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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