I need to evaluate the following integral: \begin{equation} \int_{z_1}^{z_2} d z \sqrt{g(z)}, \end{equation} where the function $g(z)$ is given by \begin{equation} g(z)=-\left(\alpha-\frac{\beta}{z^2}+\frac{\gamma}{z^4}\right)=\frac{\alpha\,(z^2-z_1^2)(z_2^2-z^2)}{z^4}, \end{equation} and $z_1$ and $z_2$ are two zeros of $g(z)$ with positive real parts.
When $z_1$ and $z_2$ are real numbers, by the change of variable $z^2=z_2^2-(z_2^2-z_1^2)\sin^2(\theta)$ I write the integral in the form of elliptic functions. The result is:
\begin{equation}
2\sqrt{\alpha} z_2 \left((1-\frac{m^2}{2})K(m)-E(m)\right),
\end{equation}
where $m:=(z_2^2-z_!^2)/z_2^2$ and $K(m)$ and $E(m)$ are respectively the complete elliptic integrals of the first and second kind.
The problem arises when $z_1=z_2^*$ are complex conjugate roots. In this case, in the above integral the path lies along the imaginary axis and the branch of $\sqrt{g(z)}$ is real and positive. I don't know how to perform the integration in this case. But, surprisingly, for some values of $z_1=z_2^*$, the numerical value obtained by Mathematica is exactly identical with the result obtained in the case of real zeros of $g(z)$.
My questions are:
(1) Which appropriate contour I should take to perform the integral in the case $z_1=z_2^*$?
(2) Is there any analytical result for the integral in this case? If yes, how can I find it?
(3) Two special choices of the parameters may be helpful. $(\alpha, \beta, \gamma)=(1.5, 0.75, 0.25)$ and $(\alpha, \beta, \gamma)=(4, -0.05, 0.1225)$.
Any help will be appreciated.