Evaluate $\int \frac{7x^4+2}{x^8-e^{7x}}dx$

Question

Evaluate $$I=\int \frac{7x^4+2}{x^8-e^{7x}}dx$$

My try: we have $$I=\int e^{-7x} \times \frac{7x^4+2}{x^8e^{-7x}-1}dx$$

I tried to use substitution $xe^{-x}=t$ but of no use.

any clue?

Answer 1

I don't think it has a closed form. A series form solution to a more general integral in terms of upper incomplete gamma functions is: $$\int \frac {at^4+2b}{t^8-be^t} dt = \sum_{k=0}^∞ \frac {1}{b^k(k+1)^{8k+1}}(\frac {a}{b}\frac {\Gamma [8k+5, (k+1)t]}{(k+1)^4} + 2\Gamma [8k+1,(k+1)t])$$