I would like to solve the following equation for $x$:
$$\int_{0}^{x}\frac{p\frac{(2(n-x)+1)^t}{(2(n-x)+2)^x}}{p\frac{(2(n-x)+1)^t}{(2(n-x)+2)^x}+(1-p)\frac{((n-x)+1)^t}{((n-x)+2)^x}}\left[\frac{(2(n-x)+1)^t}{(2(n-x)+2)^x}-\frac{((n-x)+1)^t}{((n-x)+2)^x}\right]\left[\frac{\Gamma(1+x)}{\Gamma(1+t)\Gamma(1+x-t)}\right]dt=K$$
Is there any way I can get an expression for $x$?
Is there any approximation I can do to get a simpler expression?
All tips are much appreciated.
Thanks a lot!