The integration question contains lower incomplete gamma function as well as complete gamma function. I used the following code in Mathematica but it does not solve it. Your support would be greatly appreciated.
Integrate[(1 - (Gamma[y, 0, x * Sqrt[a/((1 - (v^2 + u^2) * a) * p)]])/Gamma[z])/(1 + a), {a, 0, Infinity}]
