Is there a closed-form/analytical solution to the integral?
$$ \int \left(\frac{Ax^3 + Bx^2 + Cx + D}{Ex^3 + Fx^2 + Gx + H}\right)^4 dx $$
The function does not seem to fit any of the patterns in the list of rational functions in tables of integrals (for example).
See this Wikipedia entry.
Every rational function $\dfrac{f(x)}{g(x)}$ can be writtten as
$p(x)+\displaystyle\sum_j\dfrac{f_j(x)}{g_j(x)}$
where $p,g_j,f_j$ are polynomials such that $g_j$ is the power of an irreducible polynomial and $f_j$ has degree less than the irreducible polynomial of which $g_j$ is the power.
Note that over $\mathbb{R}$, the only irreducible polynomials are quadratic or linear.
Furthermore, the closed forms of $\displaystyle\int \dfrac{1}{(ax+b)^n}\;\text{d}x$, of $\displaystyle\int \dfrac{1}{(cx^2+dx+e)^n}\;\text{d}x$, and of $\displaystyle\int \dfrac{ax+b}{(cx^2+dx+e)^n}\;\text{d}x$
are known.
Therefore, the integral of every rational function has a closed-form expression, which you can find by first finding the partial fraction decomposition of the function.