Algebra with a complicated function

Homework Statement

$$\frac{\pi(1+2\alpha)}{t}=x \&\& ((\alpha \geq 0 \&\& 1+ 2\alpha < 4\beta \&\& \pi \sqrt{-(1+2\alpha)^2+16\beta}=2t)\\ ||((1+2\alpha>0 \&\& 2\alpha < 1 +4 \beta \&\& \beta \geq 0 \&\& \pi \sqrt{(-1+2\alpha -4\beta) (3+2 \alpha +4\beta)}=2t)$$

$$\alpha and \beta$$are integers.

This is a solution I obtained from Mathematica, it's ugly as you can tell. How can I generalize this concisely?
How can I find a general expression for x from this equation?

The Attempt at a Solution

If we look at the first condition, $$\alpha \geq 0$$ which is nice
\alpha \beta 1+2*alpha < 4*beta
0 1 1 < 4
1 2 3 < 8
2 3 5 < 12

This is false when
\alpha \beta 1+2*alpha < 4*beta
0 0 1 < FALSE
1 1 3 < 4
2 2 5 < 8

Any suggestions?

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Based on the equality for condition 1, we also know that $$\beta >0$$ but how do you generalize the order?