3.890 \(\int (2+e x)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=109 \[ -\frac{6 \sqrt{3} (2-e x)^{13/2}}{13 e}+\frac{96 \sqrt{3} (2-e x)^{11/2}}{11 e}-\frac{64 \sqrt{3} (2-e x)^{9/2}}{e}+\frac{1536 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{1536 \sqrt{3} (2-e x)^{5/2}}{5 e} \]

[Out]

(-1536*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (1536*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (
64*Sqrt[3]*(2 - e*x)^(9/2))/e + (96*Sqrt[3]*(2 - e*x)^(11/2))/(11*e) - (6*Sqrt[3
]*(2 - e*x)^(13/2))/(13*e)

_______________________________________________________________________________________

Rubi [A]  time = 0.106008, antiderivative size = 109, 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 \[ -\frac{6 \sqrt{3} (2-e x)^{13/2}}{13 e}+\frac{96 \sqrt{3} (2-e x)^{11/2}}{11 e}-\frac{64 \sqrt{3} (2-e x)^{9/2}}{e}+\frac{1536 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{1536 \sqrt{3} (2-e x)^{5/2}}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

(-1536*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (1536*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (
64*Sqrt[3]*(2 - e*x)^(9/2))/e + (96*Sqrt[3]*(2 - e*x)^(11/2))/(11*e) - (6*Sqrt[3
]*(2 - e*x)^(13/2))/(13*e)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.7826, size = 90, normalized size = 0.83 \[ - \frac{6 \sqrt{3} \left (- e x + 2\right )^{\frac{13}{2}}}{13 e} + \frac{96 \sqrt{3} \left (- e x + 2\right )^{\frac{11}{2}}}{11 e} - \frac{64 \sqrt{3} \left (- e x + 2\right )^{\frac{9}{2}}}{e} + \frac{1536 \sqrt{3} \left (- e x + 2\right )^{\frac{7}{2}}}{7 e} - \frac{1536 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*sqrt(3)*(-e*x + 2)**(13/2)/(13*e) + 96*sqrt(3)*(-e*x + 2)**(11/2)/(11*e) - 64
*sqrt(3)*(-e*x + 2)**(9/2)/e + 1536*sqrt(3)*(-e*x + 2)**(7/2)/(7*e) - 1536*sqrt(
3)*(-e*x + 2)**(5/2)/(5*e)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0609638, size = 67, normalized size = 0.61 \[ -\frac{2 (e x-2)^2 \sqrt{12-3 e^2 x^2} \left (1155 e^4 x^4+12600 e^3 x^3+56840 e^2 x^2+133600 e x+154928\right )}{5005 e \sqrt{e x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2]*(154928 + 133600*e*x + 56840*e^2*x^2 + 126
00*e^3*x^3 + 1155*e^4*x^4))/(5005*e*Sqrt[2 + e*x])

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 60, normalized size = 0.6 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( 1155\,{e}^{4}{x}^{4}+12600\,{e}^{3}{x}^{3}+56840\,{e}^{2}{x}^{2}+133600\,ex+154928 \right ) }{15015\,e} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{{\frac{3}{2}}} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(e*x-2)*(1155*e^4*x^4+12600*e^3*x^3+56840*e^2*x^2+133600*e*x+154928)*(-3
*e^2*x^2+12)^(3/2)/e/(e*x+2)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.798836, size = 126, normalized size = 1.16 \[ -\frac{{\left (2310 i \, \sqrt{3} e^{6} x^{6} + 15960 i \, \sqrt{3} e^{5} x^{5} + 22120 i \, \sqrt{3} e^{4} x^{4} - 86720 i \, \sqrt{3} e^{3} x^{3} - 304224 i \, \sqrt{3} e^{2} x^{2} - 170624 i \, \sqrt{3} e x + 1239424 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{5005 \,{\left (e^{2} x + 2 \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5005*(2310*I*sqrt(3)*e^6*x^6 + 15960*I*sqrt(3)*e^5*x^5 + 22120*I*sqrt(3)*e^4*
x^4 - 86720*I*sqrt(3)*e^3*x^3 - 304224*I*sqrt(3)*e^2*x^2 - 170624*I*sqrt(3)*e*x
+ 1239424*I*sqrt(3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

_______________________________________________________________________________________

Fricas [A]  time = 0.215294, size = 116, normalized size = 1.06 \[ \frac{6 \,{\left (1155 \, e^{8} x^{8} + 7980 \, e^{7} x^{7} + 6440 \, e^{6} x^{6} - 75280 \, e^{5} x^{5} - 196352 \, e^{4} x^{4} + 88128 \, e^{3} x^{3} + 1228160 \, e^{2} x^{2} + 341248 \, e x - 2478848\right )}}{5005 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/5005*(1155*e^8*x^8 + 7980*e^7*x^7 + 6440*e^6*x^6 - 75280*e^5*x^5 - 196352*e^4*
x^4 + 88128*e^3*x^3 + 1228160*e^2*x^2 + 341248*e*x - 2478848)/(sqrt(-3*e^2*x^2 +
12)*sqrt(e*x + 2)*e)

_______________________________________________________________________________________

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)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.3302, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done