If the branches are independent, then the BER is the multiplication of the BER of all branches. For the special case where all branches are independent and indentically distributed, the BER will be \[P^L(E)\], where \[L\] is the number of branches, and \[P(E)\] is the probability of error in each branch.
Suppose we have \[L\] independent not necessarily identically distributed fading channels between the transmitter and receiver. Employing MRC at the receiver side, the total SNR can be written as:
\[\gamma_{eq}=\gamma_1+\gamma_2+\ldots+\gamma_L\]
where \[\gamma_l=\frac{|h_l|^2\,P_S}{N_{0,l}}\] is the instantaneous SNR of branch l, \[\{h_l\}_{l=1}^L\] defined as independent complex Guanssian Random Variables (RV) with zero-mean, and variance \[\Omega_l/2=E[|h_l|^2]/2\] per dimension, which gives us the case of independent not necessarilly identically distributed Rayleigh fading channels.
Since the branches are independent, then the Moment Generating Function (MGF) of the equivalent SNR equals the multiplication of the (MGF) of all branches. Mathematically: