I am just wondering about an integral, which has the form $$\int_0^\infty\frac{dx}{\sqrt{x(x^2+ax+1)}}$$ which also equivalent to $$2\int_0^\infty\frac{dx}{\sqrt{x^4+ax^2+1}}$$ do these Integrals have a solution?
I have tried
- completing the square
- trig sub
and failed
If $A,B$ have a positive real part, $$ \int_{0}^{+\infty}\frac{dx}{\sqrt{(x^2+A)(x^2+B)}} = \frac{\pi}{2\,\text{AGM}(\sqrt{A},\sqrt{B})}=\tfrac{1}{\sqrt{A}}K\left(1-\tfrac{B}{A}\right)=\tfrac{1}{\sqrt{B}}K\left(1-\tfrac{A}{B}\right)$$ where $K$ is the complete elliptic integral of the first kind (denoted according to Mathematica's notation, such that the argument of $K$ is the elliptic modulus) and $\text{AGM}$ is the arithmetic-geometric mean, defined by the common limit of $a_{n+1}=\frac{a_n+b_n}{2},b_{n+1}=\sqrt{a_n b_n}$.
$\Gamma\left(\frac{1}{3}\right)$ and $\Gamma\left(\frac{1}{4}\right)$ can be written in terms of the $\text{AGM}$ (some links on the associated Wikipedia page point back to MSE), hence for particular values of $a$ your integral is related to the $\Gamma$ function. It is not in full generality.