$\begin{aligned} & \sqrt{1-\left(x^2-3 x\right)} d x \\=& \int \sqrt{\left(\frac{\sqrt{13}}{2}\right)^2-\left(x-\frac{3}{2}\right)^2} d x \\=& \frac{2 x-3}{4} \sqrt{1+3 x-x^2}+\frac{13}{8} \sin ^{-1}\left(\frac{2 x-3}{\sqrt{3}}\right)+c \end{aligned}$