A few days ago I had never heard of elliptic integrals. However, this morning I came across an elliptic integral in my work. The integral has the following form:
$$g(x)=\int \sqrt{x^{4}+f^{2}-2fx^{2}+h}\ dx $$
This can be simplified to:
$$g(x)=\int \sqrt{(x^{2}-f)^{2}+h}\ dx $$
In my question, $f$ and $h$ are constants. Having never worked with elliptic integrals I was (and still am) very lost on how to continue. I tried finding pieces of information online, which could help me understand how to solve the integral. I found a post of a different user on this StackExchange who had a question with a very similar integral. From the comments there I read that using the Handbook from Byrd and Friedman would help to find a solution to my question.
I found that book and I looked through it but I still cannot not come any closer to finding a solution to my question. After looking through all sorts of information on elliptic integrals I feel as lost as ever on how to solve my integral. Could someone please assist me in this?
From my understanding I somehow I have to split up the current function so that it resembles the elliptic integral of the first, maybe the second kind but I am struggling on how I could do that.
As a note, I saw in other questions programs like Wolfram Alpha can solve the elliptic integral for you. But for my case, it is important I show the solution analytically.
I think you're going to need symbolic integration system, such as Mathematica, which of course gives the answer "analytically":
$$\int \sqrt{(x^2 - f)^2 + h}\ dx =\frac{1}{3} x \sqrt{\left(f-x^2\right)^2+h}-\frac{2 i \sqrt{\frac{1}{\sqrt{-h}-f}} \left(f^2+h\right) \sqrt{\frac{f+\sqrt{-h}-x^2}{f+\sqrt{-h}}} \sqrt{\frac{-f+\sqrt{-h}+x^2}{\sqrt{-h}-f}} \left(\sqrt{-h} F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{\sqrt{-h}-f}} x\right)|\frac{f-\sqrt{-h}}{f+\sqrt{-h}}\right)-f E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{\sqrt{-h}-f}} x\right)|\frac{f-\sqrt{-h}}{f+\sqrt{-h}}\right)\right)}{3 \sqrt{\left(f-x^2\right)^2+h}}$$