First of all some remarks. I am pretty sure that your code gives a wrong integrand.
Your formula involves the square of vectors. But these are given by dot products. If you do it as you do you get a list of squares, and my ( I write m ) added to it gives another list or vector with m added to each component, because Plus is distributed over lists:
In[1]:= v = {a, b, c};
v^2
m + v^2
Out[2]= {a^2, b^2, c^2}
Out[3]= {a^2 + m, b^2 + m, c^2 + m}
But actually you can't add a scalar to a vector, these types don't fit together. Mathematica does it because the functions used are listable.
What about the squareroot? Sqrt is also distributed and so you get a vector (or list) of squareroots:
In[4]:= Sqrt[m + v^2]
Out[4]= {Sqrt[a^2 + m], Sqrt[b^2 + m], Sqrt[c^2 + m]}
And you use this in an expression as denominator with numerator 1. This gives a vector of fractions:
In[5]:= 1/Sqrt[m + v^2]
Out[5]= {1/Sqrt[a^2 + m], 1/Sqrt[b^2 + m], 1/Sqrt[c^2 + m]}
I am pretty sure the real interpretation of your formula is this
In[6]:= 1/Sqrt[m^2 + v.v]
Out[6]= 1/Sqrt[a^2 + b^2 + c^2 + m^2]
Having said this I redefine your integrand (note that I omit the underscripts. It was often remarked in this forum that variables with underscripts may behave sometimes in a faulty way)
q3v = Table[q3[i], {i, 3}]
p3v = Table[p3[i], {i, 3}]
qv = Table[q[i], {i, 3}]
pv = Table[p[i], {i, 3}]
Out[10]= {q3[1], q3[2], q3[3]}
Out[11]= {p3[1], p3[2], p3[3]}
Out[12]= {q[1], q[2], q[3]}
Out[13]= {p[1], p[2], p[3]}
a := 1/Sqrt[m^2 + q3v.q3v]
b := 1/Sqrt[m^2 + (pv - p3v - q3v).(pv - p3v - q3v)]
c := 1/Sqrt[m^2 + (p3v - qv + q3v).(p3v + q3v + qv)]
d := 1/(Sqrt[m^2 + (p3v + q3v - qv).(p3v + q3v - qv)] + Sqrt[m^2 + q3v.q3v])
It seems always to be ( p3v + q3v plus or minus something ) if you take minus-signs out of the brackets.
With this you have as integrand
In[19]:= jj = a b c d // FullSimplify
Out[19]= 1/(Sqrt[
m^2 + q3[1]^2 + q3[2]^2 +
q3[3]^2] \[Sqrt](m^2 - q[1]^2 - q[2]^2 -
q[3]^2 + (p3[1] + q3[1])^2 + (p3[2] + q3[2])^2 + (p3[3] +
q3[3])^2) \[Sqrt](m^2 + (-p[1] + p3[1] + q3[1])^2 + (-p[2] +
p3[2] + q3[2])^2 + (-p[3] + p3[3] + q3[3])^2) (Sqrt[
m^2 + q3[1]^2 + q3[2]^2 +
q3[3]^2] + \[Sqrt](m^2 + (p3[1] - q[1] + q3[1])^2 + (p3[2] -
q[2] + q3[2])^2 + (p3[3] - q[3] + q3[3])^2)))
ok. Now for the integration variables. I write
In[49]:= intvariables = Flatten[Join[{q3v}, {p3v}, {qv}, {pv}]]
Out[49]= {q3[1], q3[2], q3[3], p3[1], p3[2], p3[3], q[1], q[2], q[3], p[1], p[2], p[3]}
and then
Integrate[jj, intvariables]
I had run this for about two hours without answer. It seems that this twelvefold integration is too complicated to be done symbolically as Frank remarked above.
However, you can do it numerically. Define the integrand by specifying my
jjm = jj /. m -> 2.
and fix some Integration limits (I do it here for all variables between 1 and 2. You may of course choose others for each variable
In[20]:= intvariables2 = {#, 1, 2} & /@ intvariables
Out[20]= {{q3[1], 1, 2}, {q3[2], 1, 2}, {q3[3], 1, 2}, {p3[1], 1, 2}, {p3[2], 1, 2}, {p3[3], 1, 2}, {q[1], 1, 2}, {q[2], 1, 2},
{q[3],1, 2}, {p[1], 1, 2}, {p[2], 1, 2}, {p[3], 1, 2}}
Then
In[22]:= NIntegrate[jjm, Evaluate[Sequence @@ intvariables2]]
Out[22]= 0.00292856