How to solve $\frac{dx}{dt} = ax^2 + bx + c$? It's been a while since I've solved ODEs analytically, but this looks like a first order nonlinear ODE that is separable.
When I separate the terms, I see
\begin{align}
    \frac{1}{ax^2 + bx + c}dx = dt \\
\end{align}
The LHS it'll be quite difficult to evaluate. Is there a simple solution to this problem that I am not seeing?
 A: For integration techniques without using $i$:
If $ax^2+bx+c$ has repetitive real roots, then it may be written as $$\dfrac{1}{(jx+k)^2}$$ which is a well known integration, j and k are constants and does not deserve recognition in this answer.
If $ax^2+bx+c$ has repetitive non-real roots, then it may be written as $$\dfrac{1}{u} \dfrac{1}{x^2+v^2}$$ which upon integration yields, $$\frac{1}{u}\cdot\frac{1}{v}\arctan\left(\frac{x}{v}\right)$$ where u and v are constants.
If $ax^2+bx+c$ has distinct real roots, it's easy to factor them out and write $$\frac{1}{ax^{2}+bx+c}=\frac{A}{f\left(x\right)}+\frac{B}{g\left(x\right)}$$
where A,B are constants and f(x) and g(x) are linear functions. (the factors of the quadratic equation). The resultant form for the above would consist of logarithms of the form $$\frac{A}{p}\ln\left(f\left(x\right)\right)+\frac{B}{q}\ln\left(g\left(x\right)\right)+C_{onstant}$$ where p,q are the coefficients of x in the expression f(x) and g(x) respectively.
Iff $4ac-b^2>0$, then the roots are imaginary and the problem can be solved by expressing $$\frac{1}{ax^{2}+bx+c}=\frac{1}{\left(h\left(x\right)\right)^{2}+d}$$
where h(x) is again linear, d is a positive constant, but we just completed the whole square.
The resultant integral comes out to be:
$${\dfrac{2\arctan\left(\frac{2ax+b}{\sqrt{4ac-b^2}}\right)}{\sqrt{4ac-b^2}}}+C_{onstant}$$
and iff $d<0$, the integral is of form $$\dfrac{1}{\left(ax+b\right)^2-r^2}$$ which is $$-\dfrac{\ln\left(\left|ax+r+b\right|\right)-\ln\left(\left|ax-r+b\right|\right)}{2ar}+C_{onstant}$$ upon integration.
For integration techniques using $i$:
Make it a form of the third $\text{if}$ statement in the above body and treat $i$ as a constant.
