3.907 \(\int \frac{(2+e x)^{11/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=111 \[ -\frac{2 (2-e x)^{7/2}}{21 \sqrt{3} e}+\frac{32 (2-e x)^{5/2}}{15 \sqrt{3} e}-\frac{64 (2-e x)^{3/2}}{3 \sqrt{3} e}+\frac{512 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{512}{3 \sqrt{3} e \sqrt{2-e x}} \]

[Out]

512/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (512*Sqrt[2 - e*x])/(3*Sqrt[3]*e) - (64*(2 - e
*x)^(3/2))/(3*Sqrt[3]*e) + (32*(2 - e*x)^(5/2))/(15*Sqrt[3]*e) - (2*(2 - e*x)^(7
/2))/(21*Sqrt[3]*e)

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Rubi [A]  time = 0.108605, antiderivative size = 111, 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 \[ -\frac{2 (2-e x)^{7/2}}{21 \sqrt{3} e}+\frac{32 (2-e x)^{5/2}}{15 \sqrt{3} e}-\frac{64 (2-e x)^{3/2}}{3 \sqrt{3} e}+\frac{512 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{512}{3 \sqrt{3} e \sqrt{2-e x}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + e*x)^(11/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

512/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (512*Sqrt[2 - e*x])/(3*Sqrt[3]*e) - (64*(2 - e
*x)^(3/2))/(3*Sqrt[3]*e) + (32*(2 - e*x)^(5/2))/(15*Sqrt[3]*e) - (2*(2 - e*x)^(7
/2))/(21*Sqrt[3]*e)

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Rubi in Sympy [A]  time = 16.2517, size = 88, normalized size = 0.79 \[ - \frac{64 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{27 e} - \frac{2 \sqrt{3} \left (- e x + 2\right )^{\frac{7}{2}}}{63 e} + \frac{32 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{45 e} + \frac{512 \sqrt{3} \sqrt{- e x + 2}}{9 e} + \frac{512 \sqrt{3}}{9 e \sqrt{- e x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+2)**(11/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

-64*(-3*e*x + 6)**(3/2)/(27*e) - 2*sqrt(3)*(-e*x + 2)**(7/2)/(63*e) + 32*sqrt(3)
*(-e*x + 2)**(5/2)/(45*e) + 512*sqrt(3)*sqrt(-e*x + 2)/(9*e) + 512*sqrt(3)/(9*e*
sqrt(-e*x + 2))

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Mathematica [A]  time = 0.0655284, size = 60, normalized size = 0.54 \[ -\frac{2 \sqrt{e x+2} \left (5 e^4 x^4+72 e^3 x^3+568 e^2 x^2+5664 e x-23216\right )}{105 e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + e*x)^(11/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[2 + e*x]*(-23216 + 5664*e*x + 568*e^2*x^2 + 72*e^3*x^3 + 5*e^4*x^4))/(1
05*e*Sqrt[12 - 3*e^2*x^2])

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Maple [A]  time = 0.01, size = 60, normalized size = 0.5 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( 5\,{e}^{4}{x}^{4}+72\,{e}^{3}{x}^{3}+568\,{e}^{2}{x}^{2}+5664\,ex-23216 \right ) }{35\,e} \left ( ex+2 \right ) ^{{\frac{3}{2}}} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+2)^(11/2)/(-3*e^2*x^2+12)^(3/2),x)

[Out]

2/35*(e*x-2)*(5*e^4*x^4+72*e^3*x^3+568*e^2*x^2+5664*e*x-23216)*(e*x+2)^(3/2)/e/(
-3*e^2*x^2+12)^(3/2)

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Maxima [A]  time = 0.788431, size = 78, normalized size = 0.7 \[ \frac{10 i \, \sqrt{3} e^{4} x^{4} + 144 i \, \sqrt{3} e^{3} x^{3} + 1136 i \, \sqrt{3} e^{2} x^{2} + 11328 i \, \sqrt{3} e x - 46432 i \, \sqrt{3}}{315 \, \sqrt{e x - 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(11/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="maxima")

[Out]

1/315*(10*I*sqrt(3)*e^4*x^4 + 144*I*sqrt(3)*e^3*x^3 + 1136*I*sqrt(3)*e^2*x^2 + 1
1328*I*sqrt(3)*e*x - 46432*I*sqrt(3))/(sqrt(e*x - 2)*e)

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Fricas [A]  time = 0.236291, size = 84, normalized size = 0.76 \[ -\frac{2 \,{\left (5 \, e^{5} x^{5} + 82 \, e^{4} x^{4} + 712 \, e^{3} x^{3} + 6800 \, e^{2} x^{2} - 11888 \, e x - 46432\right )}}{105 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(11/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="fricas")

[Out]

-2/105*(5*e^5*x^5 + 82*e^4*x^4 + 712*e^3*x^3 + 6800*e^2*x^2 - 11888*e*x - 46432)
/(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)**(11/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.651443, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(11/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="giac")

[Out]

sage0*x