FCPartialFractionForm[n, {{f1,x-r1,p1},{f2,x-r2,p2}, ...}, x] is a special way of representing sums of rational functions of x given by n + \frac{f_1}{[x-r_1]^p_1} + \frac{f_2}{[x-r_2]^p_2} + \ldots
It is inspired by the parfracform from Maple and its usage in E. Panzer’s HyperInt for the integration of multiple polylogarithms.
Use ToFCPartialFractionForm to convert the given expression to this notation and FromFCPartialFractionForm to return back to the usual representation.
Overview, ToFCPartialFractionForm, FromFCPartialFractionForm.
Apart[c + x^2/(x - 1), x]c+x+\frac{1}{x-1}+1
ex1 = ToFCPartialFractionForm[c + x^2/(x - 1), x]\text{FCPartialFractionForm}\left(c+x+1,\left( \begin{array}{cc} \{x-1,-1\} & 1 \\ \end{array} \right),x\right)
FromFCPartialFractionForm[ex1]c+x+\frac{1}{x-1}+1
ex2 = ToFCPartialFractionForm[(-64*(-1 + z^2))/(15*(1 + z^2 + z^4)), z]\text{FCPartialFractionForm}\left(0,\left( \begin{array}{cc} \left\{z-\sqrt[3]{-1},-1\right\} & -\frac{32}{15} \\ \left\{z+\sqrt[3]{-1},-1\right\} & \frac{32}{15} \\ \left\{z-(-1)^{2/3},-1\right\} & \frac{32}{15} \\ \left\{z+(-1)^{2/3},-1\right\} & -\frac{32}{15} \\ \end{array} \right),z\right)
FromFCPartialFractionForm[ex2]\frac{32}{15 \left(z+\sqrt[3]{-1}\right)}+\frac{32}{15 \left(z-(-1)^{2/3}\right)}-\frac{32}{15 \left(z+(-1)^{2/3}\right)}-\frac{32}{15 \left(z-\sqrt[3]{-1}\right)}
FromFCPartialFractionForm[ex2, Factoring -> Simplify]-\frac{64 \left(z^2-1\right)}{15 \left(z^4+z^2+1\right)}