$$\int{ \sqrt{ 1+ \frac{ m }{ { a }^{ 2 }- { x }^{ 2 } } } }~\mathrm{d} x.$$ $m$ and $a$ are both constants and I have already simplified $m$ and $a$ all the way. And I can't simplify it anymore. Is this integration even possible? My calculator just shows a math error.
I saw that an ellipse didn't have any formula for its circumference and so I wanted to try making a formula for the circumference of an ellipse using some calculus. Here, $m = ((ba)^2)/(a^2-b^2)$
Where $a$ and $b$ are the two radius of the ellipse There are some other constants but I was able to put it outside the integration and so this is the only part that I don't know how to proceed further with.
I personally don't think that this integration is possible cause if it was then people way smarter than me would've already calculated a formula for the circumference already. But I still wanted to try anyway.
The starting integral is $$2 \int{ \sqrt{ \frac{ b ^ { 2 } x ^ { 2 } }{ a ^ { 2 } (a ^ { 2 } -x ^ { 2 } ) } +1 } }\,d x $$
I got this starting integral using the formula of an ellipse and the formula for the length of a graph
This smells like an Elliptic integral... Let's see.
Call $$x = a\sin(p) ~~~~~ \text{d}x = a\cos(p)\ \text{d}p$$
The integral becomes
$$\int \sqrt{1 - \frac{m}{a^2\cos^2(p)}}\ a\cos(p)\ \text{d}p$$
Which can be easily arranged like
$$a\int \sqrt{\cos^2(p) - \beta}\ \text{d}t ~~~~~~~ \beta \equiv \frac{m}{a^2}$$
The result of this is indeed an Elliptic integral:
$$a\frac{\sqrt{-2 \beta+\cos (2 p)+1} \color{red}{E\left(p\left|\frac{1}{1-\beta}\right.\right)}}{\sqrt{-\frac{-2 \beta+\cos (2 p)+1}{\beta-1}}}$$
Consider that being $\beta$ unknown, we can simplify the previous result iff $\beta < 1$ which would yield
$$\sqrt{1 - \beta}\color{red}{E\left(p\left|\frac{1}{1-\beta}\right.\right)}$$
You can pullback to $p\to x$ easily now.
As I smelled, $\color{red}{\text{Elliptic integral}}$.
More on Elliptic integrals here: https://en.wikipedia.org/wiki/Elliptic_integral