3.910 \(\int \frac{(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{2 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{8}{3 \sqrt{3} e \sqrt{2-e x}} \]

[Out]

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

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Rubi [A]  time = 0.0691016, antiderivative size = 45, 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 \[ \frac{2 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{8}{3 \sqrt{3} e \sqrt{2-e x}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + e*x)^(5/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.7602, size = 36, normalized size = 0.8 \[ \frac{2 \sqrt{3} \sqrt{- e x + 2}}{9 e} + \frac{8 \sqrt{3}}{9 e \sqrt{- e x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+2)**(5/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0340932, size = 35, normalized size = 0.78 \[ -\frac{2 (e x-6) \sqrt{e x+2}}{3 e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + e*x)^(5/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(-2*(-6 + e*x)*Sqrt[2 + 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 \[ 2\,{\frac{ \left ( ex-2 \right ) \left ( ex-6 \right ) \left ( ex+2 \right ) ^{3/2}}{e \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+2)^(5/2)/(-3*e^2*x^2+12)^(3/2),x)

[Out]

2*(e*x-2)*(e*x-6)*(e*x+2)^(3/2)/e/(-3*e^2*x^2+12)^(3/2)

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Maxima [A]  time = 0.786654, size = 27, normalized size = 0.6 \[ \frac{2 i \, \sqrt{3}{\left (e x - 6\right )}}{9 \, \sqrt{e x - 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(5/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="maxima")

[Out]

2/9*I*sqrt(3)*(e*x - 6)/(sqrt(e*x - 2)*e)

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Fricas [A]  time = 0.23071, size = 50, normalized size = 1.11 \[ -\frac{2 \,{\left (e^{2} x^{2} - 4 \, e x - 12\right )}}{3 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(5/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="fricas")

[Out]

-2/3*(e^2*x^2 - 4*e*x - 12)/(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)**(5/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.57891, size = 4, normalized size = 0.09 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(5/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="giac")

[Out]

sage0*x