Optimal. Leaf size=67 \[ -\frac{2 (2-e x)^{3/2}}{9 \sqrt{3} e}+\frac{16 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{32}{3 \sqrt{3} e \sqrt{2-e x}} \]
[Out]
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Rubi [A] time = 0.0861922, antiderivative size = 67, 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 \[ -\frac{2 (2-e x)^{3/2}}{9 \sqrt{3} e}+\frac{16 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{32}{3 \sqrt{3} e \sqrt{2-e x}} \]
Antiderivative was successfully verified.
[In] Int[(2 + e*x)^(7/2)/(12 - 3*e^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 12.3931, size = 51, normalized size = 0.76 \[ - \frac{2 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{81 e} + \frac{16 \sqrt{3} \sqrt{- e x + 2}}{9 e} + \frac{32 \sqrt{3}}{9 e \sqrt{- e x + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+2)**(7/2)/(-3*e**2*x**2+12)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0454613, size = 43, normalized size = 0.64 \[ -\frac{2 \sqrt{e x+2} \left (e^2 x^2+20 e x-92\right )}{9 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + e*x)^(7/2)/(12 - 3*e^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.009, size = 43, normalized size = 0.6 \[{\frac{ \left ( 2\,ex-4 \right ) \left ({e}^{2}{x}^{2}+20\,ex-92 \right ) }{3\,e} \left ( ex+2 \right ) ^{{\frac{3}{2}}} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(3/2),x)
[Out]
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Maxima [A] time = 0.789736, size = 49, normalized size = 0.73 \[ \frac{2 i \, \sqrt{3} e^{2} x^{2} + 40 i \, \sqrt{3} e x - 184 i \, \sqrt{3}}{27 \, \sqrt{e x - 2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + 2)^(7/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.214621, size = 61, normalized size = 0.91 \[ -\frac{2 \,{\left (e^{3} x^{3} + 22 \, e^{2} x^{2} - 52 \, e x - 184\right )}}{9 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + 2)^(7/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="fricas")
[Out]
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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)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.590542, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + 2)^(7/2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="giac")
[Out]