ClearAll[p, b];
$Assumptions = {b > 0};
p[m_, n_] := p[n, m] = If[ m < 1 || n < 1 || n > m + 1, 0 # &,
            Module[{s}, Integrate[Exp[(# - s) b n]
                (KroneckerDelta[m, 1] KroneckerDelta[n, 2] b
                 + b n p[m, n + 1][s] + If[m < 2, 0,
                   b n (n - 1) (p[m - 1, n - 2][s] + p[m - 1, n - 1][s]
                                +p[m - 1, n][s])]),
                {s, 0, #}] &]];

In[5]:= p[5, 6][t] // AbsoluteTiming

Out[5]= {83.3974,
 15 + 86 E^(b t) - 47350 E^(5 b t) + 5034 E^(6 b t) +
  3240 E^(3 b t) (34 + 15 b t) + 10 E^(2 b t) (361 + 30 b t) +
  5 E^(4 b t) (-14311 + 21216 b t)}