Optimal. Leaf size=46 \[ \frac{2 \sqrt{3} \sqrt{2-e x}}{e}-\frac{4 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]
[Out]
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Rubi [A] time = 0.0894362, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 \sqrt{3} \sqrt{2-e x}}{e}-\frac{4 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 10.328, size = 39, normalized size = 0.85 \[ \frac{2 \sqrt{- 3 e x + 6}}{e} - \frac{4 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-3*e**2*x**2+12)**(1/2)/(e*x+2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0536746, size = 63, normalized size = 1.37 \[ \frac{2 \sqrt{12-3 e^2 x^2} \left (\sqrt{e x-2}-2 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )\right )}{e \sqrt{e x-2} \sqrt{e x+2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.023, size = 66, normalized size = 1.4 \[ -2\,{\frac{\sqrt{-{e}^{2}{x}^{2}+4} \left ( 2\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -\sqrt{-3\,ex+6} \right ) \sqrt{3}}{\sqrt{ex+2}\sqrt{-3\,ex+6}e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(3/2),x)
[Out]
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Maxima [A] time = 0.802196, size = 38, normalized size = 0.83 \[ \frac{2 i \, \sqrt{3}{\left (\sqrt{e x - 2} - 2 \, \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right )\right )}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228745, size = 154, normalized size = 3.35 \[ -\frac{2 \,{\left (3 \, e^{2} x^{2} - \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 12\right )}}{\sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \sqrt{3} \int \frac{\sqrt{- e^{2} x^{2} + 4}}{e x \sqrt{e x + 2} + 2 \sqrt{e x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-3*e**2*x**2+12)**(1/2)/(e*x+2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-3 \, e^{2} x^{2} + 12}}{{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(3/2),x, algorithm="giac")
[Out]