Does Trigonometric Substitution Works?

Algebra Level 5

$\large \dfrac { \sqrt { b^{ 2 }+2{ a }^{ 2 } } }{ ab } +\dfrac { \sqrt { { a }^{ 2 }+2c^{ 2 } } }{ ac } +\dfrac { \sqrt { { c }^{ 2 }+2{ b }^{ 2 } } }{ bc }$

Let $a$ , $b$ and $c$ be positive reals satisfying $ab+bc+ca=abc$ .

If the minimum value of the expression above can be expressed as $\sqrt m$ , where $m$ is a positive integer, find $m$ .

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