Mathematica will evaluate this integral:
$$ \int_1^x\frac{1}{(1+C_1(\eta)(-1+x))^3}dx $$
but not this one:
$$ \int_1^x\frac{1}{(1+C_1(\eta)(-1+x)+C_2(\eta)(-1+x)^2)^3}dx $$
I'm trying to solve this, and it is really holding me up. Any ideas why mathematica just hangs on trying to evaluate this?
It could have been you you add the syntax you used.
Changing notations $$I=\int \frac{dx}{\left(1+a (x-1)+b (x-1)^2\right)^3}$$ what I obtained on Wolfram Cloud for $$J=2 \left(a^2-4 b\right)^2\, I$$ is $$J=\frac{24 b^2 \tan ^{-1}\left(\frac{a+2 b (x-1)}{\sqrt{4 b-a^2}}\right)}{\sqrt{4 b-a^2}}-\frac{\left(a^2-4 b\right) (a+2 b (x-1))}{\left(1+a (x-1)+b (x-1)^2\right)^2}+\frac{6 b (a+2 b (x-1))}{1+a (x-1)+b (x-1)^2}$$ and, if $b\to 0$, $$I=-\frac{1}{2 a (1+a (x-1))^2}$$ which, I suppose, is what you obtained for the first antiderivative.