I am interested in evaluating the following type of integrals. Here we a polynomial $q(x)$ of degree $d \geq 2$ with no non-negative roots. Then is there a name for integrals of the shape $$\int_0^\infty \frac{dx}{x^{\frac{2}{d+2}} q(x)^{\frac{2}{d+2}}}?$$ when $d = 2$, these are complete elliptic integrals. Is there a name for them in general? I tried searching for 'generalized elliptic integrals' but they seem to emphasize on generalizations based on the hypergeometric function rather than with the original integrals.

Any help would be appreciated.

I am interested in evaluating the following type of integrals. Here we a polynomial $q(x)$ of degree $d \geq 2$ with no non-negative roots. Then is there a name for integrals of the shape $$\int_0^\infty \frac{dx}{x^{\frac{2}{d+2}} q(x)^{\frac{2}{d+2}}}?$$ when $d = 2$, these are complete elliptic integrals. Is there a name for them in general? I tried searching for 'generalized elliptic integrals' but they seem to emphasize on generalizations based on the hypergeometric function rather than with the original integrals.