jmc

By AM-GM, $$\begin{array}{rl}\frac{(x-1{)}^7+3(x-1{)}^6+(x-1{)}^5+1}{(x-1{)}^5}& =(x-1{)}^2+3(x-1)+1+\frac{1}{(x-1{)}^5}\\ & =(x-1{)}^2+(x-1)+(x-1)+(x-1)+1+\frac{1}{(x-1{)}^5}\\ & \ge 6\sqrt[6]{(x-1{)}^2\cdot (x-1)\cdot (x-1)\cdot (x-1)\cdot 1\cdot \frac{1}{(x-1{)}^5}}\\ & =6.\end{array}$$Equality occurs when $x=2,$ so the minimum value is $\boxed{6}.$