Answer
$$ - 5x\left| { - x + 3} \right| - \frac{5}{2}\left( { - x + 3} \right)\left| { - x + 3} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {5x\frac{{\left| { - x + 3} \right|}}{{ - x + 3}}} dx \cr
& {\text{Integrate by parts }} \cr
& {\text{Let }}u = 5x,{\text{ }}du = 5dx \cr
& dv = \frac{{\left| { - x + 3} \right|}}{{ - x + 3}}dx,{\text{ }}v = \int {\frac{{\left| { - x + 3} \right|}}{{ - x + 3}}} dx \cr
& {\text{By the given table of integrals }} \cr
& v = \int {\frac{{\left| { - x + 3} \right|}}{{ - x + 3}}} dx = - \left| { - x + 3} \right| \cr
& {\text{Using the formula of integration by parts}} \cr
& \int u dv = uv - \int v du \cr
& \int {5x\frac{{\left| { - x + 3} \right|}}{{ - x + 3}}} dx = - 5x\left| { - x + 3} \right| + \int {\left| { - x + 3} \right|\left( 5 \right)} dx \cr
& \int {5x\frac{{\left| { - x + 3} \right|}}{{ - x + 3}}} dx = - 5x\left| { - x + 3} \right| + 5\int {\left| { - x + 3} \right|} dx \cr
& {\text{By the given table of integrals we solve }}\int {\left| { - x + 3} \right|} dx \cr
& \int {\left| { - x + 3} \right|} dx = - \frac{1}{2}\left( { - x + 3} \right)\left| { - x + 3} \right| + C \cr
& {\text{Therefore,}} \cr
& = - 5x\left| { - x + 3} \right| - \frac{5}{2}\left( { - x + 3} \right)\left| { - x + 3} \right| + C \cr} $$
