3.2416 \(\int \frac{x}{\sqrt{-2+4 x-3 x^2}} \, dx\)

Optimal. Leaf size=54 \[ -\frac{1}{3} \sqrt{-3 x^2+4 x-2}-\frac{2 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0152533, antiderivative size = 54, 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, Rules used = {640, 621, 204} \[ -\frac{1}{3} \sqrt{-3 x^2+4 x-2}-\frac{2 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]

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

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{-2+4 x-3 x^2}} \, dx &=-\frac{1}{3} \sqrt{-2+4 x-3 x^2}+\frac{2}{3} \int \frac{1}{\sqrt{-2+4 x-3 x^2}} \, dx\\ &=-\frac{1}{3} \sqrt{-2+4 x-3 x^2}+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-12-x^2} \, dx,x,\frac{4-6 x}{\sqrt{-2+4 x-3 x^2}}\right )\\ &=-\frac{1}{3} \sqrt{-2+4 x-3 x^2}-\frac{2 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-2+4 x-3 x^2}}\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0189733, size = 49, normalized size = 0.91 \[ \frac{1}{9} \left (-3 \sqrt{-3 x^2+4 x-2}-2 \sqrt{3} \tan ^{-1}\left (\frac{2-3 x}{\sqrt{-9 x^2+12 x-6}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[-2 + 4*x - 3*x^2],x]

[Out]

(-3*Sqrt[-2 + 4*x - 3*x^2] - 2*Sqrt[3]*ArcTan[(2 - 3*x)/Sqrt[-6 + 12*x - 9*x^2]])/9

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Maple [A]  time = 0.043, size = 41, normalized size = 0.8 \begin{align*} -{\frac{1}{3}\sqrt{-3\,{x}^{2}+4\,x-2}}+{\frac{2\,\sqrt{3}}{9}\arctan \left ({\sqrt{3} \left ( x-{\frac{2}{3}} \right ){\frac{1}{\sqrt{-3\,{x}^{2}+4\,x-2}}}} \right ) } \end{align*}

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

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Maxima [C]  time = 1.4471, size = 42, normalized size = 0.78 \begin{align*} -\frac{2}{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-3*x^2+4*x-2)^(1/2),x, algorithm="maxima")

[Out]

-2/9*I*sqrt(3)*arcsinh(1/2*sqrt(2)*(3*x - 2)) - 1/3*sqrt(-3*x^2 + 4*x - 2)

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Fricas [C]  time = 2.05362, size = 230, normalized size = 4.26 \begin{align*} -\frac{1}{9} i \, \sqrt{3} \log \left (\frac{2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) + \frac{1}{9} i \, \sqrt{3} \log \left (\frac{-2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-3*x^2+4*x-2)^(1/2),x, algorithm="fricas")

[Out]

-1/9*I*sqrt(3)*log((2*I*sqrt(3)*sqrt(-3*x^2 + 4*x - 2) - 6*x + 4)/x) + 1/9*I*sqrt(3)*log((-2*I*sqrt(3)*sqrt(-3
*x^2 + 4*x - 2) - 6*x + 4)/x) - 1/3*sqrt(-3*x^2 + 4*x - 2)

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

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 [C]  time = 1.11854, size = 42, normalized size = 0.78 \begin{align*} -\frac{2}{9} i \, \sqrt{3} \arcsin \left (\frac{1}{2} \, \sqrt{2}{\left (3 i \, x - 2 i\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-3*x^2+4*x-2)^(1/2),x, algorithm="giac")

[Out]

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