By AM-GM, \begin{align*} \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{align*}Equality occurs when $ x = 2, $ so the minimum value is $ \boxed{6} $.