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]

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

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.73, size = 35, normalized size = 1.00 \[ \frac {1}{2} \, \sqrt {-x^{2} + 6 \, x} {\left (x - 3\right )} - 9 \, \arctan \left (\frac {\sqrt {-x^{2} + 6 \, x}}{x}\right ) \]

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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giac [A]  time = 0.36, size = 25, normalized size = 0.71 \[ \frac {1}{2} \, \sqrt {-x^{2} + 6 \, x} {\left (x - 3\right )} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, x - 1\right ) \]

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)

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maple [A]  time = 0.05, size = 28, normalized size = 0.80 \[ \frac {9 \arcsin \left (\frac {x}{3}-1\right )}{2}-\frac {\left (-2 x +6\right ) \sqrt {-x^{2}+6 x}}{4} \]

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 = 2.97, size = 36, normalized size = 1.03 \[ \frac {1}{2} \, \sqrt {-x^{2} + 6 \, x} x - \frac {3}{2} \, \sqrt {-x^{2} + 6 \, x} - \frac {9}{2} \, \arcsin \left (-\frac {1}{3} \, x + 1\right ) \]

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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mupad [B]  time = 0.05, size = 26, normalized size = 0.74 \[ \frac {9\,\mathrm {asin}\left (\frac {x}{3}-1\right )}{2}+\left (\frac {x}{2}-\frac {3}{2}\right )\,\sqrt {6\,x-x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x - x^2)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- x^{2} + 6 x}\, dx \]

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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