3.892 \(\int \sqrt{2+e x} \left (12-3 e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 (2-e x)^{9/2}}{\sqrt{3} e}+\frac{48 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e} \]

[Out]

(-96*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (48*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (2*(2
- e*x)^(9/2))/(Sqrt[3]*e)

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Rubi [A]  time = 0.0841616, antiderivative size = 65, 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 \[ -\frac{2 (2-e x)^{9/2}}{\sqrt{3} e}+\frac{48 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

(-96*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (48*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (2*(2
- e*x)^(9/2))/(Sqrt[3]*e)

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Rubi in Sympy [A]  time = 12.4895, size = 54, normalized size = 0.83 \[ - \frac{2 \sqrt{3} \left (- e x + 2\right )^{\frac{9}{2}}}{3 e} + \frac{48 \sqrt{3} \left (- e x + 2\right )^{\frac{7}{2}}}{7 e} - \frac{96 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(3)*(-e*x + 2)**(9/2)/(3*e) + 48*sqrt(3)*(-e*x + 2)**(7/2)/(7*e) - 96*sqr
t(3)*(-e*x + 2)**(5/2)/(5*e)

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Mathematica [A]  time = 0.048807, size = 52, normalized size = 0.8 \[ -\frac{2 (e x-2)^2 \sqrt{4-e^2 x^2} \left (35 e^2 x^2+220 e x+428\right )}{35 e \sqrt{3 e x+6}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-2 + e*x)^2*Sqrt[4 - e^2*x^2]*(428 + 220*e*x + 35*e^2*x^2))/(35*e*Sqrt[6 +
3*e*x])

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Maple [A]  time = 0.009, size = 44, normalized size = 0.7 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( 35\,{e}^{2}{x}^{2}+220\,ex+428 \right ) }{315\,e} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{{\frac{3}{2}}} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(e*x-2)*(35*e^2*x^2+220*e*x+428)*(-3*e^2*x^2+12)^(3/2)/e/(e*x+2)^(3/2)

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Maxima [A]  time = 0.825459, size = 96, normalized size = 1.48 \[ -\frac{{\left (70 i \, \sqrt{3} e^{4} x^{4} + 160 i \, \sqrt{3} e^{3} x^{3} - 624 i \, \sqrt{3} e^{2} x^{2} - 1664 i \, \sqrt{3} e x + 3424 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{105 \,{\left (e^{2} x + 2 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(70*I*sqrt(3)*e^4*x^4 + 160*I*sqrt(3)*e^3*x^3 - 624*I*sqrt(3)*e^2*x^2 - 1
664*I*sqrt(3)*e*x + 3424*I*sqrt(3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

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Fricas [A]  time = 0.221609, size = 95, normalized size = 1.46 \[ \frac{2 \,{\left (35 \, e^{6} x^{6} + 80 \, e^{5} x^{5} - 452 \, e^{4} x^{4} - 1152 \, e^{3} x^{3} + 2960 \, e^{2} x^{2} + 3328 \, e x - 6848\right )}}{35 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(35*e^6*x^6 + 80*e^5*x^5 - 452*e^4*x^4 - 1152*e^3*x^3 + 2960*e^2*x^2 + 3328
*e*x - 6848)/(sqrt(-3*e^2*x^2 + 12)*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((e*x+2)**(1/2)*(-3*e**2*x**2+12)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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