If we write $\sqrt{2}+\frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{2}}$ in the form $\dfrac{a\sqrt{2} + b\sqrt{3}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$?

\(\quad\displaystyle \sqrt{2}+\frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{2}} \\= \sqrt 2 + \sqrt 3 + \dfrac 2{\sqrt 2} \\= 2 \sqrt 2 + \sqrt3\)

Now, the values of a, b, and c are obvious.