Optimal. Leaf size=55 \[ \frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{2 e}-\frac{\sqrt{3} \sqrt{2-e x}}{e (e x+2)} \]
[Out]
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Rubi [A] time = 0.0944513, antiderivative size = 55, 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 \[ \frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{2 e}-\frac{\sqrt{3} \sqrt{2-e x}}{e (e x+2)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 11.1173, size = 42, normalized size = 0.76 \[ - \frac{\sqrt{- 3 e x + 6}}{e \left (e x + 2\right )} + \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-3*e**2*x**2+12)**(1/2)/(e*x+2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0618818, size = 71, normalized size = 1.29 \[ \frac{\sqrt{12-3 e^2 x^2} \left ((e x+2) \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )-2 \sqrt{e x-2}\right )}{2 e \sqrt{e x-2} (e x+2)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(5/2),x]
[Out]
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Maple [B] time = 0.027, size = 88, normalized size = 1.6 \[{\frac{\sqrt{3}}{2\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ({\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ) \sqrt{3}xe+2\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -2\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{3}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(5/2),x)
[Out]
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Maxima [A] time = 0.812804, size = 50, normalized size = 0.91 \[ -\frac{i \, \sqrt{3}{\left (\frac{2 \, \sqrt{e x - 2}}{e x + 2} - \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right )\right )}}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218794, size = 157, normalized size = 2.85 \[ \frac{\sqrt{3}{\left (e^{2} x^{2} + 4 \, e x + 4\right )} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x - 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{4 \,{\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(5/2),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((-3*e**2*x**2+12)**(1/2)/(e*x+2)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-3 \, e^{2} x^{2} + 12}}{{\left (e x + 2\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(5/2),x, algorithm="giac")
[Out]