3.1.10 \(\int \sqrt {6 x-x^2} \, dx\) [10]

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

[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 = {626, 633, 222} \begin {gather*} -\frac {9}{2} \text {ArcSin}\left (1-\frac {x}{3}\right )-\frac {1}{2} \sqrt {6 x-x^2} (3-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[6*x - x^2],x]

[Out]

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

Rule 222

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 626

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 633

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} \text {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 = 0.06, size = 43, normalized size = 1.23 \begin {gather*} \frac {1}{2} \sqrt {-((-6+x) x)} \left (-3+x-\frac {18 \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-6+x}{x}}}\right )}{\sqrt {-6+x} \sqrt {x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[6*x - x^2],x]

[Out]

(Sqrt[-((-6 + x)*x)]*(-3 + x - (18*ArcTanh[1/Sqrt[(-6 + x)/x]])/(Sqrt[-6 + x]*Sqrt[x])))/2

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Maple [A]
time = 0.40, size = 28, normalized size = 0.80

method	result	size
risch	\(-\frac {\left (x -3\right ) x \left (x -6\right )}{2 \sqrt {-x \left (x -6\right )}}+\frac {9 \arcsin \left (-1+\frac {x}{3}\right )}{2}\)	\(27\)
default	\(-\frac {\left (-2 x +6\right ) \sqrt {-x^{2}+6 x}}{4}+\frac {9 \arcsin \left (-1+\frac {x}{3}\right )}{2}\)	\(28\)
meijerg	\(-\frac {18 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {6}\, \left (3-x \right ) \sqrt {-\frac {x}{6}+1}}{36}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {6}\, \sqrt {x}}{6}\right )}{2}\right )}{\sqrt {\pi }}\)	\(47\)
trager	\(\left (\frac {x}{2}-\frac {3}{2}\right ) \sqrt {-x^{2}+6 x}+\frac {9 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-x \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}+6 x}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{2}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.55, size = 36, normalized size = 1.03 \begin {gather*} \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 {gather*}

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 = 1.31, size = 35, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 6 \, x} {\left (x - 3\right )} - 9 \, \arctan \left (\frac {\sqrt {-x^{2} + 6 \, x}}{x}\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- x^{2} + 6 x}\, dx \end {gather*}

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 = 1.58, size = 25, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 6 \, x} {\left (x - 3\right )} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, x - 1\right ) \end {gather*}

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

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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