Let F be a rational function of x over a field K of characteristic 0. The RationalCanonicalForm[i](F,x) calling sequence constructs the ith differential rational canonical forms for F, $i=\left\{1\,2\right\}$.

If the RationalCanonicalForm command is called without an index, the first differential rational canonical form is constructed.

The output is a sequence of 2 elements $R,V$, called RationalCanonicalForm(F), where $R,V$ are rational functions over K such that

$F=R+\frac{\frac{\ⅆV}{\ⅆx}}{V}$.

 $\gcd \left(\mathrm{denom}\left(R\right),\mathrm{numer}\left(R\right)-i\frac{\ⅆ}{\ⅆx}\left(\mathrm{denom}\left(R\right)\right)\right)=1\mathrm{for\; all}\mathrm{integers}i\.$ 

If the third optional argument, which is the name 'polyform', is given, the output is a sequence of 4 elements $a,b,c,d$, where $a,b,c,d$ are polynomials over K, $b,c,d$ monic such that $R=\frac{a}{b}$, $V=\frac{c}{d}$.

The use of RationalCanonicalForm[1] is for testing similarity of two given hyperexponential functions. For RationalCanonicalForm[2], the polynomials $b,c,d$ are also pairwise relatively prime. RationalCanonicalForm[2] is used in a reduction algorithm for hyperexponential functions.

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

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

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

$\mathrm{R1},\mathrm{V1}≔\mathrm{RationalCanonicalForm}\left[1\right]\left(F\,x\right)$

$\mathrm{R1},\mathrm{V1}≔\frac{-12x^8-12x^7-108x^6-48x^5-239x^4+48x^3-50x^2+144x-47}{{\left(x+1\right)}^2{\left(x-1\right)}^2{\left(x^3+4x-2\right)}^2},\frac{{\left(x+1\right)}^4{\left(x-2\right)}^4}{{\left(x^3+4x-2\right)}^3}$

$\mathrm{R2},\mathrm{V2}≔\mathrm{RationalCanonicalForm}\left[2\right]\left(F\,x\right)$

$\mathrm{R2},\mathrm{V2}≔\frac{-5x^9-16x^8-14x^7-134x^6+39x^5-331x^4+96x^3+32x^2+16x-7}{{\left(x+1\right)}^2{\left(x-1\right)}^2{\left(x^3+4x-2\right)}^2},{\left(x-2\right)}^4$

$\mathrm{a1},\mathrm{b1},\mathrm{c1},\mathrm{d1}≔\mathrm{RationalCanonicalForm}\left[1\right]\left(F\,x\,'\mathrm{polyform}'\right)$

$\mathrm{a1},\mathrm{b1},\mathrm{c1},\mathrm{d1}≔-12x^8-12x^7-108x^6-48x^5-239x^4+48x^3-50x^2+144x-47,{\left(x+1\right)}^2{\left(x-1\right)}^2{\left(x^3+4x-2\right)}^2,{\left(x+1\right)}^4{\left(x-2\right)}^4,{\left(x^3+4x-2\right)}^3$

Geddes, Keith; Le, Ha; and Li, Ziming. "Differential rational canonical forms and a reduction algorithm for hyperexponential functions." Proceedings of ISSAC 2004. ACM Press, (2004): 183-190.