How to convert $\frac{x^2}{\sqrt{ax-x^2}}$ to $\frac{x^2}{\sqrt{(\frac{a}{x})^2-(x-\frac{a}{2})^2}}$

Question

When I was doing integral problems recently， I often see conversions like this.

such as: $$\int{\frac{dx}{\sqrt{(x-a)(b-x)}}}= \int{\frac{dx}{\sqrt{(a+b)x-x^2-ab}}}=\int{\frac{dx}{\sqrt{\frac{(a-b)^2}{4}-(x-\frac{a+b}{2})^2}}}$$

Is there any trick to solve this problem?

Thanks for your help!

Answer 1

$\displaystyle \int\frac{dx}{\sqrt{e^2 - (fx + g)^2}}$ will succumb to

$\displaystyle fx + g = e\sin t.$