Does Trigonometric Substitution Works?

Algebra Level 5

b2+2a2ab+a2+2c2ac+c2+2b2bc\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 aa, bb and cc be positive reals satisfying ab+bc+ca=abcab+bc+ca=abc.

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

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