~~Is it possible to simplify ~~Simplify expressions with more variables related to each other?

Is it possible to simplify expressions with more variables such like

(x^3 + 2x + 1)/y where y^2 = x^3 + x +1?

Using the relation between x and y the expression then becomes

(x^3 + x + 1 + x)/y = (y^2 + x)/y = y + x/y

Is it possible in Sage? Is Sage able to check in which form the expression (only in x, only in y, mixing) is the simplest?