See please https://en.wikipedia.org/wiki/Euler_substitution
Let us look at the most general indefinite integral
$$\int f(x) dx \tag{1} $$ where $x \in D$ and $D$ is some interval.
It is my understanding that to do any substitution $$x = \phi(t)$$
in $(1)$, it must be the case that $\phi(t)$ is either strictly increasing or strictly decreasing i.e. that $\phi$ maps in a bijective manner some interval $D_1$ into the interval $D$ and then of course $x$ takes values in $D$.
In other words to have a valid substitution, the derivative $\phi'$
must be either strictly positive or strictly negative in the whole interval $D_1$
But in the first Euler substitution, it is not always the case that
$$\phi(t) = \frac{t^2-c}{b-2t\sqrt{a}}$$
has either a positive or a negative derivative (for all values of $t$).
So... why for this substitution there is no additional requirement which would ensure that?
I mean... in this sense is it a valid substitution in general?
I assume it is after being checked by so many people. But why?
I guess the same question applies to the other two Euler substitutions, but I haven't yet calculated $\phi'$ for their cases. I wonder if for the other two substitutions $\phi'$ is strictly positive/negative, or it is but only under certain conditions, and I wonder what these conditions are. I will check that.
But let's say my question is ONLY about the first substitution in particular.
According to [Fich, Ch. VIII, §1, 268], a substitution of a variable in an indefinite integral is based on a local rule of a differentiability of a composite function at a point. Namely, if $\int g(t)dt=G(t)+C$ then $\int g(\omega(x))\omega’(x)dx =G(\omega(x))+C$, provided functions $g(t)$, $\omega(x)$, $\omega’(x)$ are continuous. This straightforwardly follows from an equality
$$\frac {\partial d}{dx} G(\omega(x))=G’(\omega(x))\omega’(x)= g(\omega(x))\omega’(x),$$ taking into account that $G’(t)=g(t)$.
I copied a more detailed explanation and an example from [Fich] here and here.
References
[Fich] (Grigorii Fichtenholz, Differential and Integral Calculus, v. II, 7-th edition, M.: Nauka, 1970 (in Russian).