3.107 \(\int \sqrt{2+4 x-3 x^2} \, dx\)

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]

-((2 - 3*x)*Sqrt[2 + 4*x - 3*x^2])/6 - (5*ArcSin[(2 - 3*x)/Sqrt[10]])/(3*Sqrt[3]
)

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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]

-((2 - 3*x)*Sqrt[2 + 4*x - 3*x^2])/6 - (5*ArcSin[(2 - 3*x)/Sqrt[10]])/(3*Sqrt[3]
)

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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]

-(-6*x + 4)*sqrt(-3*x**2 + 4*x + 2)/12 - 5*sqrt(3)*atan(sqrt(3)*(-6*x + 4)/(6*sq
rt(-3*x**2 + 4*x + 2)))/9

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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]

(-1/3 + x/2)*Sqrt[2 + 4*x - 3*x^2] - (5*ArcSin[(2 - 3*x)/Sqrt[10]])/(3*Sqrt[3])

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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]

-1/12*(-6*x+4)*(-3*x^2+4*x+2)^(1/2)+5/9*3^(1/2)*arcsin(3/10*10^(1/2)*(x-2/3))

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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]

1/2*sqrt(-3*x^2 + 4*x + 2)*x - 5/9*sqrt(3)*arcsin(-1/10*sqrt(10)*(3*x - 2)) - 1/
3*sqrt(-3*x^2 + 4*x + 2)

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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]

1/18*sqrt(3)*(sqrt(3)*sqrt(-3*x^2 + 4*x + 2)*(3*x - 2) + 10*arctan(1/3*sqrt(3)*(
3*x - 2)/sqrt(-3*x^2 + 4*x + 2)))

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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]

Integral(sqrt(-3*x**2 + 4*x + 2), x)

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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]

1/6*sqrt(-3*x^2 + 4*x + 2)*(3*x - 2) + 5/9*sqrt(3)*arcsin(1/10*sqrt(10)*(3*x - 2
))