3.933 \(\int \frac{1}{(2+e x)^{9/2} \sqrt [4]{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{2 \left (4-e^2 x^2\right )^{3/4}}{1155 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac{2 \left (4-e^2 x^2\right )^{3/4}}{385 \sqrt [4]{3} e (e x+2)^{5/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{55 \sqrt [4]{3} e (e x+2)^{7/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{15 \sqrt [4]{3} e (e x+2)^{9/2}} \]

[Out]

-(4 - e^2*x^2)^(3/4)/(15*3^(1/4)*e*(2 + e*x)^(9/2)) - (4 - e^2*x^2)^(3/4)/(55*3^
(1/4)*e*(2 + e*x)^(7/2)) - (2*(4 - e^2*x^2)^(3/4))/(385*3^(1/4)*e*(2 + e*x)^(5/2
)) - (2*(4 - e^2*x^2)^(3/4))/(1155*3^(1/4)*e*(2 + e*x)^(3/2))

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Rubi [A]  time = 0.196809, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 \left (4-e^2 x^2\right )^{3/4}}{1155 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac{2 \left (4-e^2 x^2\right )^{3/4}}{385 \sqrt [4]{3} e (e x+2)^{5/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{55 \sqrt [4]{3} e (e x+2)^{7/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{15 \sqrt [4]{3} e (e x+2)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((2 + e*x)^(9/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

-(4 - e^2*x^2)^(3/4)/(15*3^(1/4)*e*(2 + e*x)^(9/2)) - (4 - e^2*x^2)^(3/4)/(55*3^
(1/4)*e*(2 + e*x)^(7/2)) - (2*(4 - e^2*x^2)^(3/4))/(385*3^(1/4)*e*(2 + e*x)^(5/2
)) - (2*(4 - e^2*x^2)^(3/4))/(1155*3^(1/4)*e*(2 + e*x)^(3/2))

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Rubi in Sympy [A]  time = 14.0579, size = 105, normalized size = 0.74 \[ - \frac{2 \left (- 3 e^{2} x^{2} + 12\right )^{\frac{3}{4}}}{3465 e \left (e x + 2\right )^{\frac{3}{2}}} - \frac{2 \left (- 3 e^{2} x^{2} + 12\right )^{\frac{3}{4}}}{1155 e \left (e x + 2\right )^{\frac{5}{2}}} - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{3}{4}}}{165 e \left (e x + 2\right )^{\frac{7}{2}}} - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{3}{4}}}{45 e \left (e x + 2\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+2)**(9/2)/(-3*e**2*x**2+12)**(1/4),x)

[Out]

-2*(-3*e**2*x**2 + 12)**(3/4)/(3465*e*(e*x + 2)**(3/2)) - 2*(-3*e**2*x**2 + 12)*
*(3/4)/(1155*e*(e*x + 2)**(5/2)) - (-3*e**2*x**2 + 12)**(3/4)/(165*e*(e*x + 2)**
(7/2)) - (-3*e**2*x**2 + 12)**(3/4)/(45*e*(e*x + 2)**(9/2))

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Mathematica [A]  time = 0.0733999, size = 57, normalized size = 0.4 \[ \frac{(e x-2) \left (2 e^3 x^3+18 e^2 x^2+69 e x+159\right )}{1155 e (e x+2)^{7/2} \sqrt [4]{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((2 + e*x)^(9/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

((-2 + e*x)*(159 + 69*e*x + 18*e^2*x^2 + 2*e^3*x^3))/(1155*e*(2 + e*x)^(7/2)*(12
- 3*e^2*x^2)^(1/4))

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Maple [A]  time = 0.008, size = 52, normalized size = 0.4 \[{\frac{ \left ( ex-2 \right ) \left ( 2\,{e}^{3}{x}^{3}+18\,{e}^{2}{x}^{2}+69\,ex+159 \right ) }{1155\,e} \left ( ex+2 \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+2)^(9/2)/(-3*e^2*x^2+12)^(1/4),x)

[Out]

1/1155*(e*x-2)*(2*e^3*x^3+18*e^2*x^2+69*e*x+159)/(e*x+2)^(7/2)/e/(-3*e^2*x^2+12)
^(1/4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\left (e x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(9/2)),x, algorithm="maxima")

[Out]

integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(9/2)), x)

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Fricas [A]  time = 0.223176, size = 105, normalized size = 0.74 \[ \frac{2 \, e^{4} x^{4} + 14 \, e^{3} x^{3} + 33 \, e^{2} x^{2} + 21 \, e x - 318}{1155 \,{\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(9/2)),x, algorithm="fricas")

[Out]

1/1155*(2*e^4*x^4 + 14*e^3*x^3 + 33*e^2*x^2 + 21*e*x - 318)/((e^4*x^3 + 6*e^3*x^
2 + 12*e^2*x + 8*e)*(-3*e^2*x^2 + 12)^(1/4)*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)**(9/2)/(-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. \[ \int \frac{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\left (e x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(9/2)),x, algorithm="giac")

[Out]

integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(9/2)), x)