I am faced with an expression of the form: $\frac{F(x)}{(1+x)G(x)}=y(\theta).$ Given some set of parameters $\theta$, I can easily compute $y$. The functionals $F(x)$ and $G(x)$ are integrals of functions $f(x')$ and $g(x')$ and the limits of integration depend on $x$.
I know that the left hand side is monotonically decreasing with $x$. What is the quickest way to find the value of $x$ that satisfies the equation with Python? Is there any method that doesn't require many, many guesses with a numerical integration routine?