Optimal. Leaf size=43 \[ \frac{2 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{8 \sqrt{2-e x}}{\sqrt{3} e} \]
[Out]
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Rubi [A] time = 0.0688459, antiderivative size = 43, 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 \[ \frac{2 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{8 \sqrt{2-e x}}{\sqrt{3} e} \]
Antiderivative was successfully verified.
[In] Int[(2 + e*x)^(3/2)/Sqrt[12 - 3*e^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 9.51388, size = 32, normalized size = 0.74 \[ \frac{2 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{27 e} - \frac{8 \sqrt{3} \sqrt{- e x + 2}}{3 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+2)**(3/2)/(-3*e**2*x**2+12)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0344961, size = 40, normalized size = 0.93 \[ \frac{2 (e x-2) \sqrt{e x+2} (e x+10)}{3 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + e*x)^(3/2)/Sqrt[12 - 3*e^2*x^2],x]
[Out]
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Maple [A] time = 0.005, size = 35, normalized size = 0.8 \[{\frac{ \left ( 2\,ex-4 \right ) \left ( ex+10 \right ) }{3\,e}\sqrt{ex+2}{\frac{1}{\sqrt{-3\,{e}^{2}{x}^{2}+12}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(1/2),x)
[Out]
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Maxima [A] time = 0.791039, size = 38, normalized size = 0.88 \[ -\frac{2 i \, \sqrt{3}{\left (e^{2} x^{2} + 8 \, e x - 20\right )}}{9 \, \sqrt{e x - 2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + 2)^(3/2)/sqrt(-3*e^2*x^2 + 12),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.214562, size = 61, normalized size = 1.42 \[ \frac{2 \,{\left (e^{3} x^{3} + 10 \, e^{2} x^{2} - 4 \, e x - 40\right )}}{3 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + 2)^(3/2)/sqrt(-3*e^2*x^2 + 12),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)**(3/2)/(-3*e**2*x**2+12)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + 2\right )}^{\frac{3}{2}}}{\sqrt{-3 \, e^{2} x^{2} + 12}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + 2)^(3/2)/sqrt(-3*e^2*x^2 + 12),x, algorithm="giac")
[Out]