Let F be a rational function of x over a field K of characteristic 0. The PolynomialNormalForm(F,x) command constructs the differential polynomial normal form for F.

The output is a sequence of 3 elements $a,b,c$ where $a,b,c$ are polynomials over K such that:

 $F\=\frac{a}{b}\+\frac{\frac{\ⅆ}{\ⅆx}c}{c}\.$ 

$\gcd \left(b\,a-i\left(\frac{\ⅆb}{\ⅆx}\right)\right)=1$ for all non-negative integers $i$.

$\gcd \left(b\,c\right)=1$.

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$F≔\frac{4}{x-2}+\frac{4}{x+1}-\frac{3}{{\left(x+1\right)}^2}-\frac{9}{{\left(x-1\right)}^2}-\frac{9x^2+12}{x^3+4x-2}+\frac{1}{{\left(x^3+4x-2\right)}^2}$

$F≔\frac{4}{x-2}+\frac{4}{x+1}-\frac{3}{{\left(x+1\right)}^2}-\frac{9}{{\left(x-1\right)}^2}-\frac{9x^2+12}{x^3+4x-2}+\frac{1}{{\left(x^3+4x-2\right)}^2}$

$a,b,c≔\mathrm{PolynomialNormalForm}\left(F\,x\right)$

$a,b,c≔-5x^9-16x^8-14x^7-134x^6+39x^5-331x^4+96x^3+32x^2+16x-7,{\left(x+1\right)}^2{\left(x-1\right)}^2{\left(x^3+4x-2\right)}^2,{\left(x-2\right)}^4$

$\mathrm{nF}≔\frac{a}{b}+\frac{\mathrm{diff}\left(c\,x\right)}{c}\:$

$\mathrm{Testzero}\left(\mathrm{normal}\left(F-\mathrm{nF}\right)\right)$

$\mathrm{true}$

$\mathrm{res}≔\mathrm{resultant}\left(b\,a-j\mathrm{diff}\left(b\,x\right)\,x\right)\:$

$H≔\mathrm{select}\left(\mathrm{type}\,\left\{\mathrm{solve}\left(\mathrm{res}\,j\right)\right\}\,'\mathrm{nonnegint}'\right)\:$

$\mathrm{evalb}\left(H=\varnothing \mathbf{and}\gcd \left(b\,c\right)=1\right)$

$\mathrm{true}$

Almkvist, G, and Zeilberger, D. "The method of differentiating under the integral sign." Journal of Symbolic Computation. Vol. 10. (1990): 571-591.