Feynman parameter for a simple and a squared propagator ($1/(A{B}^2)=2{\int }_0^1(1-x)[xA+(1-x)B{]}^{-3}dx$)

Differentiate $1/(AB)={\int }_0^1[xA+(1-x)B{]}^{-2}dx$ with respect to $B$ to obtain the displayed identity. The squared denominator supplies the factor $2(1-x)$. Changing the parameter to $1-x$ also changes which denominator carries each weight; its use must be consistent with the shifted mass parameter.