From WolframAlpha or Mathematica $$\int_0^1 \frac{1}{\sqrt{x(1-x)(x+c) }}=\frac{2K(-\frac{1}{c})}{\sqrt{c}}$$ for $c>0$ where $$K(x)$$ was the complete elliptic integral of the first kind.
However, I could not find out where does this expression come from. How to show that $\int_0^1 \frac{1}{\sqrt{x(1-x)(x+c) }}=\frac{2K(-\frac{1}{c})}{\sqrt{c}}$?