num=7 (3764+3673 √2) x^4-12 (19302+10727 √2) x^3 y+756 (-44+31 √2) x^2 y^2+864 (37+51 √2) x y^3-5184 (2+√2) y^4; den=((7+10 √2) x-6 y) ((9+32 √2) x+12 (1+√2) y); MinimalBy[z /. Simplify@First@Solve[Reduce[z==num/den^2, #], z, Method->Rational] & /@ {x, y}, LeafCount] $\frac{\left(6-5 \sqrt{2}\right) x-12 \left(\sqrt{2}-2\right) y}{\left(23 \sqrt{2}-55\right) x-12 y}$
Another way
factor = Factor[num/den^2, Extension -> Automatic] With[{expr = (a x + b y)/(c x + d y)}, expr /. RootReduce@SolveAlways[factor == expr, {x, y}]] // FullSimplify