How do I evaluate the following Integral

Question

The integral is $$\int_{0}^{\infty}dx\frac{1}{\sqrt{1+a(1+x^2)^m+b(1+x^2)^{m-2}x^2}},$$ where $m, a$ and $b$ are real numbers such that the integral is definitely convergent. Any ideas on how to solve this?

Answer 1

• The integral converges for $m>1$.

• If the integrand has a pole on $(0,\infty)$, i.e., if its denominator vanishes for some $x>0$, see Cauchy principal value.

• In particular, for $m=2$, if the quartic polynomial can be factored into a product of two coprime quadratics, then, for positive values of A and B, we have the following results, in
  terms of elliptic integrals:

$$\begin{align} \int_0^\infty\frac{dx}{\sqrt{\Big(x^2+A\Big)\Big(x^2+B\Big)}}~&=~\frac1{\sqrt{+A}}~K\bigg(\sqrt{1-\frac BA}~\bigg) \\\\\ \int_0^\infty\frac{dx}{\sqrt{\Big(x^2-A\Big)\Big(x^2+B\Big)}}~&=~\frac{-1}{\sqrt{-A}}~\overline K\bigg(\sqrt{1+\frac BA}~\bigg) \\\\ \int_0^\infty\frac{dx}{\sqrt{\Big(x^2-A\Big)\Big(x^2-B\Big)}}~&=~\frac1{\sqrt{-A}}~K\bigg(\sqrt{1-\frac BA}~\bigg)+\frac2{\sqrt B}~\Re\bigg[K\bigg(\sqrt{\frac AB}~\bigg)\bigg] \end{align}$$