The missing branch I believe is right at the beginning it should be:

prod2=5/2*(3200/4507... and so on. Sorry for the typo.

Indeed, MATLAB cannot calculate the integral of an exponent including a square root. The reason we have this exponent in the first place is from the kernel function which must be included in the analysis in order to separate different the parts submmittted to the damping effect of the beam, so it cant be excluded. 

Since all kernal functions include an exponent with square root I cannot replace by an easier function, I think my only possibility s to simplify the function using a Taylor series and lose degree of accuracy.

Any other suggestions is welll appreciated. 

Truthly thankful for the comments.