Optimal. Leaf size=59 \[ \frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}}-\frac{1}{6} (2-3 x) \sqrt{-3 x^2+4 x-2} \]
[Out]
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Rubi [A] time = 0.0328287, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}}-\frac{1}{6} (2-3 x) \sqrt{-3 x^2+4 x-2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-2 + 4*x - 3*x^2],x]
[Out]
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Rubi in Sympy [A] time = 2.00817, size = 53, normalized size = 0.9 \[ - \frac{\left (- 6 x + 4\right ) \sqrt{- 3 x^{2} + 4 x - 2}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 4\right )}{6 \sqrt{- 3 x^{2} + 4 x - 2}} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-3*x**2+4*x-2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0487513, size = 54, normalized size = 0.92 \[ \frac{1}{6} \sqrt{-3 x^2+4 x-2} (3 x-2)+\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{-9 x^2+12 x-6}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-2 + 4*x - 3*x^2],x]
[Out]
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Maple [A] time = 0.007, size = 46, normalized size = 0.8 \[ -{\frac{-6\,x+4}{12}\sqrt{-3\,{x}^{2}+4\,x-2}}-{\frac{\sqrt{3}}{9}\arctan \left ({\sqrt{3} \left ( x-{\frac{2}{3}} \right ){\frac{1}{\sqrt{-3\,{x}^{2}+4\,x-2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-3*x^2+4*x-2)^(1/2),x)
[Out]
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Maxima [A] time = 0.917032, size = 62, normalized size = 1.05 \[ \frac{1}{2} \, \sqrt{-3 \, x^{2} + 4 \, x - 2} x + \frac{1}{9} i \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-3*x^2 + 4*x - 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220191, size = 115, normalized size = 1.95 \[ \frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2}{\left (3 \, x - 2\right )} + i \, \log \left (\frac{2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) - i \, \log \left (\frac{-2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-3*x^2 + 4*x - 2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- 3 x^{2} + 4 x - 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-3*x**2+4*x-2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.211172, size = 51, normalized size = 0.86 \[ \frac{1}{9} \, \sqrt{3} i \arcsin \left (\frac{1}{2} \, \sqrt{2} i{\left (3 \, x - 2\right )}\right ) + \frac{1}{6} \, \sqrt{-3 \, x^{2} + 4 \, x - 2}{\left (3 \, x - 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-3*x^2 + 4*x - 2),x, algorithm="giac")
[Out]