3.930 \(\int \frac{1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0452638, 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 \[ -\frac{\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (e x+2)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.57203, size = 26, normalized size = 0.74 \[ - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{3}{4}}}{9 e \left (e x + 2\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-3*e**2*x**2 + 12)**(3/4)/(9*e*(e*x + 2)**(3/2))

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Mathematica [A]  time = 0.0386363, size = 35, normalized size = 1. \[ \frac{e x-2}{3 e \sqrt{e x+2} \sqrt [4]{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2 + e*x)/(3*e*Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(1/4))

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Maple [A]  time = 0.007, size = 30, normalized size = 0.9 \[{\frac{ex-2}{3\,e}{\frac{1}{\sqrt{ex+2}}}{\frac{1}{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.217322, size = 39, normalized size = 1.11 \[ \frac{e x - 2}{3 \,{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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