In[3]:= LaplaceTransform[(1 + 3 t)^-n, t, s]
Out[3]= 1/3 E^(s/3) ExpIntegralE[n, s/3]
In[4]:= Integrate[(1 + 3 t)^-n Exp[-s t], {t, 0, \[Infinity]}]
Out[4]= ConditionalExpression[1/3 E^(s/3) ExpIntegralE[n, s/3],
Re[s] > 0 || (Re[s] >= 0 && Re[n] > 1)]