How to speed up plotting of Sophomore's Dream function

Reformulation as a ParametricNDSolve[] problem is very fast:

ff = 10;
pfun = ParametricNDSolveValue[{y'[t] == (t + $MachineEpsilon)^(a t), 
   y[0] == 1}, y, {t, 0, ff}, {a}]

LogPlot[Evaluate[
  Table[pfun[a][
    x], {a, {-50, -30, -20, -10, -5, -1, 0, 0.5, 1, 2, 3, 5, 
     10}}]], {x, 0, 10}, PlotRange -> {1, 10^10}]

Perhaps slightly faster:

f[a_, x_] := NIntegrate[t^(a t), {t, 0, x}]
al = {-50, -30, -20, -10, -5, -1, 0, 0.5, 1, 2, 3, 5, 10};
tab[a_] := Table[{j, f[a, j]}, {j, 0, 10, 0.1}]
ListLogPlot[tab /@ al, Joined -> True, PlotRange -> {0.01, 10^10}, 
 PlotLegends -> LineLegend[Automatic, al], 
 GridLines -> {Range[10], PowerRange[1, 10^10, 10]}, Frame -> True]