3.900 \(\int \frac{(2+e x)^{7/2}}{\sqrt{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{2 (2-e x)^{7/2}}{7 \sqrt{3} e}-\frac{8 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}-\frac{128 \sqrt{2-e x}}{\sqrt{3} e} \]

[Out]

(-128*Sqrt[2 - e*x])/(Sqrt[3]*e) + (32*(2 - e*x)^(3/2))/(Sqrt[3]*e) - (8*Sqrt[3]
*(2 - e*x)^(5/2))/(5*e) + (2*(2 - e*x)^(7/2))/(7*Sqrt[3]*e)

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Rubi [A]  time = 0.097166, antiderivative size = 85, 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 \[ \frac{2 (2-e x)^{7/2}}{7 \sqrt{3} e}-\frac{8 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}-\frac{128 \sqrt{2-e x}}{\sqrt{3} e} \]

Antiderivative was successfully verified.

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

[Out]

(-128*Sqrt[2 - e*x])/(Sqrt[3]*e) + (32*(2 - e*x)^(3/2))/(Sqrt[3]*e) - (8*Sqrt[3]
*(2 - e*x)^(5/2))/(5*e) + (2*(2 - e*x)^(7/2))/(7*Sqrt[3]*e)

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Rubi in Sympy [A]  time = 14.3387, size = 70, normalized size = 0.82 \[ \frac{32 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{9 e} + \frac{2 \sqrt{3} \left (- e x + 2\right )^{\frac{7}{2}}}{21 e} - \frac{8 \sqrt{3} \left (- e x + 2\right )^{\frac{5}{2}}}{5 e} - \frac{128 \sqrt{3} \sqrt{- e x + 2}}{3 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+2)**(7/2)/(-3*e**2*x**2+12)**(1/2),x)

[Out]

32*(-3*e*x + 6)**(3/2)/(9*e) + 2*sqrt(3)*(-e*x + 2)**(7/2)/(21*e) - 8*sqrt(3)*(-
e*x + 2)**(5/2)/(5*e) - 128*sqrt(3)*sqrt(-e*x + 2)/(3*e)

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Mathematica [A]  time = 0.0596336, size = 57, normalized size = 0.67 \[ \frac{2 (e x-2) \sqrt{e x+2} \left (5 e^3 x^3+54 e^2 x^2+284 e x+1416\right )}{35 e \sqrt{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-2 + e*x)*Sqrt[2 + e*x]*(1416 + 284*e*x + 54*e^2*x^2 + 5*e^3*x^3))/(35*e*Sqr
t[12 - 3*e^2*x^2])

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Maple [A]  time = 0.009, size = 52, normalized size = 0.6 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( 5\,{e}^{3}{x}^{3}+54\,{e}^{2}{x}^{2}+284\,ex+1416 \right ) }{35\,e}\sqrt{ex+2}{\frac{1}{\sqrt{-3\,{e}^{2}{x}^{2}+12}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(1/2),x)

[Out]

2/35*(e*x-2)*(5*e^3*x^3+54*e^2*x^2+284*e*x+1416)*(e*x+2)^(1/2)/e/(-3*e^2*x^2+12)
^(1/2)

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Maxima [A]  time = 0.787437, size = 61, normalized size = 0.72 \[ -\frac{2 i \, \sqrt{3}{\left (5 \, e^{4} x^{4} + 44 \, e^{3} x^{3} + 176 \, e^{2} x^{2} + 848 \, e x - 2832\right )}}{105 \, \sqrt{e x - 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*I*sqrt(3)*(5*e^4*x^4 + 44*e^3*x^3 + 176*e^2*x^2 + 848*e*x - 2832)/(sqrt(e
*x - 2)*e)

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Fricas [A]  time = 0.2225, size = 84, normalized size = 0.99 \[ \frac{2 \,{\left (5 \, e^{5} x^{5} + 54 \, e^{4} x^{4} + 264 \, e^{3} x^{3} + 1200 \, e^{2} x^{2} - 1136 \, e x - 5664\right )}}{35 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*e^5*x^5 + 54*e^4*x^4 + 264*e^3*x^3 + 1200*e^2*x^2 - 1136*e*x - 5664)/(sq
rt(-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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + 2\right )}^{\frac{7}{2}}}{\sqrt{-3 \, e^{2} x^{2} + 12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + 2)^(7/2)/sqrt(-3*e^2*x^2 + 12), x)