The IsHyperexponential(H,x) command returns true if $H$ is a hyperexponential function of x. Otherwise, it returns false.

A function H is hyperexponential of x if $\frac{\frac{\ⅆ}{\ⅆx}H\left(x\right)}{H\left(x\right)}=R\left(x\right)$, a rational function of x. $R\left(x\right)$ is the certificate of $H\left(x\right)$. If the third optional argument is included, it is assigned the certificate $R\left(x\right)$.

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

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

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

$\mathrm{IsHyperexponential}\left(H\,x\,'R'\right)$

$\mathrm{true}$

$R$

$-\frac{25x^8+473x^7+3748x^6+16441x^5+46985x^4+87685x^3+89797x^2+40832x+17764}{\left(x^6+16x^5+103x^4+327x^3+647x^2+737x+194\right)\left(x+2\right)\left(x-1\right){\left(x+4\right)}^2}$