3.2401 \(\int \frac{5-4 x}{\sqrt{-8+12 x-4 x^2}} \, dx\)

Optimal. Leaf size=25 \[ \sqrt{-4 x^2+12 x-8}+\frac{1}{2} \sin ^{-1}(3-2 x) \]

[Out]

Sqrt[-8 + 12*x - 4*x^2] + ArcSin[3 - 2*x]/2

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Rubi [A]  time = 0.0107881, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {640, 619, 216} \[ \sqrt{-4 x^2+12 x-8}+\frac{1}{2} \sin ^{-1}(3-2 x) \]

Antiderivative was successfully verified.

[In]

Int[(5 - 4*x)/Sqrt[-8 + 12*x - 4*x^2],x]

[Out]

Sqrt[-8 + 12*x - 4*x^2] + ArcSin[3 - 2*x]/2

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 619

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{5-4 x}{\sqrt{-8+12 x-4 x^2}} \, dx &=\sqrt{-8+12 x-4 x^2}-\int \frac{1}{\sqrt{-8+12 x-4 x^2}} \, dx\\ &=\sqrt{-8+12 x-4 x^2}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{16}}} \, dx,x,12-8 x\right )\\ &=\sqrt{-8+12 x-4 x^2}+\frac{1}{2} \sin ^{-1}(3-2 x)\\ \end{align*}

Mathematica [A]  time = 0.009147, size = 25, normalized size = 1. \[ \sqrt{-4 x^2+12 x-8}+\frac{1}{2} \sin ^{-1}(3-2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - 4*x)/Sqrt[-8 + 12*x - 4*x^2],x]

[Out]

Sqrt[-8 + 12*x - 4*x^2] + ArcSin[3 - 2*x]/2

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Maple [A]  time = 0.045, size = 24, normalized size = 1. \begin{align*} -{\frac{\arcsin \left ( -3+2\,x \right ) }{2}}+2\,\sqrt{-{x}^{2}+3\,x-2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(5-4*x)/(-x^2+3*x-2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.44415, size = 31, normalized size = 1.24 \begin{align*} 2 \, \sqrt{-x^{2} + 3 \, x - 2} - \frac{1}{2} \, \arcsin \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(5-4*x)/(-x^2+3*x-2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.39347, size = 120, normalized size = 4.8 \begin{align*} 2 \, \sqrt{-x^{2} + 3 \, x - 2} + \frac{1}{2} \, \arctan \left (\frac{\sqrt{-x^{2} + 3 \, x - 2}{\left (2 \, x - 3\right )}}{2 \,{\left (x^{2} - 3 \, x + 2\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(5-4*x)/(-x^2+3*x-2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{4 x}{\sqrt{- x^{2} + 3 x - 2}}\, dx + \int - \frac{5}{\sqrt{- x^{2} + 3 x - 2}}\, dx}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(5-4*x)/(-x**2+3*x-2)**(1/2),x)

[Out]

-(Integral(4*x/sqrt(-x**2 + 3*x - 2), x) + Integral(-5/sqrt(-x**2 + 3*x - 2), x))/2

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Giac [A]  time = 1.58267, size = 31, normalized size = 1.24 \begin{align*} 2 \, \sqrt{-x^{2} + 3 \, x - 2} - \frac{1}{2} \, \arcsin \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(5-4*x)/(-x^2+3*x-2)^(1/2),x, algorithm="giac")

[Out]

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