I have the following equation:
$$\frac{1}{p_1^{1/(\gamma-1)}}\left[\frac{1}{\gamma}\left(a_1-\frac{\gamma-1}{2}U\right)^2\right]^\frac{\gamma}{\gamma-1}=p_5\left(1+\frac{\gamma(\gamma+1)}{4}\frac{U^2}{a_5^2}+\gamma\frac{U}{a_5}\sqrt{\frac{(\gamma+1)^2}{16}\frac{U^2}{a_5^2}+1}\right)$$
which I have to solve with respect to $U$ (all the other parameters are known).
I tried to use Mathcad but it was effortless. If I try to solve manually I'm stuck because it's highly irrational in the left part.
Yet I need the exact solution. Could someone propose an advice how to solve the equation?