3.932 \(\int \frac{1}{(2+e x)^{7/2} \sqrt [4]{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{2 \left (4-e^2 x^2\right )^{3/4}}{231 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac{2 \left (4-e^2 x^2\right )^{3/4}}{77 \sqrt [4]{3} e (e x+2)^{5/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{11 \sqrt [4]{3} e (e x+2)^{7/2}} \]

[Out]

-(4 - e^2*x^2)^(3/4)/(11*3^(1/4)*e*(2 + e*x)^(7/2)) - (2*(4 - e^2*x^2)^(3/4))/(7
7*3^(1/4)*e*(2 + e*x)^(5/2)) - (2*(4 - e^2*x^2)^(3/4))/(231*3^(1/4)*e*(2 + e*x)^
(3/2))

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Rubi [A]  time = 0.142917, antiderivative size = 106, 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 \[ -\frac{2 \left (4-e^2 x^2\right )^{3/4}}{231 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac{2 \left (4-e^2 x^2\right )^{3/4}}{77 \sqrt [4]{3} e (e x+2)^{5/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{11 \sqrt [4]{3} e (e x+2)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((2 + e*x)^(7/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

-(4 - e^2*x^2)^(3/4)/(11*3^(1/4)*e*(2 + e*x)^(7/2)) - (2*(4 - e^2*x^2)^(3/4))/(7
7*3^(1/4)*e*(2 + e*x)^(5/2)) - (2*(4 - e^2*x^2)^(3/4))/(231*3^(1/4)*e*(2 + e*x)^
(3/2))

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Rubi in Sympy [A]  time = 10.6315, size = 80, normalized size = 0.75 \[ - \frac{2 \left (- 3 e^{2} x^{2} + 12\right )^{\frac{3}{4}}}{693 e \left (e x + 2\right )^{\frac{3}{2}}} - \frac{2 \left (- 3 e^{2} x^{2} + 12\right )^{\frac{3}{4}}}{231 e \left (e x + 2\right )^{\frac{5}{2}}} - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{3}{4}}}{33 e \left (e x + 2\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+2)**(7/2)/(-3*e**2*x**2+12)**(1/4),x)

[Out]

-2*(-3*e**2*x**2 + 12)**(3/4)/(693*e*(e*x + 2)**(3/2)) - 2*(-3*e**2*x**2 + 12)**
(3/4)/(231*e*(e*x + 2)**(5/2)) - (-3*e**2*x**2 + 12)**(3/4)/(33*e*(e*x + 2)**(7/
2))

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Mathematica [A]  time = 0.0618972, size = 49, normalized size = 0.46 \[ \frac{(e x-2) \left (2 e^2 x^2+14 e x+41\right )}{231 e (e x+2)^{5/2} \sqrt [4]{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((2 + e*x)^(7/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

((-2 + e*x)*(41 + 14*e*x + 2*e^2*x^2))/(231*e*(2 + e*x)^(5/2)*(12 - 3*e^2*x^2)^(
1/4))

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Maple [A]  time = 0.008, size = 44, normalized size = 0.4 \[{\frac{ \left ( ex-2 \right ) \left ( 2\,{e}^{2}{x}^{2}+14\,ex+41 \right ) }{231\,e} \left ( ex+2 \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+2)^(7/2)/(-3*e^2*x^2+12)^(1/4),x)

[Out]

1/231*(e*x-2)*(2*e^2*x^2+14*e*x+41)/(e*x+2)^(5/2)/e/(-3*e^2*x^2+12)^(1/4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\left (e x + 2\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(7/2)),x, algorithm="maxima")

[Out]

integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(7/2)), x)

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Fricas [A]  time = 0.219824, size = 84, normalized size = 0.79 \[ \frac{2 \, e^{3} x^{3} + 10 \, e^{2} x^{2} + 13 \, e x - 82}{231 \,{\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(7/2)),x, algorithm="fricas")

[Out]

1/231*(2*e^3*x^3 + 10*e^2*x^2 + 13*e*x - 82)/((e^3*x^2 + 4*e^2*x + 4*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)**(7/2)/(-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. \[ \int \frac{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\left (e x + 2\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(7/2)),x, algorithm="giac")

[Out]

integrate(1/((-3*e^2*x^2 + 12)^(1/4)*(e*x + 2)^(7/2)), x)