I am trying to solve the following integral involving a cubic polynomial in the denominator of the integrand: $$\int\frac{r^ndr}{r^3+ar^2+br+c}$$ Solving this integral in Mathematica gives the result
Here, one can observe that the integral is expressed as a sum of three terms corresponding to the three roots of the cubic polynomial. However, I tried to solve the integral analytically by "completion of cube" similar to the completion of squares method, but I was unsuccessful.
My question is whether such an approach could be used to solve the integral? I could not find any resources regarding "completion of cube" method. Also, if the approach is not a suitable one, are there any other method to solve such an integral?
$$\int\frac{r^n}{r^3+ar^2+br+c} \,dr=\int\frac{r^n}{(r-r_1)(r-r_2)(r-r_3)}\,dr$$ Now, using partial fraction decomposition $$\frac{r^n}{(r-r_1)(r-r_2)(r-r_3)}=a_1\frac {r^n}{r-r_1}+a_2\frac {r^n}{r-r_2}+a_3\frac {r^n}{r-r_3}$$ making that you face integrals $$I=\int \frac {r^n}{r-k}\,dr=-\frac{r^{n+1}}{k (n+1)} \, _2F_1\left(1,n+1;n+2;\frac{r}{k}\right)$$