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Using Assumptions in Expressions that Evaluate to be Elliptic Integrals

I have the following integral that I am trying to evaluate in Mathematica:

$\int \sqrt{\alpha + m g l \cos(q)} dq$. If $\alpha > m g l$, then the result is a complete elliptic integral of the second kind and Mathematica outputs EllipticE[m]. If $\alpha < m g l$, then Mathematica outputs and incomplete elliptic integral of the second kind, i.e. EllipticE[$\phi$,m].

I am trying to obtain a general expression for the integral when $\alpha > m g l$, but Mathematica keeps giving me a conditional expression, even after using the Assumptions command with $\alpha > m g l$ and $\alpha \in$ Reals. Does anyone know what's going on here?