I have tried many different ways of solving this integral but always seem to get stuck. Any help would be greatly appreciated.
$\displaystyle \int \frac{x}{\sqrt{(x-6)(3-x)}}\,dx=$ $\displaystyle \int \frac{x}{\sqrt{-x^2+9x-18}}\,dx=\frac{1}{2}\int\frac{-2x+9}{\sqrt{-x^2+9x-18}}\,dx-\frac{9}{2}\int\frac{dx}{\sqrt{-\left(x-\frac{9}{2}\right)^2+\frac{9}{4}}}$ .
The first integral above is a root, and the second one an arcsine.
Tonio