3.886 \(\int \frac{\sqrt{12-3 e^2 x^2}}{\sqrt{2+e x}} \, dx\)

Optimal. Leaf size=20 \[ -\frac{2 (2-e x)^{3/2}}{\sqrt{3} e} \]

[Out]

(-2*(2 - e*x)^(3/2))/(Sqrt[3]*e)

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Rubi [A]  time = 0.0396517, antiderivative size = 20, 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 \[ -\frac{2 (2-e x)^{3/2}}{\sqrt{3} e} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(2 - e*x)^(3/2))/(Sqrt[3]*e)

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Rubi in Sympy [A]  time = 4.22721, size = 15, normalized size = 0.75 \[ - \frac{2 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{9 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-3*e*x + 6)**(3/2)/(9*e)

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Mathematica [A]  time = 0.0285834, size = 34, normalized size = 1.7 \[ \frac{2 (e x-2) \sqrt{4-e^2 x^2}}{e \sqrt{3 e x+6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 30, normalized size = 1.5 \[{\frac{2\,ex-4}{3\,e}\sqrt{-3\,{e}^{2}{x}^{2}+12}{\frac{1}{\sqrt{ex+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(e*x-2)*(-3*e^2*x^2+12)^(1/2)/e/(e*x+2)^(1/2)

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Maxima [A]  time = 0.786002, size = 34, normalized size = 1.7 \[ \frac{{\left (2 i \, \sqrt{3} e x - 4 i \, \sqrt{3}\right )} \sqrt{e x - 2}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(2*I*sqrt(3)*e*x - 4*I*sqrt(3))*sqrt(e*x - 2)/e

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Fricas [A]  time = 0.225133, size = 61, normalized size = 3.05 \[ -\frac{2 \,{\left (e^{3} x^{3} - 2 \, e^{2} x^{2} - 4 \, e x + 8\right )}}{\sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(e^3*x^3 - 2*e^2*x^2 - 4*e*x + 8)/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \sqrt{3} \int \frac{\sqrt{- e^{2} x^{2} + 4}}{\sqrt{e x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*Integral(sqrt(-e**2*x**2 + 4)/sqrt(e*x + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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