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Algebra Level 4

A=12x3y+108xy3+81y4+x4+1+2y+y2+54x2y2x2+x2y+6xy+9y3+6xy2+9y2A = \dfrac{12 x^{3} y + 108x y^{3} + 81y^{4} + x^{4} + 1 + 2y + y^{2} + 54x^{2} y^{2} }{x^{2} + x^{2} y + 6xy + 9y^{3} + 6xy^{2} + 9y^{2}}

Let TT be the minimum value of AA for positive real numbers x,yx,y. Find the value of 5T5T.

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