Analytic approximation $\int_0^1 \frac{P_3(t)}{\sqrt{1-k^2 P_3^2(t)}}dt$

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 $k$).

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.

• Are any of the constants in your problem very large or very small? Otherwise, there's not much that analytical methods could do. – Igor Khavkine Aug 5 '15 at 18:11
• Google hyperelliptic integral? – Jason Starr Aug 5 '15 at 18:15
• @IgorKhavkine. 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. – Mark Aug 6 '15 at 6:21
• Then probably a power series expansion, like in Robert Israel's answer, is your best bet. Though, given what you wrote, $(P(t)-a)/a$ might be a better expansion parameter. – Igor Khavkine Aug 6 '15 at 12:22

You could expand in a series in powers of $k$:
$$\dfrac{P(t)}{\sqrt{1-k^2 P(t)^2}} = \sum_{j=0}^\infty \dfrac{(2j)! 4^{-j}}{j!^2} k^{2j} P(t)^{2j+1}$$