Elliptic integrals are integrals of the form $\int R\left[ t, \sqrt{P(t)} \,\right] \, dt$, where $P(t)$ is a polynomial of the third or fourth degree and $R$ is a rational function.
The above is the definition of an elliptic integral, I understand the definition. But then based on the definition why is the following function an elliptic integral:
$$f(x)=\int^{\frac{L}{15}}_{0} \sqrt{{(x^{2}-A)}^{2}+B}\ dx$$
I have been told the above function is an elliptical integral, however according to the definition it does not satisfy all criteria. The integrand is a polynomial of the fourth degree under a square root. However, the function is not rational. So then how can it be an elliptic integral?
Does it get somehow expanded so that integrand becomes rational? My end goal is to find the solution to that integral analytically, but currently I am stuck.
I could not find anything helpful online, I would appreciate some help. Thanks :)
The function is rational when $\sqrt{P(t)}$ is treated like a variable, so it is elliptic. (There's also the condition that $P(t)$ has no repeated roots, otherwise a factor may be taken out and the integral becomes elementary.)