3.909 \(\int \frac{(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{2 (2-e x)^{3/2}}{9 \sqrt{3} e}+\frac{16 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{32}{3 \sqrt{3} e \sqrt{2-e x}} \]

[Out]

32/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (16*Sqrt[2 - e*x])/(3*Sqrt[3]*e) - (2*(2 - e*x)
^(3/2))/(9*Sqrt[3]*e)

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Rubi [A]  time = 0.0861922, antiderivative size = 67, 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 \[ -\frac{2 (2-e x)^{3/2}}{9 \sqrt{3} e}+\frac{16 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{32}{3 \sqrt{3} e \sqrt{2-e x}} \]

Antiderivative was successfully verified.

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

[Out]

32/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (16*Sqrt[2 - e*x])/(3*Sqrt[3]*e) - (2*(2 - e*x)
^(3/2))/(9*Sqrt[3]*e)

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Rubi in Sympy [A]  time = 12.3931, size = 51, normalized size = 0.76 \[ - \frac{2 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{81 e} + \frac{16 \sqrt{3} \sqrt{- e x + 2}}{9 e} + \frac{32 \sqrt{3}}{9 e \sqrt{- e x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-3*e*x + 6)**(3/2)/(81*e) + 16*sqrt(3)*sqrt(-e*x + 2)/(9*e) + 32*sqrt(3)/(9*
e*sqrt(-e*x + 2))

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Mathematica [A]  time = 0.0454613, size = 43, normalized size = 0.64 \[ -\frac{2 \sqrt{e x+2} \left (e^2 x^2+20 e x-92\right )}{9 e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[2 + e*x]*(-92 + 20*e*x + e^2*x^2))/(9*e*Sqrt[12 - 3*e^2*x^2])

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Maple [A]  time = 0.009, size = 43, normalized size = 0.6 \[{\frac{ \left ( 2\,ex-4 \right ) \left ({e}^{2}{x}^{2}+20\,ex-92 \right ) }{3\,e} \left ( ex+2 \right ) ^{{\frac{3}{2}}} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(e*x-2)*(e^2*x^2+20*e*x-92)*(e*x+2)^(3/2)/e/(-3*e^2*x^2+12)^(3/2)

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Maxima [A]  time = 0.789736, size = 49, normalized size = 0.73 \[ \frac{2 i \, \sqrt{3} e^{2} x^{2} + 40 i \, \sqrt{3} e x - 184 i \, \sqrt{3}}{27 \, \sqrt{e x - 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27*(2*I*sqrt(3)*e^2*x^2 + 40*I*sqrt(3)*e*x - 184*I*sqrt(3))/(sqrt(e*x - 2)*e)

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Fricas [A]  time = 0.214621, size = 61, normalized size = 0.91 \[ -\frac{2 \,{\left (e^{3} x^{3} + 22 \, e^{2} x^{2} - 52 \, e x - 184\right )}}{9 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(e^3*x^3 + 22*e^2*x^2 - 52*e*x - 184)/(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)**(7/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.590542, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x