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I have this integral $$\int\frac{dz}{\sqrt{(z^{2}-\rho^{2})(\lambda^{2} - z^{2})}}$$ and parameters obey the following conditions

$$z= \exp[k\varphi],$$ $$\lambda^{2} = \frac{b + \sqrt{b^{2} - 4ac}}{-2a} > 0,$$ $$\rho^{2} = \frac{b- \sqrt{b^{2} - 4ac}}{-2a} > 0,$$ $ \rho^{2} < \lambda^{2}$, $\rho \leq z \leq \lambda$, $a <0$, $c<0$, $b>0$.

I have tried to evaluate the integral using the elliptic integral of the first kind, the obtained result is complex.

Is there a way to find an analytical (real-value) solution to the integral?

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