3.2401 \(\int \frac{x}{\sqrt{2+4 x-3 x^2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{1}{3} \sqrt{-3 x^2+4 x+2}-\frac{2 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.060417, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{1}{3} \sqrt{-3 x^2+4 x+2}-\frac{2 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[2 + 4*x - 3*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.59302, size = 51, normalized size = 1.27 \[ - \frac{\sqrt{- 3 x^{2} + 4 x + 2}}{3} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 4\right )}{6 \sqrt{- 3 x^{2} + 4 x + 2}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-3*x**2+4*x+2)**(1/2),x)

[Out]

-sqrt(-3*x**2 + 4*x + 2)/3 - 2*sqrt(3)*atan(sqrt(3)*(-6*x + 4)/(6*sqrt(-3*x**2 +
4*x + 2)))/9

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Mathematica [A]  time = 0.0311823, size = 40, normalized size = 1. \[ -\frac{1}{3} \sqrt{-3 x^2+4 x+2}-\frac{2 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[2 + 4*x - 3*x^2],x]

[Out]

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

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Maple [A]  time = 0.007, size = 30, normalized size = 0.8 \[ -{\frac{1}{3}\sqrt{-3\,{x}^{2}+4\,x+2}}+{\frac{2\,\sqrt{3}}{9}\arcsin \left ({\frac{3\,\sqrt{10}}{10} \left ( x-{\frac{2}{3}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-3*x^2+4*x+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.750742, size = 42, normalized size = 1.05 \[ -\frac{2}{9} \, \sqrt{3} \arcsin \left (-\frac{1}{10} \, \sqrt{10}{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(-3*x^2 + 4*x + 2),x, algorithm="maxima")

[Out]

-2/9*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 = 0.222223, size = 63, normalized size = 1.58 \[ -\frac{1}{9} \, \sqrt{3}{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x + 2} - 2 \, \arctan \left (\frac{\sqrt{3}{\left (3 \, x - 2\right )}}{3 \, \sqrt{-3 \, x^{2} + 4 \, x + 2}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(-3*x^2 + 4*x + 2),x, algorithm="fricas")

[Out]

-1/9*sqrt(3)*(sqrt(3)*sqrt(-3*x^2 + 4*x + 2) - 2*arctan(1/3*sqrt(3)*(3*x - 2)/sq
rt(-3*x^2 + 4*x + 2)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{- 3 x^{2} + 4 x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-3*x**2+4*x+2)**(1/2),x)

[Out]

Integral(x/sqrt(-3*x**2 + 4*x + 2), x)

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GIAC/XCAS [A]  time = 0.214811, size = 42, normalized size = 1.05 \[ \frac{2}{9} \, \sqrt{3} \arcsin \left (\frac{1}{10} \, \sqrt{10}{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(-3*x^2 + 4*x + 2),x, algorithm="giac")

[Out]

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