Zorich Exercise 5.7.3: Reduction of integrals of the form $\int R(x,\sqrt{ax^2+bx+c})\,dx$

Question

Consider integrals of the form $$\int R(x,\sqrt{ax^2+bx+c})\,dx,\tag{1}$$ where $R$ is a rational function of $x$ and $\sqrt{ax^2+bx+c}$. Show that the integral $(1)$ can always be reduced to computing integrals of the following three types: $$\int \frac{P(x)}{\sqrt{ax^2+bx+c}}\,dx,\qquad \int\frac{dx}{(x-x_0)^k\sqrt{ax^2+bx+c}},\qquad \int\frac{(Ax+B)\,dx}{(x^2+px+a)^m\sqrt{ax^2+bx+c}}.$$

This is an exercise in Zorich's Mathematical Analysis I textbook, chapter 5, section 5.7.

My attempt: Assume that $R(x,y)=\frac{P(x,y)}{Q(x,y)}$, where $P, Q$ are polynomials. We write $$P(x,y)=P_1(x,y^2)y+P_2(x,y^2),\qquad Q(x,y)=Q_1(x,y^2)y+Q_2(x,y^2),$$ where $P_1, P_2$ and $Q_1, Q_2$ are all polynomials. Then \begin{align*} R(x,\sqrt{ax^2+bx+c})&=\frac{P_1(x, ax^2+bx+c)\sqrt{ax^2+bx+c}+P_2(x, ax^2+bx+c)}{Q_1(x, ax^2+bx+c)\sqrt{ax^2+bx+c}+Q_2(x, ax^2+bx+c)}\\ &=\frac{F_1(x)\sqrt{ax^2+bx+c}+F_2(x)}{G_1(x)\sqrt{ax^2+bx+c}+G_2(x)}, \end{align*} where $F_1, F_2, G_1, G_2$ are polynomials.

And I'm stuck here.

Answer 1

Hint: \begin{align} \frac{F_1(x)\sqrt{ax^2+bx+c}+F_2(x)}{G_1(x)\sqrt{ax^2+bx+c}+G_2(x)} = \frac{F_1(x)\sqrt{ax^2+bx+c}+F_2(x)}{G_1(x)^2(ax^2+bx+c)-G_2(x)^2}(G_1(x)\sqrt{ax^2+bx+c}-G_2(x)) \end{align}