Answer
$$\frac{3}{2}{\sin ^{ - 1}}\left( {x - 1} \right) + \frac{1}{2}\left( {x - 1} \right)\sqrt {2x - {x^2}} - 2\sqrt {2x - {x^2}} + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{{x^2}}}{{\sqrt {2x - {x^2}} }}} dx \cr
& {\text{Completing the square}} \cr
& 2x - {x^2} = - \left( {{x^2} - 2x} \right) \cr
& 2x - {x^2} = - \left( {{x^2} - 2x + 1 - 1} \right) \cr
& 2x - {x^2} = - \left[ {{{\left( {x - 1} \right)}^2} - 1} \right] = 1 - {\left( {x - 1} \right)^2},{\text{ then}} \cr
& \int {\frac{{{x^2}}}{{\sqrt {2x - {x^2}} }}} dx = \int {\frac{{{x^2}}}{{\sqrt {1 - {{\left( {x - 1} \right)}^2}} }}} dx \cr
& {\text{Let }}u = x - 1,{\text{ }}x = u + 1,{\text{ }}dx = du \cr
& \int {\frac{{{x^2}}}{{\sqrt {1 - {{\left( {x - 1} \right)}^2}} }}} dx = \int {\frac{{{{\left( {u + 1} \right)}^2}}}{{\sqrt {1 - {u^2}} }}} du \cr
& = \int {\frac{{{u^2} + 2u + 1}}{{\sqrt {1 - {u^2}} }}} du \cr
& {\text{Let }}u = \sin \theta ,{\text{ }}du = \cos \theta d\theta \cr
& = \int {\frac{{{{\sin }^2}\theta + 2\sin \theta + 1}}{{\sqrt {1 - {{\sin }^2}\theta } }}\left( {\cos \theta } \right)} d\theta \cr
& = \int {\frac{{{{\sin }^2}\theta + 2\sin \theta + 1}}{{\cos \theta }}\left( {\cos \theta } \right)} d\theta \cr
& = \int {\left( {{{\sin }^2} + 2\sin \theta + 1} \right)} d\theta \cr
& = \int {\left( {\frac{1}{2} + \frac{1}{2}\cos 2\theta + 2\sin \theta + 1} \right)} d\theta \cr
& = \int {\left( {\frac{3}{2} + \frac{1}{2}\cos 2\theta + 2\sin \theta } \right)} d\theta \cr
& {\text{Integrating}} \cr
& = \frac{3}{2}\theta + \frac{1}{4}\sin 2\theta - 2\cos \theta + C \cr
& = \frac{3}{2}\theta + \frac{1}{2}\sin \theta \cos \theta - 2\cos \theta + C \cr
& {\text{Substitute back }}\sin \theta = u{\text{ and }}\cos \theta = \sqrt {1 - {u^2}} \cr
& = \frac{3}{2}{\sin ^{ - 1}}u + \frac{1}{2}u\sqrt {1 - {u^2}} - 2\sqrt {1 - {u^2}} + C \cr
& {\text{Substitute }}u = x - 1 \cr
& = \frac{3}{2}{\sin ^{ - 1}}\left( {x - 1} \right) + \frac{1}{2}\left( {x - 1} \right)\sqrt {1 - {{\left( {x - 1} \right)}^2}} - 2\sqrt {1 - {{\left( {x - 1} \right)}^2}} + C \cr
& = \frac{3}{2}{\sin ^{ - 1}}\left( {x - 1} \right) + \frac{1}{2}\left( {x - 1} \right)\sqrt {2x - {x^2}} - 2\sqrt {2x - {x^2}} + C \cr} $$
