The Polynomial(f, k) command computes a closed form of the indefinite sum of f with respect to k.

Polynomials are summed using a formula based on Bernoulli polynomials.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{IndefiniteSum}\right]\right)\:$

$p≔68y^2-17x^2{y}^2-98x{y}^3-36x^{4}y+40x^3{y}^2+22x^2{y}^3$

$p≔-36x^{4}y+40x^3{y}^2+22x^2{y}^3-17x^2{y}^2-98x{y}^3+68y^2$

$\mathrm{dpx}≔\mathrm{Polynomial}\left(p\,x\right)$

$\mathrm{dpx}≔\frac{391}{6}{y}^{2}x-60x^2{y}^3+\frac{158}{3}x{y}^3+\frac{22}{3}{x}^3{y}^3-\frac{77}{3}{x}^3{y}^2+\frac{37}{2}{x}^2{y}^2+10y^2{x}^4-\frac{36}{5}y{x}^5+18x^{4}y-12y{x}^3+\frac{6}{5}yx$

$\mathrm{expand}\left(p-\left(\mathrm{eval}\left(\mathrm{dpx}\,x=x+1\right)-\mathrm{dpx}\right)\right)$

$0$

$\mathrm{dpy}≔\mathrm{Polynomial}\left(p\,y\right)$

$\mathrm{dpy}≔-18y^2{x}^4+18x^{4}y+\frac{40}{3}{x}^3{y}^3-\frac{50}{3}{x}^2{y}^3+\frac{68}{3}{y}^3-20x^3{y}^2+14x^2{y}^2-34y^2+\frac{20}{3}y{x}^3-\frac{17}{6}{x}^{2}y+\frac{34}{3}y+\frac{11}{2}{x}^2{y}^4-\frac{49}{2}x{y}^4+49x{y}^3-\frac{49}{2}{y}^{2}x$

$\mathrm{expand}\left(p-\left(\mathrm{eval}\left(\mathrm{dpy}\,y=y+1\right)-\mathrm{dpy}\right)\right)$

$0$