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

Optimal. Leaf size=87 \[ \frac{6 \sqrt{3} (2-e x)^{11/2}}{11 e}-\frac{8 \sqrt{3} (2-e x)^{9/2}}{e}+\frac{288 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{384 \sqrt{3} (2-e x)^{5/2}}{5 e} \]

[Out]

(-384*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (288*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (8*
Sqrt[3]*(2 - e*x)^(9/2))/e + (6*Sqrt[3]*(2 - e*x)^(11/2))/(11*e)

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Rubi [A]  time = 0.0947895, 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 \[ \frac{6 \sqrt{3} (2-e x)^{11/2}}{11 e}-\frac{8 \sqrt{3} (2-e x)^{9/2}}{e}+\frac{288 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{384 \sqrt{3} (2-e x)^{5/2}}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

(-384*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (288*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (8*
Sqrt[3]*(2 - e*x)^(9/2))/e + (6*Sqrt[3]*(2 - e*x)^(11/2))/(11*e)

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Rubi in Sympy [A]  time = 14.723, size = 71, normalized size = 0.82 \[ \frac{6 \sqrt{3} \left (- e x + 2\right )^{\frac{11}{2}}}{11 e} - \frac{8 \sqrt{3} \left (- e x + 2\right )^{\frac{9}{2}}}{e} + \frac{288 \sqrt{3} \left (- e x + 2\right )^{\frac{7}{2}}}{7 e} - \frac{384 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(3)*(-e*x + 2)**(11/2)/(11*e) - 8*sqrt(3)*(-e*x + 2)**(9/2)/e + 288*sqrt(3
)*(-e*x + 2)**(7/2)/(7*e) - 384*sqrt(3)*(-e*x + 2)**(5/2)/(5*e)

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Mathematica [A]  time = 0.0564399, size = 59, normalized size = 0.68 \[ -\frac{2 (e x-2)^2 \sqrt{12-3 e^2 x^2} \left (105 e^3 x^3+910 e^2 x^2+3020 e x+4264\right )}{385 e \sqrt{e x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2]*(4264 + 3020*e*x + 910*e^2*x^2 + 105*e^3*x
^3))/(385*e*Sqrt[2 + e*x])

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Maple [A]  time = 0.008, size = 52, normalized size = 0.6 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( 105\,{e}^{3}{x}^{3}+910\,{e}^{2}{x}^{2}+3020\,ex+4264 \right ) }{1155\,e} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{{\frac{3}{2}}} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1155*(e*x-2)*(105*e^3*x^3+910*e^2*x^2+3020*e*x+4264)*(-3*e^2*x^2+12)^(3/2)/e/(
e*x+2)^(3/2)

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Maxima [A]  time = 0.797592, size = 111, normalized size = 1.28 \[ -\frac{{\left (210 i \, \sqrt{3} e^{5} x^{5} + 980 i \, \sqrt{3} e^{4} x^{4} - 400 i \, \sqrt{3} e^{3} x^{3} - 8352 i \, \sqrt{3} e^{2} x^{2} - 9952 i \, \sqrt{3} e x + 34112 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{385 \,{\left (e^{2} x + 2 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/385*(210*I*sqrt(3)*e^5*x^5 + 980*I*sqrt(3)*e^4*x^4 - 400*I*sqrt(3)*e^3*x^3 -
8352*I*sqrt(3)*e^2*x^2 - 9952*I*sqrt(3)*e*x + 34112*I*sqrt(3))*(e*x + 2)*sqrt(e*
x - 2)/(e^2*x + 2*e)

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Fricas [A]  time = 0.215717, size = 105, normalized size = 1.21 \[ \frac{6 \,{\left (105 \, e^{7} x^{7} + 490 \, e^{6} x^{6} - 620 \, e^{5} x^{5} - 6136 \, e^{4} x^{4} - 4176 \, e^{3} x^{3} + 33760 \, e^{2} x^{2} + 19904 \, e x - 68224\right )}}{385 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/385*(105*e^7*x^7 + 490*e^6*x^6 - 620*e^5*x^5 - 6136*e^4*x^4 - 4176*e^3*x^3 + 3
3760*e^2*x^2 + 19904*e*x - 68224)/(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)**(3/2)*(-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. \[ \int{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}{\left (e x + 2\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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