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Is it possible to 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 simplest?

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?