3.261 \(\int x \sqrt{2 r x-x^2} \, dx\)

Optimal. Leaf size=64 \[ r^3 \tan ^{-1}\left (\frac{x}{\sqrt{2 r x-x^2}}\right )-\frac{1}{2} r (r-x) \sqrt{2 r x-x^2}-\frac{1}{3} \left (2 r x-x^2\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0172441, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {640, 612, 620, 203} \[ r^3 \tan ^{-1}\left (\frac{x}{\sqrt{2 r x-x^2}}\right )-\frac{1}{2} r (r-x) \sqrt{2 r x-x^2}-\frac{1}{3} \left (2 r x-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[2*r*x - x^2],x]

[Out]

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

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]

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 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x \sqrt{2 r x-x^2} \, dx &=-\frac{1}{3} \left (2 r x-x^2\right )^{3/2}+r \int \sqrt{2 r x-x^2} \, dx\\ &=-\frac{1}{2} r (r-x) \sqrt{2 r x-x^2}-\frac{1}{3} \left (2 r x-x^2\right )^{3/2}+\frac{1}{2} r^3 \int \frac{1}{\sqrt{2 r x-x^2}} \, dx\\ &=-\frac{1}{2} r (r-x) \sqrt{2 r x-x^2}-\frac{1}{3} \left (2 r x-x^2\right )^{3/2}+r^3 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{x}{\sqrt{2 r x-x^2}}\right )\\ &=-\frac{1}{2} r (r-x) \sqrt{2 r x-x^2}-\frac{1}{3} \left (2 r x-x^2\right )^{3/2}+r^3 \tan ^{-1}\left (\frac{x}{\sqrt{2 r x-x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.109978, size = 72, normalized size = 1.12 \[ \frac{1}{6} \sqrt{-x (x-2 r)} \left (\frac{6 r^{5/2} \sin ^{-1}\left (\frac{\sqrt{x}}{\sqrt{2} \sqrt{r}}\right )}{\sqrt{x} \sqrt{2-\frac{x}{r}}}-3 r^2-r x+2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[2*r*x - x^2],x]

[Out]

(Sqrt[-(x*(-2*r + x))]*(-3*r^2 - r*x + 2*x^2 + (6*r^(5/2)*ArcSin[Sqrt[x]/(Sqrt[2]*Sqrt[r])])/(Sqrt[x]*Sqrt[2 -
x/r])))/6

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Maple [A]  time = 0.007, size = 73, normalized size = 1.1 \begin{align*} -{\frac{1}{3} \left ( 2\,rx-{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{rx}{2}\sqrt{2\,rx-{x}^{2}}}-{\frac{{r}^{2}}{2}\sqrt{2\,rx-{x}^{2}}}+{\frac{{r}^{3}}{2}\arctan \left ({(x-r){\frac{1}{\sqrt{2\,rx-{x}^{2}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*r*x-x^2)^(1/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7791, size = 108, normalized size = 1.69 \begin{align*} -r^{3} \arctan \left (\frac{\sqrt{2 \, r x - x^{2}}}{x}\right ) - \frac{1}{6} \,{\left (3 \, r^{2} + r x - 2 \, x^{2}\right )} \sqrt{2 \, r x - x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{- x \left (- 2 r + x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(-x*(-2*r + x)), x)

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Giac [A]  time = 1.08457, size = 61, normalized size = 0.95 \begin{align*} -\frac{1}{2} \, r^{3} \arcsin \left (\frac{r - x}{r}\right ) \mathrm{sgn}\left (r\right ) - \frac{1}{6} \,{\left (3 \, r^{2} +{\left (r - 2 \, x\right )} x\right )} \sqrt{2 \, r x - x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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