<s>[INST] How can I integrate $\int\sqrt{\frac{x^2+bx+c}{x^2+ex+f}}dx?$ How can I integrate
$$\int\sqrt{\dfrac{x^2+bx+c}{x^2+ex+f}}\,dx?$$ I was thinking a substitution $$t=\frac{x^2+bx+c}{x^2+ex+f},$$ which inverts as follows:
$$(x^2+ex+f)t=x^2+bx+c$$
$$(t-1)x^2+(et-b)x+ft-c=0$$
$$x=\dfrac{b-et+\sqrt{(b-et)^2-4(t-1)(ft-c)}}{2(t-1)},$$ but I've never really dealt with integrals of this complexity with the aim of finding a closed form in terms of fundamental integral functions. The best I could do is expand the integrand as a not-so-nice power series.
 [/INST] Going off of Travis Willse's answer, I notice that
$$\int \frac{x^2+ax+b}{\sqrt{x^4+\alpha x^2+\beta x+\gamma}}dx$$ can be addressed as follows:
Since $$\frac{1}{\sqrt{x^4+\alpha x^2+\beta x+\gamma}}=\sum_{p,q,r\geqslant 0 }\binom{-\frac{1}{2}}{p,q,r,-\frac{1}{2}-p-q-r}\alpha^{q}\beta^r\gamma^{-\frac{1}{2}-p-q-r}x^{4p+2q+r},$$ we have
$$\int \frac{x^2+ax+b}{\sqrt{x^4+\alpha x^2+\beta x+\gamma}}dx=\int \sum_{p,q,r\geqslant 0 }\binom{-\frac{1}{2}}{p,q,r,-\frac{1}{2}-p-q-r}\alpha^{q}\beta^r\gamma^{-\frac{1}{2}-p-q-r}x^{4p+2q+r}(x^2+ax+b) dx\\=\boxed{\sum_{p,q,r\geqslant 0 }\binom{-\frac{1}{2}}{p,q,r,-\frac{1}{2}-p-q-r}\alpha^{q}\beta^r\gamma^{-\frac{1}{2}-p-q-r}\left(\frac{x^{4p+2q+r+3}}{4p+2q+r+3}+\frac{ax^{4p+2q+r+2}}{4p+2q+r+2}+\frac{bx^{4p+2q+r}}{4p+2q+r}\right) +C}$$ However, this is not the answer I'm looking for, but I put it here just so that it's out there to inspire someone.
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