3.931 \(\int \frac{1}{(2+e x)^{5/2} \sqrt [4]{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{\left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (e x+2)^{5/2}} \]

[Out]

-(4 - e^2*x^2)^(3/4)/(7*3^(1/4)*e*(2 + e*x)^(5/2)) - (4 - e^2*x^2)^(3/4)/(21*3^(
1/4)*e*(2 + e*x)^(3/2))

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Rubi [A]  time = 0.0942859, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (e x+2)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(4 - e^2*x^2)^(3/4)/(7*3^(1/4)*e*(2 + e*x)^(5/2)) - (4 - e^2*x^2)^(3/4)/(21*3^(
1/4)*e*(2 + e*x)^(3/2))

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Rubi in Sympy [A]  time = 7.14834, size = 51, normalized size = 0.72 \[ - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{3}{4}}}{63 e \left (e x + 2\right )^{\frac{3}{2}}} - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{3}{4}}}{21 e \left (e x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-3*e**2*x**2 + 12)**(3/4)/(63*e*(e*x + 2)**(3/2)) - (-3*e**2*x**2 + 12)**(3/4)
/(21*e*(e*x + 2)**(5/2))

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Mathematica [A]  time = 0.0569384, size = 40, normalized size = 0.56 \[ \frac{(e x-2) (e x+5)}{21 e (e x+2)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2 + e*x)*(5 + e*x))/(21*e*(2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(1/4))

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Maple [A]  time = 0.007, size = 35, normalized size = 0.5 \[{\frac{ \left ( ex-2 \right ) \left ( ex+5 \right ) }{21\,e} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*(e*x-2)*(e*x+5)/(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. \[ \int \frac{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\left (e x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230575, size = 61, normalized size = 0.86 \[ \frac{e^{2} x^{2} + 3 \, e x - 10}{21 \,{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\left (e^{2} x + 2 \, e\right )} \sqrt{e x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*(e^2*x^2 + 3*e*x - 10)/((-3*e^2*x^2 + 12)^(1/4)*(e^2*x + 2*e)*sqrt(e*x + 2)
)

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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)**(5/2)/(-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. \[ \int \frac{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\left (e x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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