To sum it is to damp it

Square both sides will gives $2a^2+3b^2+6c^2+2ab\sqrt{6}+4ac\sqrt{3}+6bc\sqrt{2} = 2009+918\sqrt{2}+272\sqrt{3}+72\sqrt{6}$ By observing the coefficient $ab = 36$ , $ac = 68$ , $bc = 153$ . Multiply them all to $(abc)^2 = (6\times2\times3\times17)^2$ . As all variables are positive, then $abc = 612$ . Solve for each gives $(a, b, c) = (4, 9, 17)$ so $a+b+c = 4+9+17 = 30$

Squaring both sides,

$2a^{2}+3b^{2}+6c^{2}+2ab\sqrt{6}+2ac\sqrt{12}+2bc\sqrt{18}=2009+918\sqrt{2}+272\sqrt{3}+72\sqrt{6}$

$2a^{2}+3b^{2}+6c^{2}+2ab\sqrt{6}+4ac\sqrt{3}+6bc\sqrt{2}=2009+918\sqrt{2}+272\sqrt{3}+72\sqrt{6}$

Since $a$ , $b$ and $c$ are positive integers,

$2a^{2}+3b^{2}+6c^{2}=2009$ $(*)$

$2ab\sqrt{6}=72\sqrt{6}$

$4ac\sqrt{3}=272\sqrt{3}$

$6bc\sqrt{2}=918\sqrt{2}$

$ab=36=4\times9$

$ac=68=4\times17$

$bc=153=9\times17$

$a=4$ , $b=9$ , $c=17$

Checking our solution against $(*)$ ,

$2a^{2}+3b^{2}+6c^{2}=2\times16+3\times81+6\times289=2009$

Therefore,

$a+b+c=\boxed{30}$