Is there a way in Mathematica to test whether the integral converges, i.e. numerical integration is kosher? The variables are

q3v = Table[q3[i], {i, 3}]; 
p3v = Table[p3[i], {i, 3}]; 
qv = Table[q[i], {i, 3}]; 
pv = Table[p[i], {i, 3}];

The limits are:

intvariables = Flatten[Join[{q3v}, {p3v}, {qv}, {pv}]]
intvariables2 = {#, -1, 1} & /@ intvariables

The integral to be evaulated is

1/((1 + p[1]^2 + p[2]^2 + p[3]^2)^2*
   (1 + q[1]^2 + q[2]^2 + q[3]^2)^2*
   Sqrt[Sqrt[1. + p[1]^2 + p[2]^2 + p[3]^2]*
     Sqrt[1. + p3[1]^2 + p3[2]^2 + p3[3]^2]*
     Sqrt[1. + q3[1]^2 + q3[2]^2 + q3[3]^2]*
     Sqrt[1. + (-p[1] + p3[1] + q3[1])^2 + 
       (-p[2] + p3[2] + q3[2])^2 + (-p[3] + p3[3] + q3[3])^2]]*
   Sqrt[Sqrt[1. + p3[1]^2 + p3[2]^2 + p3[3]^2]*
     Sqrt[1. + q[1]^2 + q[2]^2 + q[3]^2]*
     Sqrt[1. + q3[1]^2 + q3[2]^2 + q3[3]^2]*
     Sqrt[1. + (p3[1] - q[1] + q3[1])^2 + (p3[2] - q[2] + q3[2])^
        2 + (p3[3] - q[3] + q3[3])^2]]*
   (Sqrt[1. + q3[1]^2 + q3[2]^2 + q3[3]^2] + 
    Sqrt[1. + (p3[1] - q[1] + q3[1])^2 + (p3[2] - q[2] + q3[2])^
       2 + (p3[3] - q[3] + q3[3])^2]))

The results are different if the integration limits are changed to -10, 10 for example. Is there a way to test that the integral converges, without being able to perform a symbolic integration (which Mathematica will not do)?