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Let $q(x)$ be a quadratic polynomial with no positive real roots with positive leading coefficient. Is there a name for the following integral?

$$\displaystyle \int_0^\infty \frac{dx}{\sqrt{x q(x)}}.$$

This integral will converge by our hypothesis on $q$, since in a neighbourhood around $0$ the function is bounded from above by a multiple of $x^{-1/2}$, and is asymptotically bounded by $x^{-3/2}$. If anyone knows a name for this type of integral, or better yet show that it can be evaluated in terms of well-known functions in mathematics, that would be deeply appreciated. Also, there is like a distinction between the case when $q$ has real roots or has no real roots.

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    $\begingroup$ this are ellepitc intergrals $\endgroup$ – tired Oct 15 '15 at 22:03
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$$\int_0^\infty\frac{dx}{\sqrt{x(x^2+a^2)}}=\frac8{\sqrt{a\pi}}\cdot\Gamma^2\bigg(\frac54\bigg).$$

However, if the coefficient of x does not vanish, then we are dealing with elliptic integrals.

$$\int_0^\infty\frac{dx}{\sqrt{x(x^2+ax+b)}}=\frac2{\sqrt[4]b}\cdot K\bigg(\sqrt{\frac12-\frac a{4\sqrt b}}\bigg).$$

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