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

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

[Out]

(-2*Sqrt[2 - e*x])/(Sqrt[3]*e)

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Rubi [A]  time = 0.0405082, 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 \[ -\frac{2 \sqrt{2-e x}}{\sqrt{3} e} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[2 - e*x])/(Sqrt[3]*e)

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Rubi in Sympy [A]  time = 4.20868, size = 15, normalized size = 0.75 \[ - \frac{2 \sqrt{- 3 e x + 6}}{3 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-3*e*x + 6)/(3*e)

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Mathematica [A]  time = 0.0266031, size = 33, normalized size = 1.65 \[ \frac{2 (e x-2) \sqrt{e x+2}}{e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 30, normalized size = 1.5 \[ 2\,{\frac{ \left ( ex-2 \right ) \sqrt{ex+2}}{e\sqrt{-3\,{e}^{2}{x}^{2}+12}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.788501, size = 34, normalized size = 1.7 \[ -\frac{2 i \, \sqrt{3} e x - 4 i \, \sqrt{3}}{3 \, \sqrt{e x - 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + 2)/sqrt(-3*e^2*x^2 + 12),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 = 0.22895, size = 45, normalized size = 2.25 \[ \frac{2 \,{\left (e^{2} x^{2} - 4\right )}}{\sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(e^2*x^2 - 4)/(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. \[ \frac{\sqrt{3} \int \frac{\sqrt{e x + 2}}{\sqrt{- e^{2} x^{2} + 4}}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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