An isosceles triangle has base $b$ (where $b$ is the unique side length) and height $h$. Inside the triangle, there is an inscribed square of side length $5$. The positive integer solutions for $(b, h)$ can be expressed as $(p, q)$, $(r, r)$, and $(q, p)$ where $p, q, r$ are distinct positive integers. Find $p + q + r$.