Quick Way to Integrate Quadratic Denominators?

Question

Is there a quick way to evaluate

$\int \frac{dx}{ax^2+bx+c}$

$\int \frac{dx}{(ax^2+bx+c)^n}$

without memorizing any inverse sines or tans, or ending up with crazy reduction formulae - preferably with some intuitive reason why we get the answer we get?

Answer 1

Preposition: No.

Proof: $$\int_{-\infty}^\infty\frac{dx}{x^2+1}=\pi$$

Answer 2

you can express the quadratic as a product of 2 monomials over $\mathbb{C}$, and then express it as an integral of a sum of the inverse of monomials. without boundaries on the integrals this is pretty quick.