Real Analysis – Integrate ? 1/?(-ax^2 + bx + c) dx

Question

Is there a way to solve this integral?

$$\int \frac{dx}{\sqrt{-ax^2 + bx +c}}, \;\;\;\;\; \forall\; a,b,c \in (0,+\infty) \;in\;R$$

I have tried it by changing of variable and by parts, but no result. I don't know what else to do.

Thanks in advance.

Answer 1

Hint #1: $ax^2+bx+c=a(x^2+\frac{b}{a}x+\frac{c}{a})$

Hint #2: it is possible to find $\quad p, q \quad$ such that: $\quad (x^2+\frac{b}{a}x+\frac{c}{a}) = (x+p)^2 \pm q$

Hint #3: a substitution $\quad x+p = t \quad$ will work...