Optimal. Leaf size=45 \[ -\frac{1}{6} \sqrt{-3 x^2+4 x+2} (2-3 x)-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0425203, antiderivative size = 45, 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 \[ -\frac{1}{6} \sqrt{-3 x^2+4 x+2} (2-3 x)-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 + 4*x - 3*x^2],x]
[Out]
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Rubi in Sympy [A] time = 2.02748, size = 56, normalized size = 1.24 \[ - \frac{\left (- 6 x + 4\right ) \sqrt{- 3 x^{2} + 4 x + 2}}{12} - \frac{5 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 4\right )}{6 \sqrt{- 3 x^{2} + 4 x + 2}} \right )}}{9} \]
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 = 0.0345582, size = 46, normalized size = 1.02 \[ \left (\frac{x}{2}-\frac{1}{3}\right ) \sqrt{-3 x^2+4 x+2}-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 + 4*x - 3*x^2],x]
[Out]
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Maple [A] time = 0.005, size = 35, normalized size = 0.8 \[ -{\frac{-6\,x+4}{12}\sqrt{-3\,{x}^{2}+4\,x+2}}+{\frac{5\,\sqrt{3}}{9}\arcsin \left ({\frac{3\,\sqrt{10}}{10} \left ( x-{\frac{2}{3}} \right ) } \right ) } \]
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 = 0.831942, size = 62, normalized size = 1.38 \[ \frac{1}{2} \, \sqrt{-3 \, x^{2} + 4 \, x + 2} x - \frac{5}{9} \, \sqrt{3} \arcsin \left (-\frac{1}{10} \, \sqrt{10}{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x + 2} \]
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 = 0.221357, size = 70, normalized size = 1.56 \[ \frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x + 2}{\left (3 \, x - 2\right )} + 10 \, \arctan \left (\frac{\sqrt{3}{\left (3 \, x - 2\right )}}{3 \, \sqrt{-3 \, x^{2} + 4 \, x + 2}}\right )\right )} \]
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 = 0.20963, size = 49, normalized size = 1.09 \[ \frac{1}{6} \, \sqrt{-3 \, x^{2} + 4 \, x + 2}{\left (3 \, x - 2\right )} + \frac{5}{9} \, \sqrt{3} \arcsin \left (\frac{1}{10} \, \sqrt{10}{\left (3 \, x - 2\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-3*x^2 + 4*x + 2),x, algorithm="giac")
[Out]