KöMaL 2024 C1798

Let parameter $p$ be a given integer. Find all integer solutions of the equation $\big(p+\frac{1}{p}\big)\cdot\big(x-\frac{1}{x}\big)+\big(p-\frac{1}{p}\big)\cdot\big(x+\frac{1}{x}\big)=4px+5+\frac{1}{p}$.