The Gosper(T,x) command solves the problem of indefinite hyperexponential integration, that is, for the input hyperexponential function T of x, it constructs another hyperexponential function G of x such that $T\left(x\right)=\frac{\ⅆ}{\ⅆx}G\left(x\right)$, provided that such a G exists. Otherwise, the function returns the error message ``no polynomial solution found''.

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

$T≔\frac{-21x^2+116x-94}{-73-35x^4-14x^3-9x^2-51x}\exp \left(-\frac{91}{-86+5x}\right)-\frac{-68-7x^3+58x^2-94x}{{\left(-73-35x^4-14x^3-9x^2-51x\right)}^2}\exp \left(-\frac{91}{-86+5x}\right)\left(-140x^3-42x^2-18x-51\right)+\frac{455\left(-68-7x^3+58x^2-94x\right)}{\left(-73-35x^4-14x^3-9x^2-51x\right){\left(-86+5x\right)}^2}\exp \left(-\frac{91}{-86+5x}\right)$

$T≔\frac{\left(-21x^2+116x-94\right){\ⅇ}^{-\frac{91}{-86+5x}}}{-35x^4-14x^3-9x^2-51x-73}-\frac{\left(-7x^3+58x^2-94x-68\right){\ⅇ}^{-\frac{91}{-86+5x}}\left(-140x^3-42x^2-18x-51\right)}{{\left(-35x^4-14x^3-9x^2-51x-73\right)}^2}+\frac{455\left(-7x^3+58x^2-94x-68\right){\ⅇ}^{-\frac{91}{-86+5x}}}{\left(-35x^4-14x^3-9x^2-51x-73\right){\left(-86+5x\right)}^2}$

$\mathrm{Int}\left(T\,x\right)=\mathrm{Gosper}\left(T\,x\right)$

$\int \left(\frac{\left(-21x^2+116x-94\right){\ⅇ}^{-\frac{91}{-86+5x}}}{-35x^4-14x^3-9x^2-51x-73}-\frac{\left(-7x^3+58x^2-94x-68\right){\ⅇ}^{-\frac{91}{-86+5x}}\left(-140x^3-42x^2-18x-51\right)}{{\left(-35x^4-14x^3-9x^2-51x-73\right)}^2}+\frac{455\left(-7x^3+58x^2-94x-68\right){\ⅇ}^{-\frac{91}{-86+5x}}}{\left(-35x^4-14x^3-9x^2-51x-73\right){\left(-86+5x\right)}^2}\right)\ⅆx=\frac{\left(7x^3-58x^2+94x+68\right){\ⅇ}^{-\frac{91}{-86+5x}}}{35x^4+14x^3+9x^2+51x+73}$

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