First thing to do is heck whether the x/alpha*x^2 is really what you meant. If it was, why not say x^3/alpha ?
Next, check whether you really want to integrate from 4 down to 1, instead of from 1 to 4.
Thirdly, check why you define beta but then do not use it in what is being integrated.
If your integral is exactly as shown, then there is no readily available analytic form for it. A small number of values of alpha lead to analytic forms; the others do not.