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

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

[Out]

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

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Rubi [A]  time = 0.0688459, 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 \[ \frac{2 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{8 \sqrt{2-e x}}{\sqrt{3} e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.51388, size = 32, normalized size = 0.74 \[ \frac{2 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{27 e} - \frac{8 \sqrt{3} \sqrt{- e x + 2}}{3 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-3*e*x + 6)**(3/2)/(27*e) - 8*sqrt(3)*sqrt(-e*x + 2)/(3*e)

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Mathematica [A]  time = 0.0344961, size = 40, normalized size = 0.93 \[ \frac{2 (e x-2) \sqrt{e x+2} (e x+10)}{3 e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-2 + e*x)*Sqrt[2 + e*x]*(10 + e*x))/(3*e*Sqrt[12 - 3*e^2*x^2])

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Maple [A]  time = 0.005, size = 35, normalized size = 0.8 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( ex+10 \right ) }{3\,e}\sqrt{ex+2}{\frac{1}{\sqrt{-3\,{e}^{2}{x}^{2}+12}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(e*x-2)*(e*x+10)*(e*x+2)^(1/2)/e/(-3*e^2*x^2+12)^(1/2)

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Maxima [A]  time = 0.791039, size = 38, normalized size = 0.88 \[ -\frac{2 i \, \sqrt{3}{\left (e^{2} x^{2} + 8 \, e x - 20\right )}}{9 \, \sqrt{e x - 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*I*sqrt(3)*(e^2*x^2 + 8*e*x - 20)/(sqrt(e*x - 2)*e)

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Fricas [A]  time = 0.214562, size = 61, normalized size = 1.42 \[ \frac{2 \,{\left (e^{3} x^{3} + 10 \, e^{2} x^{2} - 4 \, e x - 40\right )}}{3 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(e^3*x^3 + 10*e^2*x^2 - 4*e*x - 40)/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)*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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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