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

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {640, 619, 216} \begin {gather*} \sqrt {-4 x^2+12 x-8}+\frac {1}{2} \sin ^{-1}(3-2 x) \end {gather*}

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 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]

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 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 {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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \sqrt {-4 x^2+12 x-8}+\frac {1}{2} \sin ^{-1}(3-2 x) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.19, size = 38, normalized size = 1.52 \begin {gather*} 2 \sqrt {-x^2+3 x-2}+\tan ^{-1}\left (\frac {\sqrt {-x^2+3 x-2}}{x-1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[-2 + 3*x - x^2] + ArcTan[Sqrt[-2 + 3*x - x^2]/(-1 + x)]

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fricas [B]  time = 0.41, size = 47, normalized size = 1.88 \begin {gather*} 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 {gather*}

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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giac [A]  time = 0.24, size = 23, normalized size = 0.92 \begin {gather*} 2 \, \sqrt {-x^{2} + 3 \, x - 2} - \frac {1}{2} \, \arcsin \left (2 \, x - 3\right ) \end {gather*}

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)

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maple [A]  time = 0.06, size = 24, normalized size = 0.96 \begin {gather*} -\frac {\arcsin \left (2 x -3\right )}{2}+2 \sqrt {-x^{2}+3 x -2} \end {gather*}

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

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maxima [A]  time = 1.90, size = 23, normalized size = 0.92 \begin {gather*} 2 \, \sqrt {-x^{2} + 3 \, x - 2} - \frac {1}{2} \, \arcsin \left (2 \, x - 3\right ) \end {gather*}

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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mupad [B]  time = 1.30, size = 46, normalized size = 1.84 \begin {gather*} \frac {5\,\mathrm {asin}\left (2\,x-3\right )}{2}+2\,\sqrt {-x^2+3\,x-2}+\ln \left (x\,1{}\mathrm {i}+\sqrt {-x^2+3\,x-2}-\frac {3}{2}{}\mathrm {i}\right )\,3{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x*1i + (3*x - x^2 - 2)^(1/2) - 3i/2)*3i + (5*asin(2*x - 3))/2 + 2*(3*x - x^2 - 2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {4 x}{\sqrt {- x^{2} + 3 x - 2}}\, dx + \int \left (- \frac {5}{\sqrt {- x^{2} + 3 x - 2}}\right )\, dx}{2} \end {gather*}

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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