This integral poses a challenge.

Ordinarily integrating rational functions can be solved using the Hermite-Ostrogradski method. However, in the following integral, the coefficients ${\beta}_{0},...,{\beta}_{4}$ are not integers. (Hence, the Hermite-Ostrogradski method would not be appropriate).

Note: Expanding the integrand (trying to solve the integral using a partial fraction decomposition) is (because of the nature of the physical problem described by this integral) an inappropriate solution to this case.

$\int {\displaystyle \frac{1}{{\beta}_{0}+{\beta}_{1}x+{\beta}_{2}{x}^{2}+{\beta}_{3}{x}^{3}+{\beta}_{4}{x}^{4}}}dx$

How can this rational function be evaluated?

Ordinarily integrating rational functions can be solved using the Hermite-Ostrogradski method. However, in the following integral, the coefficients ${\beta}_{0},...,{\beta}_{4}$ are not integers. (Hence, the Hermite-Ostrogradski method would not be appropriate).

Note: Expanding the integrand (trying to solve the integral using a partial fraction decomposition) is (because of the nature of the physical problem described by this integral) an inappropriate solution to this case.

$\int {\displaystyle \frac{1}{{\beta}_{0}+{\beta}_{1}x+{\beta}_{2}{x}^{2}+{\beta}_{3}{x}^{3}+{\beta}_{4}{x}^{4}}}dx$

How can this rational function be evaluated?