3.922 \(\int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx\)

Optimal. Leaf size=35 \[ -\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{5 e (e x+2)^{5/2}} \]

[Out]

-(3^(1/4)*(4 - e^2*x^2)^(5/4))/(5*e*(2 + e*x)^(5/2))

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Rubi [A]  time = 0.0456917, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{5 e (e x+2)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(5/2),x]

[Out]

-(3^(1/4)*(4 - e^2*x^2)^(5/4))/(5*e*(2 + e*x)^(5/2))

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Rubi in Sympy [A]  time = 4.20868, size = 26, normalized size = 0.74 \[ - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{5}{4}}}{15 e \left (e x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3*e**2*x**2+12)**(1/4)/(e*x+2)**(5/2),x)

[Out]

-(-3*e**2*x**2 + 12)**(5/4)/(15*e*(e*x + 2)**(5/2))

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Mathematica [A]  time = 0.0395077, size = 35, normalized size = 1. \[ \frac{(e x-2) \sqrt [4]{12-3 e^2 x^2}}{5 e (e x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(5/2),x]

[Out]

((-2 + e*x)*(12 - 3*e^2*x^2)^(1/4))/(5*e*(2 + e*x)^(3/2))

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Maple [A]  time = 0.007, size = 30, normalized size = 0.9 \[{\frac{ex-2}{5\,e}\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(5/2),x)

[Out]

1/5*(e*x-2)/(e*x+2)^(3/2)/e*(-3*e^2*x^2+12)^(1/4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}}{{\left (e x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(5/2),x, algorithm="maxima")

[Out]

integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(5/2), x)

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Fricas [A]  time = 0.228555, size = 61, normalized size = 1.74 \[ \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}{\left (e x - 2\right )}}{5 \,{\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(5/2),x, algorithm="fricas")

[Out]

1/5*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)*(e*x - 2)/(e^3*x^2 + 4*e^2*x + 4*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((-3*e**2*x**2+12)**(1/4)/(e*x+2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230931, size = 62, normalized size = 1.77 \[ -\frac{3^{\frac{1}{4}}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}{\left (\frac{4}{x e + 2} - 1\right )} e^{\left (-1\right )}}{5 \, \sqrt{x e + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(5/2),x, algorithm="giac")

[Out]

-1/5*3^(1/4)*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)*(4/(x*e + 2) - 1)*e^(-1)/sqrt(x*e
+ 2)