What is the method to do this integration?
$$\int\frac{dz}{\sqrt{\left(z-a\right)\left(z-b\right)\left(z-c\right)}}$$
When I checked it in Mathematica it gave the result in terms of EllipticF.
The result what I got in Mathematica is
$\frac{-2}{\sqrt{b-a)}}F[sin^{-1}(\frac{\sqrt{b-a}}{\sqrt{z-a}}),\frac{a-c}{a-b}]$
It depends on what you consider an answer. If you look at DLMF section 19.16.1 This is called a symmetric elliptic integral denoted by $2R_f(-a,-b,-c)$. Another possible answer is what Mathematica gives you. There are several tables of integrals such as Gradshteyn and Ryzhik and you can find answers in them. There is no one method. There are a variety of possible methods and answers for you to choose from and you can use whatever is more familiar or convenient for you.
For this particular integral, it is clearly symmetric in $a,b,c$ but Mathematica used one particcular ordering. There are five others to choose from.