For a specified hyperexponential function H of x, the (H1, H2) := ReduceHyperexp(H, x, newH) calling sequence constructs two hyperexponential functions H1 and H2 such that $H\left(x\right)=\frac{\ⅆ}{\ⅆx}\mathrm{H1}\left(x\right)+\mathrm{H2}\left(x\right)$ and the certificate $\frac{\frac{\ⅆ}{\ⅆx}\mathrm{H2}\left(x\right)}{\mathrm{H2}\left(x\right)}$ has a differential rational normal form $r,s,u,v$ with v of minimal degree.

The output from ReduceHyperexp is a sequence of two elements $\mathrm{H1},\mathrm{H2}$ each of which is either $0$ or written in the form

(The form shown above is called a multiplicative decomposition of the hyperexponential function $H\left(x\right)$.)

ReduceHyperexp is a generalization of the reduction algorithm for rational functions by Hermite (recall that a rational function is also a hyperexponential function). It also covers the differential Gosper's algorithm.

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

$H≔\frac{\exp \left(\mathrm{Int}\left(\frac{2x-7}{{\left(x+4\right)}^2}\,x\right)\right)\left(x^6+16x^5+103x^4+327x^3+647x^2+737x+194\right)}{{\left(x-1\right)}^2{\left(x+2\right)}^4{\left(x+4\right)}^2}$

$H≔\frac{{\ⅇ}^{\int \frac{2x-7}{{\left(x+4\right)}^2}\ⅆx}\left(x^6+16x^5+103x^4+327x^3+647x^2+737x+194\right)}{{\left(x-1\right)}^2{\left(x+2\right)}^4{\left(x+4\right)}^2}$

$\mathrm{H1},\mathrm{H2}≔\mathrm{ReduceHyperexp}\left(H\,x\,'\mathrm{nH}'\right)$

$\mathrm{H1},\mathrm{H2}≔-\frac{\left(24x^3+143x^2+292x+216\right){\ⅇ}^{\int -\frac{15}{{\left(x+4\right)}^2}\ⅆx}}{\left(x-1\right){\left(x+2\right)}^3},\frac{\left(x^3+17x^2+88x-231\right){\ⅇ}^{\int \frac{-23-2x}{{\left(x+4\right)}^2}\ⅆx}}{x-1}$

$\mathrm{nH}$

$\frac{\left(x^6+16x^5+103x^4+327x^3+647x^2+737x+194\right){\ⅇ}^{\int -\frac{15}{{\left(x+4\right)}^2}\ⅆx}}{{\left(x+2\right)}^4{\left(x-1\right)}^2}$

$H≔-\frac{\exp \left(\mathrm{Int}\left(\frac{2x-7}{{\left(x+4\right)}^2}\,x\right)\right)\left(x^2+27x+62\right)}{{\left(x+2\right)}^4{\left(x+4\right)}^2}$

$H≔-\frac{{\ⅇ}^{\int \frac{2x-7}{{\left(x+4\right)}^2}\ⅆx}\left(x^2+27x+62\right)}{{\left(x+2\right)}^4{\left(x+4\right)}^2}$

$\mathrm{H1},\mathrm{H2}≔\mathrm{ReduceHyperexp}\left(H\,x\right)$

$\mathrm{H1},\mathrm{H2}≔-\frac{\left(x^2+8x+16\right){\ⅇ}^{\int -\frac{15}{{\left(x+4\right)}^2}\ⅆx}}{{\left(x+2\right)}^3},0$

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.