3.901 \(\int \frac{(2+e x)^{5/2}}{\sqrt{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 (2-e x)^{5/2}}{5 \sqrt{3} e}+\frac{16 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{32 \sqrt{2-e x}}{\sqrt{3} e} \]

[Out]

(-32*Sqrt[2 - e*x])/(Sqrt[3]*e) + (16*(2 - e*x)^(3/2))/(3*Sqrt[3]*e) - (2*(2 - e
*x)^(5/2))/(5*Sqrt[3]*e)

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Rubi [A]  time = 0.0854073, antiderivative size = 65, 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 \[ -\frac{2 (2-e x)^{5/2}}{5 \sqrt{3} e}+\frac{16 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{32 \sqrt{2-e x}}{\sqrt{3} e} \]

Antiderivative was successfully verified.

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

[Out]

(-32*Sqrt[2 - e*x])/(Sqrt[3]*e) + (16*(2 - e*x)^(3/2))/(3*Sqrt[3]*e) - (2*(2 - e
*x)^(5/2))/(5*Sqrt[3]*e)

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Rubi in Sympy [A]  time = 12.2528, size = 51, normalized size = 0.78 \[ \frac{16 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{27 e} - \frac{2 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{15 e} - \frac{32 \sqrt{3} \sqrt{- e x + 2}}{3 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+2)**(5/2)/(-3*e**2*x**2+12)**(1/2),x)

[Out]

16*(-3*e*x + 6)**(3/2)/(27*e) - 2*sqrt(3)*(-e*x + 2)**(5/2)/(15*e) - 32*sqrt(3)*
sqrt(-e*x + 2)/(3*e)

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Mathematica [A]  time = 0.0438994, size = 49, normalized size = 0.75 \[ \frac{2 (e x-2) \sqrt{e x+2} \left (3 e^2 x^2+28 e x+172\right )}{15 e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-2 + e*x)*Sqrt[2 + e*x]*(172 + 28*e*x + 3*e^2*x^2))/(15*e*Sqrt[12 - 3*e^2*x^
2])

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Maple [A]  time = 0.008, size = 44, normalized size = 0.7 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( 3\,{e}^{2}{x}^{2}+28\,ex+172 \right ) }{15\,e}\sqrt{ex+2}{\frac{1}{\sqrt{-3\,{e}^{2}{x}^{2}+12}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+2)^(5/2)/(-3*e^2*x^2+12)^(1/2),x)

[Out]

2/15*(e*x-2)*(3*e^2*x^2+28*e*x+172)*(e*x+2)^(1/2)/e/(-3*e^2*x^2+12)^(1/2)

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Maxima [A]  time = 0.793197, size = 63, normalized size = 0.97 \[ -\frac{6 i \, \sqrt{3} e^{3} x^{3} + 44 i \, \sqrt{3} e^{2} x^{2} + 232 i \, \sqrt{3} e x - 688 i \, \sqrt{3}}{45 \, \sqrt{e x - 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/45*(6*I*sqrt(3)*e^3*x^3 + 44*I*sqrt(3)*e^2*x^2 + 232*I*sqrt(3)*e*x - 688*I*sq
rt(3))/(sqrt(e*x - 2)*e)

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Fricas [A]  time = 0.215427, size = 73, normalized size = 1.12 \[ \frac{2 \,{\left (3 \, e^{4} x^{4} + 28 \, e^{3} x^{3} + 160 \, e^{2} x^{2} - 112 \, e x - 688\right )}}{15 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*e^4*x^4 + 28*e^3*x^3 + 160*e^2*x^2 - 112*e*x - 688)/(sqrt(-3*e^2*x^2 + 1
2)*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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + 2\right )}^{\frac{5}{2}}}{\sqrt{-3 \, e^{2} x^{2} + 12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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