 2.09 \[ -\frac{2 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{9 \,{\left (e^{3} x^{2} - 4 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^3*x^2 - 4*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((e*x+2)**(3/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.609622, size = 4, normalized size = 0.18 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x