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

Optimal. Leaf size=43 \[ \frac{2 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{8 (2-e x)^{3/2}}{\sqrt{3} e} \]

[Out]

(-8*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (2*Sqrt[3]*(2 - e*x)^(5/2))/(5*e)

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Rubi [A]  time = 0.0662358, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{8 (2-e x)^{3/2}}{\sqrt{3} e} \]

Antiderivative was successfully verified.

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

[Out]

(-8*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (2*Sqrt[3]*(2 - e*x)^(5/2))/(5*e)

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Rubi in Sympy [A]  time = 8.4983, size = 32, normalized size = 0.74 \[ - \frac{8 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{9 e} + \frac{2 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{5 e} \]

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]

-8*(-3*e*x + 6)**(3/2)/(9*e) + 2*sqrt(3)*(-e*x + 2)**(5/2)/(5*e)

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Mathematica [A]  time = 0.0342679, size = 42, normalized size = 0.98 \[ \frac{2 (e x-2) (3 e x+14) \sqrt{4-e^2 x^2}}{5 e \sqrt{3 e x+6}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-2 + e*x)*(14 + 3*e*x)*Sqrt[4 - e^2*x^2])/(5*e*Sqrt[6 + 3*e*x])

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( 3\,ex+14 \right ) }{15\,e}\sqrt{-3\,{e}^{2}{x}^{2}+12}{\frac{1}{\sqrt{ex+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(e*x-2)*(3*e*x+14)*(-3*e^2*x^2+12)^(1/2)/e/(e*x+2)^(1/2)

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Maxima [A]  time = 0.794361, size = 66, normalized size = 1.53 \[ \frac{{\left (6 i \, \sqrt{3} e^{2} x^{2} + 16 i \, \sqrt{3} e x - 56 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{15 \,{\left (e^{2} x + 2 \, e\right )}} \]

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]

1/15*(6*I*sqrt(3)*e^2*x^2 + 16*I*sqrt(3)*e*x - 56*I*sqrt(3))*(e*x + 2)*sqrt(e*x
- 2)/(e^2*x + 2*e)

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Fricas [A]  time = 0.212198, size = 73, normalized size = 1.7 \[ -\frac{2 \,{\left (3 \, e^{4} x^{4} + 8 \, e^{3} x^{3} - 40 \, e^{2} x^{2} - 32 \, e x + 112\right )}}{5 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

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]

-2/5*(3*e^4*x^4 + 8*e^3*x^3 - 40*e^2*x^2 - 32*e*x + 112)/(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. \[ \sqrt{3} \int \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}\, dx \]

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)

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

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)