I am trying to solve the following integration but not getting it correctly.
$$P = \int_0^\infty \left(\gamma\left(1+b,\frac{t(1+x)-1}{A\sigma^2_1}\right)\right)^N\cdot\frac{\exp\left(\frac{-x}{A_1\sigma^2_5}\right)\left(\frac x{A_1\sigma^2_5}\right)^b\gamma\left(1+b,\frac{x}{A_1\sigma^2_5}\right)^{M-1}}{A_1\sigma^2_5}\text{d}x$$
where $\gamma(\cdot,\cdot)$ is lower incomplete Gamma function, $x$ is variable of integration and all other things are constant.
Using change of variable, I obtained the following simplified equation but not getting how to proceed further.
$$P = \int_0^\infty \underbrace{\left(\gamma\left(1+b,\frac{t(1+A_1\sigma^2_5y)-1}{A\sigma^2_1}\right)\right)^N}_{P_1}\cdot\underbrace{{\exp\left(-y\right)\cdot y^b\cdot \gamma\left(1+b,y\right)^{M-1}}}_{P_2}\text{d}y$$
Any help in this regard will be highly appreciated.