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

Optimal. Leaf size=71 \[ -\frac{\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (e x+2)^{5/2}}-\frac{\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (e x+2)^{7/2}} \]

[Out]

-(4 - e^2*x^2)^(5/4)/(3*3^(3/4)*e*(2 + e*x)^(7/2)) - (4 - e^2*x^2)^(5/4)/(15*3^(
3/4)*e*(2 + e*x)^(5/2))

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Rubi [A]  time = 0.0917298, 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 \[ -\frac{\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (e x+2)^{5/2}}-\frac{\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (e x+2)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(4 - e^2*x^2)^(5/4)/(3*3^(3/4)*e*(2 + e*x)^(7/2)) - (4 - e^2*x^2)^(5/4)/(15*3^(
3/4)*e*(2 + e*x)^(5/2))

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Rubi in Sympy [A]  time = 6.74902, size = 51, normalized size = 0.72 \[ - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{5}{4}}}{135 e \left (e x + 2\right )^{\frac{5}{2}}} - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{5}{4}}}{27 e \left (e x + 2\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-3*e**2*x**2 + 12)**(5/4)/(135*e*(e*x + 2)**(5/2)) - (-3*e**2*x**2 + 12)**(5/4
)/(27*e*(e*x + 2)**(7/2))

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Mathematica [A]  time = 0.0430742, size = 48, normalized size = 0.68 \[ \frac{\sqrt [4]{4-e^2 x^2} \left (e^2 x^2+5 e x-14\right )}{15\ 3^{3/4} e (e x+2)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((4 - e^2*x^2)^(1/4)*(-14 + 5*e*x + e^2*x^2))/(15*3^(3/4)*e*(2 + e*x)^(5/2))

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Maple [A]  time = 0.008, size = 35, normalized size = 0.5 \[{\frac{ \left ( ex-2 \right ) \left ( ex+7 \right ) }{45\,e}\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \left ( ex+2 \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45*(e*x-2)*(e*x+7)*(-3*e^2*x^2+12)^(1/4)/(e*x+2)^(5/2)/e

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220619, size = 82, normalized size = 1.15 \[ \frac{{\left (e^{2} x^{2} + 5 \, e x - 14\right )}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}}{45 \,{\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45*(e^2*x^2 + 5*e*x - 14)*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)/(e^4*x^3 + 6*e
^3*x^2 + 12*e^2*x + 8*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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.237314, size = 124, normalized size = 1.75 \[ -\frac{1}{180} \cdot 3^{\frac{1}{4}}{\left (\frac{9 \,{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}{\left (\frac{4}{x e + 2} - 1\right )}}{\sqrt{x e + 2}} + \frac{5 \,{\left ({\left (x e + 2\right )}^{2} - 8 \, x e\right )}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}}{{\left (x e + 2\right )}^{\frac{5}{2}}}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/180*3^(1/4)*(9*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)*(4/(x*e + 2) - 1)/sqrt(x*e +
2) + 5*((x*e + 2)^2 - 8*x*e)*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)/(x*e + 2)^(5/2))*e
^(-1)