If $\alpha, \beta$ are the roots of the equation $x^{2} - \left(5 + 3\sqrt{\log_{3}5} - 5\sqrt{\log_{5}3}\right)x + 3\left(3^{\tfrac{1}{3}\log_{3}5} - 5^{\tfrac{2}{3}\log_{5}3} - 1\right) = 0$, then the equation, whose roots are $\alpha + \tfrac{1}{\beta}$ and $\beta + \tfrac{1}{\alpha}$, is: