\(\int x \sqrt {2 x-x^2} \, dx\) [39]

Optimal result

Integrand size = 15, antiderivative size = 50 \[ \int x \sqrt {2 x-x^2} \, dx=-\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{3} \left (2 x-x^2\right )^{3/2}-\frac {1}{2} \arcsin (1-x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {654, 626, 633, 222} \[ \int x \sqrt {2 x-x^2} \, dx=-\frac {1}{2} \arcsin (1-x)-\frac {1}{3} \left (2 x-x^2\right )^{3/2}-\frac {1}{2} (1-x) \sqrt {2 x-x^2} \]

[In]

Int[x*Sqrt[2*x - x^2],x]

[Out]

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

Rule 654

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*} \text {integral}& = -\frac {1}{3} \left (2 x-x^2\right )^{3/2}+\int \sqrt {2 x-x^2} \, dx \\ & = -\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{3} \left (2 x-x^2\right )^{3/2}+\frac {1}{2} \int \frac {1}{\sqrt {2 x-x^2}} \, dx \\ & = -\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{3} \left (2 x-x^2\right )^{3/2}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,2-2 x\right ) \\ & = -\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{3} \left (2 x-x^2\right )^{3/2}-\frac {1}{2} \sin ^{-1}(1-x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.08 \[ \int x \sqrt {2 x-x^2} \, dx=\frac {1}{6} \sqrt {-((-2+x) x)} \left (-3-x+2 x^2+\frac {6 \log \left (\sqrt {-2+x}-\sqrt {x}\right )}{\sqrt {-2+x} \sqrt {x}}\right ) \]

[In]

Integrate[x*Sqrt[2*x - x^2],x]

[Out]

(Sqrt[-((-2 + x)*x)]*(-3 - x + 2*x^2 + (6*Log[Sqrt[-2 + x] - Sqrt[x]])/(Sqrt[-2 + x]*Sqrt[x])))/6

Maple [A] (verified)

Time = 2.59 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.64

method	result	size
risch	\(-\frac {\left (2 x^{2}-x -3\right ) x \left (-2+x \right )}{6 \sqrt {-x \left (-2+x \right )}}+\frac {\arcsin \left (-1+x \right )}{2}\)	\(32\)
pseudoelliptic	\(-\arctan \left (\frac {\sqrt {-x \left (-2+x \right )}}{x}\right )+\frac {\left (2 x^{2}-x -3\right ) \sqrt {-x \left (-2+x \right )}}{6}\)	\(37\)
default	\(-\frac {\left (-x^{2}+2 x \right )^{\frac {3}{2}}}{3}-\frac {\left (2-2 x \right ) \sqrt {-x^{2}+2 x}}{4}+\frac {\arcsin \left (-1+x \right )}{2}\)	\(39\)
meijerg	\(\frac {4 i \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-10 x^{2}+5 x +15\right ) \sqrt {1-\frac {x}{2}}}{120}-\frac {i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {2}\, \sqrt {x}}{2}\right )}{4}\right )}{\sqrt {\pi }}\)	\(52\)
trager	\(\left (\frac {1}{3} x^{2}-\frac {1}{6} x -\frac {1}{2}\right ) \sqrt {-x^{2}+2 x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+2 x}+x -1\right )}{2}\)	\(54\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.84 \[ \int x \sqrt {2 x-x^2} \, dx=\frac {1}{6} \, {\left (2 \, x^{2} - x - 3\right )} \sqrt {-x^{2} + 2 \, x} - \arctan \left (\frac {\sqrt {-x^{2} + 2 \, x}}{x}\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.54 \[ \int x \sqrt {2 x-x^2} \, dx=\sqrt {- x^{2} + 2 x} \left (\frac {x^{2}}{3} - \frac {x}{6} - \frac {1}{2}\right ) + \frac {\operatorname {asin}{\left (x - 1 \right )}}{2} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int x \sqrt {2 x-x^2} \, dx=-\frac {1}{3} \, {\left (-x^{2} + 2 \, x\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} x - \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} - \frac {1}{2} \, \arcsin \left (-x + 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.58 \[ \int x \sqrt {2 x-x^2} \, dx=\frac {1}{6} \, {\left ({\left (2 \, x - 1\right )} x - 3\right )} \sqrt {-x^{2} + 2 \, x} + \frac {1}{2} \, \arcsin \left (x - 1\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.84 \[ \int x \sqrt {2 x-x^2} \, dx=-\frac {\sqrt {2\,x-x^2}\,\left (-8\,x^2+4\,x+12\right )}{24}-\frac {\ln \left (x-1-\sqrt {-x\,\left (x-2\right )}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]

[In]

int(x*(2*x - x^2)^(1/2),x)

[Out]

- (log(x - (-x*(x - 2))^(1/2)*1i - 1)*1i)/2 - ((2*x - x^2)^(1/2)*(4*x - 8*x^2 + 12))/24