3.10 \(\int \sqrt{6 x-x^2} \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{2} \sqrt{6 x-x^2} (3-x)-\frac{9}{2} \sin ^{-1}\left (1-\frac{x}{3}\right ) \]

[Out]

-((3 - x)*Sqrt[6*x - x^2])/2 - (9*ArcSin[1 - x/3])/2

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Rubi [A]  time = 0.0098087, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {612, 619, 216} \[ -\frac{1}{2} \sqrt{6 x-x^2} (3-x)-\frac{9}{2} \sin ^{-1}\left (1-\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[6*x - x^2],x]

[Out]

-((3 - x)*Sqrt[6*x - x^2])/2 - (9*ArcSin[1 - x/3])/2

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 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 \sqrt{6 x-x^2} \, dx &=-\frac{1}{2} (3-x) \sqrt{6 x-x^2}+\frac{9}{2} \int \frac{1}{\sqrt{6 x-x^2}} \, dx\\ &=-\frac{1}{2} (3-x) \sqrt{6 x-x^2}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{36}}} \, dx,x,6-2 x\right )\\ &=-\frac{1}{2} (3-x) \sqrt{6 x-x^2}-\frac{9}{2} \sin ^{-1}\left (1-\frac{x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0427556, size = 32, normalized size = 0.91 \[ \frac{1}{2} (x-3) \sqrt{-(x-6) x}-9 \sin ^{-1}\left (\sqrt{1-\frac{x}{6}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[6*x - x^2],x]

[Out]

((-3 + x)*Sqrt[-((-6 + x)*x)])/2 - 9*ArcSin[Sqrt[1 - x/6]]

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Maple [A]  time = 0.056, size = 28, normalized size = 0.8 \begin{align*} -{\frac{-2\,x+6}{4}\sqrt{-{x}^{2}+6\,x}}+{\frac{9}{2}\arcsin \left ( -1+{\frac{x}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+6*x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.78977, size = 49, normalized size = 1.4 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 6 \, x} x - \frac{3}{2} \, \sqrt{-x^{2} + 6 \, x} - \frac{9}{2} \, \arcsin \left (-\frac{1}{3} \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+6*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.17013, size = 82, normalized size = 2.34 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 6 \, x}{\left (x - 3\right )} - 9 \, \arctan \left (\frac{\sqrt{-x^{2} + 6 \, x}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+6*x)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(-x^2 + 6*x)*(x - 3) - 9*arctan(sqrt(-x^2 + 6*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- x^{2} + 6 x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+6*x)**(1/2),x)

[Out]

Integral(sqrt(-x**2 + 6*x), x)

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Giac [A]  time = 1.28916, size = 34, normalized size = 0.97 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 6 \, x}{\left (x - 3\right )} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+6*x)^(1/2),x, algorithm="giac")

[Out]

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