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

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

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Rubi [A]  time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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])

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 49, normalized size = 0.91 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [C]  time = 0.15, size = 60, normalized size = 1.11 \begin {gather*} -\frac {1}{3} \sqrt {-3 x^2+4 x-2}+\frac {2 i \log \left (\sqrt {3} \sqrt {-3 x^2+4 x-2}-3 i x+2 i\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Sqrt[-2 + 4*x - 3*x^2] + (((2*I)/3)*Log[2*I - (3*I)*x + Sqrt[3]*Sqrt[-2 + 4*x - 3*x^2]])/Sqrt[3]

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fricas [C]  time = 0.42, size = 79, normalized size = 1.46 \begin {gather*} -\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 {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {-3 \, x^{2} + 4 \, x - 2}}\,{d x} \end {gather*}

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]

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

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maple [A]  time = 0.06, size = 41, normalized size = 0.76 \begin {gather*} \frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -\frac {2}{3}\right )}{\sqrt {-3 x^{2}+4 x -2}}\right )}{9}-\frac {\sqrt {-3 x^{2}+4 x -2}}{3} \end {gather*}

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 = 2.01, size = 31, normalized size = 0.57 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.10, size = 46, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {-3\,x^2+4\,x-2}}{3}-\frac {\sqrt {3}\,\ln \left (\sqrt {-3\,x^2+4\,x-2}+\frac {\sqrt {3}\,\left (3\,x-2\right )\,1{}\mathrm {i}}{3}\right )\,2{}\mathrm {i}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(4*x - 3*x^2 - 2)^(1/2),x)

[Out]

- (3^(1/2)*log((4*x - 3*x^2 - 2)^(1/2) + (3^(1/2)*(3*x - 2)*1i)/3)*2i)/9 - (4*x - 3*x^2 - 2)^(1/2)/3

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {- 3 x^{2} + 4 x - 2}}\, dx \end {gather*}

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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