Solving an Integral - $ \int t^2\frac{\left(2t\sqrt{at^2+bt+c} \right )^{2k}}{(at^2+bt+c)} \ dt $

Question

How do we solve $ \int t^2\frac{\left(2t\sqrt{at^2+bt+c} \right )^{2k}}{(at^2+bt+c)} \ dt \tag 1 $ to a finite form?

1. $k,a,b,c$ are constants
2. $at^2+bt+c$ does not guarantee equal roots always

Answer 1

According to a CAS, there is a closed form expression $$I_k=\int t^2\frac{\left(2t\sqrt{at^2+bt+c} \right )^{2k}}{(at^2+bt+c)} \ dt $$ but it involves the Appell hypergeometric function of two variables which I do not suppose that this will be of major interest to you.

For integer values of power $k$, what you could easily show is that $$I_k=t^{2k+3}P_{2k-2}(t)$$ where $P_m(t)$ is a polynomial of $t$ of degree $m$. The expressions can be obtained using the binomial theorem.