\int \frac{1}{(x-a)^2(x-b)^3} \ dx = \frac{1}{(a-b)^3} \int \frac{(a-b)^3}{(x-a)^2(x-b)^3}\ dx = \frac{1}{(a-b)^3} \int \frac{[(x-b)-(x-a)]^3}{(x-a)^2(x-b)^3}\ dx
Now you can finish it
\int \frac{1}{(x-a)^2(x-b)^3} \ dx = \frac{1}{(a-b)^3} \int \frac{(a-b)^3}{(x-a)^2(x-b)^3}\ dx = \frac{1}{(a-b)^3} \int \frac{[(x-b)-(x-a)]^3}{(x-a)^2(x-b)^3}\ dx
Now you can finish it