Optimal. Leaf size=50 \[ \frac{1}{6 \sqrt{3} e \sqrt{2-e x}}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{12 \sqrt{3} e} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0902509, antiderivative size = 50, 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 \[ \frac{1}{6 \sqrt{3} e \sqrt{2-e x}}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{12 \sqrt{3} e} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 + e*x]/(12 - 3*e^2*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 10.5316, size = 41, normalized size = 0.82 \[ - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{36 e} + \frac{1}{6 e \sqrt{- 3 e x + 6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+2)**(1/2)/(-3*e**2*x**2+12)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0437858, size = 56, normalized size = 1.12 \[ \frac{\sqrt{e x+2} \left (\sqrt{e x-2} \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )+2\right )}{12 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 + e*x]/(12 - 3*e^2*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.02, size = 60, normalized size = 1.2 \[{\frac{1}{ \left ( 108\,ex-216 \right ) e}\sqrt{-3\,{e}^{2}{x}^{2}+12} \left ( \sqrt{3}{\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ) \sqrt{-3\,ex+6}-6 \right ){\frac{1}{\sqrt{ex+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+2)^(1/2)/(-3*e^2*x^2+12)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.820391, size = 45, normalized size = 0.9 \[ \frac{-i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - \frac{2 i \, \sqrt{3}}{\sqrt{e x - 2}}}{36 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + 2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.220725, size = 150, normalized size = 3. \[ \frac{\sqrt{3}{\left (3 \,{\left (e^{2} x^{2} - 4\right )} \log \left (-\frac{\sqrt{3}{\left (e^{2} x^{2} - 4 \, e x - 12\right )} + 4 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}\right )}}{216 \,{\left (e^{3} x^{2} - 4 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + 2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{3} \int \frac{\sqrt{e x + 2}}{- e^{2} x^{2} \sqrt{- e^{2} x^{2} + 4} + 4 \sqrt{- e^{2} x^{2} + 4}}\, dx}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+2)**(1/2)/(-3*e**2*x**2+12)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.625924, size = 4, normalized size = 0.08 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + 2)/(-3*e^2*x^2 + 12)^(3/2),x, algorithm="giac")
[Out]