3.883 \(\int (2+e x)^{5/2} \sqrt{12-3 e^2 x^2} \, dx\)

Optimal. Leaf size=87 \[ \frac{2 (2-e x)^{9/2}}{3 \sqrt{3} e}-\frac{24 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{128 (2-e x)^{3/2}}{\sqrt{3} e} \]

[Out]

(-128*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (96*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) - (24*Sq
rt[3]*(2 - e*x)^(7/2))/(7*e) + (2*(2 - e*x)^(9/2))/(3*Sqrt[3]*e)

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Rubi [A]  time = 0.0920911, antiderivative size = 87, 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 \[ \frac{2 (2-e x)^{9/2}}{3 \sqrt{3} e}-\frac{24 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{128 (2-e x)^{3/2}}{\sqrt{3} e} \]

Antiderivative was successfully verified.

[In]  Int[(2 + e*x)^(5/2)*Sqrt[12 - 3*e^2*x^2],x]

[Out]

(-128*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (96*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) - (24*Sq
rt[3]*(2 - e*x)^(7/2))/(7*e) + (2*(2 - e*x)^(9/2))/(3*Sqrt[3]*e)

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Rubi in Sympy [A]  time = 13.1175, size = 70, normalized size = 0.8 \[ - \frac{128 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{9 e} + \frac{2 \sqrt{3} \left (- e x + 2\right )^{\frac{9}{2}}}{9 e} - \frac{24 \sqrt{3} \left (- e x + 2\right )^{\frac{7}{2}}}{7 e} + \frac{96 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+2)**(5/2)*(-3*e**2*x**2+12)**(1/2),x)

[Out]

-128*(-3*e*x + 6)**(3/2)/(9*e) + 2*sqrt(3)*(-e*x + 2)**(9/2)/(9*e) - 24*sqrt(3)*
(-e*x + 2)**(7/2)/(7*e) + 96*sqrt(3)*(-e*x + 2)**(5/2)/(5*e)

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Mathematica [A]  time = 0.0569819, size = 58, normalized size = 0.67 \[ \frac{2 (e x-2) \sqrt{4-e^2 x^2} \left (35 e^3 x^3+330 e^2 x^2+1284 e x+2552\right )}{105 e \sqrt{3 e x+6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + e*x)^(5/2)*Sqrt[12 - 3*e^2*x^2],x]

[Out]

(2*(-2 + e*x)*Sqrt[4 - e^2*x^2]*(2552 + 1284*e*x + 330*e^2*x^2 + 35*e^3*x^3))/(1
05*e*Sqrt[6 + 3*e*x])

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Maple [A]  time = 0.01, size = 52, normalized size = 0.6 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( 35\,{e}^{3}{x}^{3}+330\,{e}^{2}{x}^{2}+1284\,ex+2552 \right ) }{315\,e}\sqrt{-3\,{e}^{2}{x}^{2}+12}{\frac{1}{\sqrt{ex+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+2)^(5/2)*(-3*e^2*x^2+12)^(1/2),x)

[Out]

2/315*(e*x-2)*(35*e^3*x^3+330*e^2*x^2+1284*e*x+2552)*(-3*e^2*x^2+12)^(1/2)/e/(e*
x+2)^(1/2)

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Maxima [A]  time = 0.792636, size = 96, normalized size = 1.1 \[ \frac{{\left (70 i \, \sqrt{3} e^{4} x^{4} + 520 i \, \sqrt{3} e^{3} x^{3} + 1248 i \, \sqrt{3} e^{2} x^{2} - 32 i \, \sqrt{3} e x - 10208 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{315 \,{\left (e^{2} x + 2 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2),x, algorithm="maxima")

[Out]

1/315*(70*I*sqrt(3)*e^4*x^4 + 520*I*sqrt(3)*e^3*x^3 + 1248*I*sqrt(3)*e^2*x^2 - 3
2*I*sqrt(3)*e*x - 10208*I*sqrt(3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

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Fricas [A]  time = 0.226017, size = 95, normalized size = 1.09 \[ -\frac{2 \,{\left (35 \, e^{6} x^{6} + 260 \, e^{5} x^{5} + 484 \, e^{4} x^{4} - 1056 \, e^{3} x^{3} - 7600 \, e^{2} x^{2} + 64 \, e x + 20416\right )}}{105 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2),x, algorithm="fricas")

[Out]

-2/105*(35*e^6*x^6 + 260*e^5*x^5 + 484*e^4*x^4 - 1056*e^3*x^3 - 7600*e^2*x^2 + 6
4*e*x + 20416)/(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)**(5/2)*(-3*e**2*x**2+12)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-3 \, e^{2} x^{2} + 12}{\left (e x + 2\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2), x)