I am trying to solve
$$ \int_{b_1}^{b_2}\sqrt{(b-b_1)(b_2-b)(b_3-b)(b_4-b)} db $$
where $ b1 < b2 < b3 < b4 $. I am quite sure this is some linear combination of elliptic integrals, but I cannot find the right way to find the right expression. My first idea was to make a change of variable in order to get
$$ K_0\int_{-1}^{1} \sqrt{(1-t^2)(T_3-t)(T_4-t)} $$
by $ t=(2b - b_1- b_2)/(b_2-b_1) $ and where $K_0 = (b_2-b_1)^3/8$. But I am stuck at this point.