Answer
$\dfrac{(x+2)(x-3)\sqrt{4x-x^2}}{3}+4\sin^{-1}(\dfrac{x-2}{2})+C$
Work Step by Step
Apply formula : $\int x\sqrt{2ax-x^2}dx=\dfrac{(x+a)(2x-3a)\sqrt{2ax-x^2}}{6}+\dfrac{a^3}{2}\sin^{-1} (\dfrac{x-a}{a} )+C$
Consider $I=\dfrac{(x+2)(2x-6)\sqrt{4x-x^2}}{6}+\dfrac{8}{2}\sin^{-1}[(\dfrac{x-2}{2})]+C\\=\dfrac{(x+2)(x-3)\sqrt{4x-x^2}}{3}+4\sin^{-1}(\dfrac{x-2}{2})+C$
