I have the following integral:
$$\int\_0^1 \frac{P\_3(t)}{\sqrt{1-k^2 P\_3^2(t)}}dt$$
where $P\_3(t)$ is a third-degree polynomial with all coefficients different from zero and $k$ a generic constant. This integral, once known coefficients and the constant $k$, can be integrated numerically, but I wonder if, in some research work, this integral was approximated analytically (under some assumptions on coefficients and the constant $k$).


Thanks in advance.


**Edit (suggested by IgorKhavkine's comment).**


If $P(t)=a+bt+ct^2+dt^3$, $a$ is very large respect other constants. The coefficient $d$ is $<0$ and very small respect to $c,b$. Coefficients $c,b$ are comparable.



Analytic approximation $\int_0^1 \frac{P_3(t)}{\sqrt{1-k^2 P_3^2(t)}}dt$
You could expand in a series in powers of $k$:


$$ \dfrac{P(t)}{\sqrt{1-k^2 P(t)^2}} = \sum\_{j=0}^\infty \dfrac{(2j)! 4^{-j}}{j!^2} k^{2j} P(t)^{2j+1} $$


