Good day! In my scientific research I’ve stuck upon an integral, that sims to be quiet simple yet I was not able to find a solution. $$\int_0^{\infty } \frac{\left(\theta _2+x^2\right){}^{\alpha -\beta }}{\left(x^2+1\right) \left(\theta _1+x^2\right){}^{\alpha }} \, dx, $$ with the set of conditions on arbitrary real-valued parameters: $\alpha >0\land \beta >0\land \alpha >\beta$ and $\theta _2>\theta _1>1$.
The problem comes from signal processing and is the result of complex path integral reduction. I’ve tried various approaches including Laplace or Mellin transforms after applying Shwinger or Feynman parametrizations, representing though more general function, etc. Nothing worked. Actually, it would be nice to detach the $α$ terms form others in the probable solution, since this integral is involved in some recursion procedure.