I need to solve integral:
$\int_0^{r(z)} \dfrac{\mathrm{d}z}{r^4(z)-2r^2(z)}$,
where $r(z)=r_i-z(r_i-1)$, where $r_i$ is constant, $z$ is longitudinal coordinate. Boundary condition $r(z)$ is dependent on $z$, so $r$ is not constant and I am not sure how to solve this integral.
If I change boundary immediately in this inegral I will have this shape:
$\int_0^{r(z)} \dfrac{\mathrm{d}z}{a-bz+cz^2-dz^3+ez^4}$, should I solve this integral instead of upper integral and how?
First, do a change of variable $x = r(z)$. Then do a partial fraction decomposition: $$ \frac{1}{x^4 - 2x^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - \sqrt2} +\frac{D}{x + \sqrt2}. $$ Can you continue?