I'll give my motivating problem, and then ask my general question.
So, I'm attempting to integrate the following indefinite integral:
$$\int\frac{\mathrm{d}u}{\sqrt{au^3+bu^2+cu+d}}$$ Now, I learned from poking around here that this is most likely an elliptic integral; sure enough, Byrd and Friedman have the integral $$\int^y_\alpha\frac{\mathrm{d}t}{\sqrt{(t-a)(t-b)(t-c)}}=gF(\phi,k)$$
where it gives values for $g$, $\phi$, and $k$. However, $k$ varies depending on the value of $y$ (it is different for $y>c>b>a$ than for $c\geq c>b>a$, for example). I'm not exactly sure what values $y$ will take nor what range it is in (this integral is from physics), so I can't use their recommended strategy of splitting up the integral given in the introduction.
So, here's my general question: in cases like these, how does one construct the general indefinite integral from tables of definite integrals?
Assuming $a\neq0$, write$$I=\int\frac{du}{\sqrt{au^3+bu^2+cu+d}}=\frac 1 {\sqrt a}\int\frac{du}{\sqrt{(u-r_1)(u-r_2)(u-r_3)}}$$where $r_1,r_2,r_3$ are the roots of the cubic.
Then, without any assumptions we have $$I=-\frac 2 {\sqrt a\,\sqrt{r_2-r_1}}\color{blue}{\frac{ (u-r_1)^{3/2} \sqrt{\frac{u-r_2}{u-r_1}} \sqrt{\frac{u-r_3}{u-r_1}} }{ \sqrt{(u-r_1) (u-r_2) (u-r_3)}}}F\left(\sin ^{-1}\left(\frac{\sqrt{r_2-r_1}}{\sqrt{u-r_1}}\right)|\frac{r_1-r_3}{r_1-r_2}\right)$$ which could simplify a lot depending on the bounds of integration (depending where they locate with respect to the roots. The factor is blue is not necessay equal to $1$; this depends on $\alpha$ and $y$.