I need to solve the equation $$\int \frac{1}{x^3}\frac{x^7+ax^6+bx^5+cx^4+dx^3+ex^2+fx+g}{x^6+Ax^5+Bx^4+Cx^3+Dx^2+Ex+F}\quad dx=1$$ for x, where the coefficients of the polynomials in the numerator and denominator are constants. I tried to factorize the denominator which would allow to solve the integral using partial fractions, but I was unsuccessful. So I am looking for any general formula for the same, or any method to solve the integral using partial fractions.
Is there any general formula for integration of such rational functions or any method to express the integrand as partial fractions? Thanks for any help.
N.B.: I am interested to solve the integration manually without the use of computers.