Questions on elliptic integrals, integrals that involve the square root of a cubic or quartic polynomial.

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

We 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. Is it an elliptic integral? ...