Find inverse of a complicated function

I am looking for an analytic inverse function for

u0[alpha_] := (p0 (1/2 (((m1 - m2) (m1 + m2))/p0 + p0) + ((1 - 2 alpha)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/( 2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)) - \[Sqrt](((1 - 2 alpha)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/( 4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)^2) + ((1 - 2 alpha)^2 m1^4 + m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/( p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2))))/(p0 + ((1 - 2 alpha)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/( 2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)) - \[Sqrt](((1 - 2 alpha)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/( 4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)^2) + ((1 - 2 alpha)^2 m1^4 + m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/( p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)) + (1 - 2 alpha) \[Sqrt](-m1^2 + (-(((1 - 2 alpha)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/( 2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2))) + \[Sqrt](((1 - 2 alpha)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/( 4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)^2) + ((1 - 2 alpha)^2 m1^4 + m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/( p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 alpha)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)))^2)); 

With p0 > 0, 0 < m1 < p0/2 , 0 < m2 < p0/2. I have already tried to invert this function in full generality with the help of Gröbner basis, with InverseFunction ad with Solve, but nothing succeeded so far. For InverseFunction, i used

Assuming[p0 > 0 && 0 < m1 < 1/2 && 0 < m2 < 1/2 && 0 <= xi <= 1 && 0 <= # <= 1, Simplify[InverseFunction[(p0 (1/ 2 (((m1 - m2) (m1 + m2))/p0 + p0) + ((1 - 2 #)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/( 2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)) - \[Sqrt](((1 - 2 #)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/( 4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)^2) + ((1 - 2 #)^2 m1^4 + m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/( p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2))))/(p0 + ((1 - 2 #)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/( 2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)) - \[Sqrt](((1 - 2 #)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/( 4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)^2) + ((1 - 2 #)^2 m1^4 + m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/( p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)) + (1 - 2 #)*\[Sqrt](-m1^2 + (-(((1 - 2 #)^2 m1^2 (((m1 - m2) (m1 + m2))/p0 + p0))/( 2 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2))) + \[Sqrt](((1 - 2 #)^4 m1^4 (((m1 - m2) (m1 + m2))/p0 + p0)^2)/( 4 (p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)^2) + ((1 - 2 #)^2 m1^4 + m1^2 (1/2 (-(((m1 - m2) (m1 + m2))/p0) - p0) + p0)^2)/( p0^2 - p0 (((m1 - m2) (m1 + m2))/p0 + p0) + 1/4 (1 - (1 - 2 #)^2) (((m1 - m2) (m1 + m2))/p0 + p0)^2)))^2)) &][u0]]] 

But this does not seem to produce any results at least on my machine. Do you have any suggestions how to tackle this issue? Please keep in mind that i am explicitly looking for an analytic function alpha[u0] in the end.