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

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]

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

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Rubi [A]  time = 0.0157649, 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, Rules used = {612, 621, 204} \[ \frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}}-\frac{1}{6} (2-3 x) \sqrt{-3 x^2+4 x-2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-2 + 4*x - 3*x^2],x]

[Out]

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

Rule 612

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

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 \sqrt{-2+4 x-3 x^2} \, dx &=-\frac{1}{6} (2-3 x) \sqrt{-2+4 x-3 x^2}-\frac{1}{3} \int \frac{1}{\sqrt{-2+4 x-3 x^2}} \, dx\\ &=-\frac{1}{6} (2-3 x) \sqrt{-2+4 x-3 x^2}-\frac{2}{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}{6} (2-3 x) \sqrt{-2+4 x-3 x^2}+\frac{\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 = 0.0254413, 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]

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

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Maple [A]  time = 0.047, size = 46, normalized size = 0.8 \begin{align*} -{\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 ) } \end{align*}

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)-1/9*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 = 1.72406, size = 62, normalized size = 1.05 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-3*x^2 + 4*x - 2)*x + 1/9*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 = 2.00885, size = 244, normalized size = 4.14 \begin{align*} \frac{1}{6} \, \sqrt{-3 \, x^{2} + 4 \, x - 2}{\left (3 \, x - 2\right )} + \frac{1}{18} i \, \sqrt{3} \log \left (\frac{2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) - \frac{1}{18} i \, \sqrt{3} \log \left (\frac{-2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(-3*x^2 + 4*x - 2)*(3*x - 2) + 1/18*I*sqrt(3)*log((2*I*sqrt(3)*sqrt(-3*x^2 + 4*x - 2) - 6*x + 4)/x) -
1/18*I*sqrt(3)*log((-2*I*sqrt(3)*sqrt(-3*x^2 + 4*x - 2) - 6*x + 4)/x)

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

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 [C]  time = 1.20144, size = 49, normalized size = 0.83 \begin{align*} \frac{1}{6} \, \sqrt{-3 \, x^{2} + 4 \, x - 2}{\left (3 \, x - 2\right )} + \frac{1}{9} i \, \sqrt{3} \arcsin \left (\frac{1}{2} \, \sqrt{2}{\left (3 i \, x - 2 i\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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