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

Optimal. Leaf size=45 \[ \frac{6 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{24 \sqrt{3} (2-e x)^{5/2}}{5 e} \]

[Out]

(-24*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (6*Sqrt[3]*(2 - e*x)^(7/2))/(7*e)

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Rubi [A]  time = 0.0674326, antiderivative size = 45, 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 \[ \frac{6 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{24 \sqrt{3} (2-e x)^{5/2}}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

(-24*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (6*Sqrt[3]*(2 - e*x)^(7/2))/(7*e)

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Rubi in Sympy [A]  time = 9.93868, size = 36, normalized size = 0.8 \[ \frac{6 \sqrt{3} \left (- e x + 2\right )^{\frac{7}{2}}}{7 e} - \frac{24 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(3)*(-e*x + 2)**(7/2)/(7*e) - 24*sqrt(3)*(-e*x + 2)**(5/2)/(5*e)

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Mathematica [A]  time = 0.0467985, size = 43, normalized size = 0.96 \[ -\frac{6 (e x-2)^2 (5 e x+18) \sqrt{12-3 e^2 x^2}}{35 e \sqrt{e x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*(-2 + e*x)^2*(18 + 5*e*x)*Sqrt[12 - 3*e^2*x^2])/(35*e*Sqrt[2 + e*x])

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( 5\,ex+18 \right ) }{35\,e} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{{\frac{3}{2}}} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(e*x-2)*(5*e*x+18)*(-3*e^2*x^2+12)^(3/2)/e/(e*x+2)^(3/2)

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Maxima [A]  time = 0.854869, size = 63, normalized size = 1.4 \[ -\frac{{\left (30 i \, \sqrt{3} e^{3} x^{3} - 12 i \, \sqrt{3} e^{2} x^{2} - 312 i \, \sqrt{3} e x + 432 i \, \sqrt{3}\right )} \sqrt{e x - 2}}{35 \, e} \]

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="maxima")

[Out]

-1/35*(30*I*sqrt(3)*e^3*x^3 - 12*I*sqrt(3)*e^2*x^2 - 312*I*sqrt(3)*e*x + 432*I*s
qrt(3))*sqrt(e*x - 2)/e

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Fricas [A]  time = 0.223516, size = 84, normalized size = 1.87 \[ \frac{18 \,{\left (5 \, e^{5} x^{5} - 2 \, e^{4} x^{4} - 72 \, e^{3} x^{3} + 80 \, e^{2} x^{2} + 208 \, e x - 288\right )}}{35 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

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]

18/35*(5*e^5*x^5 - 2*e^4*x^4 - 72*e^3*x^3 + 80*e^2*x^2 + 208*e*x - 288)/(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((-3*e**2*x**2+12)**(3/2)/(e*x+2)**(1/2),x)

[Out]

Timed out

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

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)