It will evaluate albeit it takes some time. The reason is that Integrate goes through some trouble to discern what conditions might allow for convergence. The expression below should give an indication of the difficulties encountered.

ConditionalExpression[-((B + A*p^4)/p^2) + B/q^2 + A*q^2 - d*Log[p] + d*Log[q], 
 ((Im[p] <= Im[q] && (Im[p] >= 0 || Im[q] <= 0 || 
     Im[q]*Re[p] >= Im[p]*Re[q])) || (Im[p] >= Im[q] && 
    (Im[p] <= 0 || Im[q] >= 0 || Im[q]*Re[p] <= Im[p]*Re[q]))) && 
  ((p/(p - q) != 0 && Re[p/(-p + q)] >= 0) || NotElement[p/(p - q), Reals] || 
   Re[p/(p - q)] > 1) && (NotElement[p/(p - q), Reals] || Re[p/(p - q)] >= 1 || 
   Re[p/(-p + q)] >= 0) && ((NotElement[q, Reals] && 
    ((Re[p] != 0 && (Element[p, Reals] || (Im[p] == Im[q] && Re[q] == 0))) || 
     (Im[p] == Im[q] && Re[q] != 0 && 2*q*Conjugate[q] != 
       q*Conjugate[p] + p*Conjugate[q]))) || (Element[q, Reals] && 
    ((NotElement[p, Reals] && q != 0) || (Re[p] < q && 
      (q < 0 || (Re[p] > 0 && q > 0))) || (Re[p] > q && 
      (q > 0 || (q < 0 && Re[p] < 0))))) || (Re[p] > 0 && Re[q] == 0 && 
    ((Im[p] < 0 && Im[q] > 0) || (Im[p] > 0 && Im[q] < 0))) || 
   (Im[p] < 0 && ((Im[p] > Im[q] && Im[q] < 0) || (Im[q] > 0 && Re[q] != 0 && 
      Im[q]*Re[p] > Im[p]*Re[q]))) || (Im[p] < Im[q] && 
    (Im[q] < 0 || (Im[p] > 0 && Im[q] > 0))) || (Im[p] > 0 && Im[q] < 0 && 
    Re[q] != 0 && Im[q]*Re[p] < Im[p]*Re[q]) || (Im[p] > Im[q] && Im[q] > 0))]

As others have noted, placing plausible assumptions in the integration will simplify matters considerably.

Stone,

I would report this as a bug. It evaluates without the limits:

in = Integrate[(2*A*r - ((2*B)/r^3) + (d/r)), r]

and then you can manually evaluate it at the limits:

(in /. r -> q) - (in /. r -> p)

My guess is that MMA is trying to figure out what values of p and q are legitimate.

I also tried Rubi for Mathematica and it returned your original integral instantly (of course Rubi did not figure out when p and q are valid.)

<< Rubi`
Int[(2*A*r - ((2*B)/r^3) + (d/r)), {r, p, q}]

but it gave the same answer:

-(B/p^2) - A p^2 + B/q^2 + A q^2 - d Log[p] + d Log[q]

Regards,

Neil