### 3.71 $$\int \frac{x^2}{\sqrt{2 x-x^2}} \, dx$$

Optimal. Leaf size=46 $-\frac{1}{2} \sqrt{2 x-x^2} x-\frac{3}{2} \sqrt{2 x-x^2}-\frac{3}{2} \sin ^{-1}(1-x)$

[Out]

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

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Rubi [A]  time = 0.0153809, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.235, Rules used = {670, 640, 619, 216} $-\frac{1}{2} \sqrt{2 x-x^2} x-\frac{3}{2} \sqrt{2 x-x^2}-\frac{3}{2} \sin ^{-1}(1-x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2/Sqrt[2*x - x^2],x]

[Out]

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

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

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

Mathematica [A]  time = 0.0435854, size = 47, normalized size = 1.02 $\frac{1}{2} \left (-\sqrt{2-x} x^{3/2}-3 \sqrt{-(x-2) x}-6 \sin ^{-1}\left (\sqrt{1-\frac{x}{2}}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2/Sqrt[2*x - x^2],x]

[Out]

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

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Maple [A]  time = 0.044, size = 35, normalized size = 0.8 \begin{align*}{\frac{3\,\arcsin \left ( -1+x \right ) }{2}}-{\frac{3}{2}\sqrt{-{x}^{2}+2\,x}}-{\frac{x}{2}\sqrt{-{x}^{2}+2\,x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.73757, size = 49, normalized size = 1.07 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} x - \frac{3}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{3}{2} \, \arcsin \left (-x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.94772, size = 84, normalized size = 1.83 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x + 3\right )} - 3 \, \arctan \left (\frac{\sqrt{-x^{2} + 2 \, x}}{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/sqrt(-x*(x - 2)), x)

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Giac [A]  time = 1.22651, size = 31, normalized size = 0.67 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x + 3\right )} + \frac{3}{2} \, \arcsin \left (x - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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