Can't Evaluate Integral (Electromagnetic Fields & Wave Integral)

[roadrunner]

Not sure if this is acceptable or desirable but with the CAS setting, simplify, set to minimum,

simplify((q*p/(4*π*ɘ*l))*∫(1/(p^2+z^2)^(3/2),z,-l/2,l/2)*√(l^2+4*p^2))/√(l^2+4*p^2)

returns almost exactly the result you want:

   

-road