Is there a way to get Mathematica to provide a meaningful answer - perhaps semi-numerically - for the following numerical integral over vectors? Note that it is OK to assume a value for \Alpha. Additionally the vector $\vec{x}$ is not being integrated over. So, if absolutely essential, different values of $\vec{x}$ could be taken for the numerical integration.

$Assumptions = Element[p1v | p3v | p4v | p5v | xv, Vectors[3, Reals]]; 

a = Simplify[ReleaseHold[Hold[E^((-I)*p1v . xv)]]]
b = Simplify[ReleaseHold[Hold[(p1v . p1v + p3v . p3v)/
           ((p3v - p1v)*(p3v - p1v)*(\[Alpha]^2*p3v . p3v + 1)^2)]]]
jj = FullSimplify[a*b]

Now, the following symbolic integral doesn't seem to work, i.e. Mathematica just spits back the input

Integrate[(p1v . p1v + p3v . p3v)/(E^(I*p1v . xv)*
    ((p1v - p3v)^2*(1 + \[Alpha]^2*p3v . p3v)^2)), 
  {p1v, -Infinity, Infinity}, {p3v, -Infinity, Infinity}]

but neither do the following NIntegrate commands work

NIntegrate[(p1v . p1v + p3v . p3v)/(E^(I*p1v . xv)*
    ((p1v - p3v)^2*(1 + p3v . p3v)^2)), {p1v, 0, 1}, {p3v, 0, 1}, 
  {xv, 0, 1}]
NIntegrate::inumr: The integrand (E^(-I p1v.xv) (p1v.p1v+p3v.p3v))/((p1v-p3v)^2 (1+p3v.p3v)^2) has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,1},{0,1},{0,1}}.

Note that above, \Alpha was taken to be zero, and a simultaneous integration over $\vec{x}$ was attempted, if Mathematica can't do any kind of semi-numerical integration.

The following NIntegrate doesn't work either - probably because I don't know how to make Mathematica perform a numerical integration with an algebraic parameter.

NIntegrate[(p1v . p1v + p3v . p3v)/(E^(I*p1v . xv)*
    ((p1v - p3v)^2*(1 + p3v . p3v)^2)), {p1v, 0, 1}, {p3v, 0, 1}]
NIntegrate::inumr: The integrand (E^(-I p1v.xv) (p1v.p1v+p3v.p3v))/((p1v-p3v)^2 (1+p3v.p3v)^2) has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,1},{0,1}}.

If there is a way to conclusively know before integration whether the integrals are non-convergent, that would be very helpful, but that's also unknown to me how to do that in Mathematica.