Integral involving ratios of polynomial and square roots Is there a simple way to compute the following integral
$$I(y,\beta,\gamma)=\int_1^y \frac{x^2}{\sqrt{\left(x^2-1\right)\left(x^2-\gamma ^2\right) \left(x^2+\beta ^2\right) }}\mathrm{d}x$$
with $y>1$, $0<\gamma<1$ and $\beta>0$?


Define the function $\mathcal{I}:\left(0,1\right)\times\mathbb{R}_{>0}\times\left(1,\infty\right)\rightarrow\mathbb{R}$ via the improper integral
$$\mathcal{I}{\left(p,q,y\right)}:=\int_{1}^{y}\mathrm{d}x\,\frac{2x^{2}}{\sqrt{\left(x^{2}-1\right)\left(x^{2}-p^{2}\right)\left(x^{2}+q^{2}\right)}}.$$

Suppose $\left(p,q,y\right)\in\left(0,1\right)\times\mathbb{R}_{>0}\times\left(1,\infty\right)$, and set $z:=y^{2}\land a:=1\land b:=p^{2}\land c:=0\land d:=-q^{2}$.
Note that $z>a>b>c>d$, and then also
$$0<\frac{\left(a-d\right)\left(b-c\right)}{\left(a-c\right)\left(b-d\right)}<1\land0<\frac{\left(b-d\right)\left(z-a\right)}{\left(a-d\right)\left(z-b\right)}<1.$$
The integral $\mathcal{I}$ can be transformed into an elliptic integral by means of a simple power substitution:
$$\begin{align}
\mathcal{I}{\left(p,q,y\right)}
&=\int_{1}^{y}\mathrm{d}x\,\frac{2x^{2}}{\sqrt{\left(x^{2}-1\right)\left(x^{2}-p^{2}\right)\left(x^{2}+q^{2}\right)}}\\
&=\int_{1}^{y^{2}}\mathrm{d}t\,\frac{t}{\sqrt{t}\sqrt{\left(t-1\right)\left(t-p^{2}\right)\left(t+q^{2}\right)}};~~~\small{\left[x=\sqrt{t}\right]}\\
&=\int_{a}^{z}\mathrm{d}t\,\frac{t}{\sqrt{\left(t-a\right)\left(t-b\right)\left(t-c\right)\left(t-d\right)}}.\\
\end{align}$$
Next, consider the linear fractional transformation given implicitly by the relation
$$\left(t-b\right)\left(1-u\right)=\left(a-b\right).$$
Then,
$$t=\frac{a-bu}{1-u}\implies dt=du\,\frac{\left(a-b\right)}{\left(1-u\right)^{2}},$$
and
$$\frac{t-a}{t-b}=u\land\frac{t-c}{t-b}=\frac{a-c}{a-b}-\frac{b-c}{a-b}u\land\frac{t-d}{t-b}=\frac{a-d}{a-b}-\frac{b-d}{a-b}u.$$
Applying this LFT to $\mathcal{I}$, we obtain
$$\begin{align}
\mathcal{I}{\left(p,q,y\right)}
&=\int_{a}^{z}\mathrm{d}t\,\frac{t}{\sqrt{\left(t-a\right)\left(t-b\right)\left(t-c\right)\left(t-d\right)}}\\
&=\int_{a}^{z}\mathrm{d}t\,\frac{t}{\left(t-b\right)^{2}\sqrt{\left(\frac{t-a}{t-b}\right)\left(\frac{t-c}{t-b}\right)\left(\frac{t-d}{t-b}\right)}}\\
&=\int_{0}^{\frac{z-a}{z-b}}\mathrm{d}u\,\frac{\left(a-b\right)}{\left(1-u\right)^{2}}\cdot\frac{a-bu}{1-u}\cdot\frac{\left(1-u\right)^{2}}{\left(a-b\right)^{2}}\\
&~~~~~\times\frac{1}{\sqrt{u\left(\frac{a-c}{a-b}-\frac{b-c}{a-b}u\right)\left(\frac{a-d}{a-b}-\frac{b-d}{a-b}u\right)}};~~~\small{\left[t=\frac{a-bu}{1-u}\right]}\\
&=\int_{0}^{\frac{z-a}{z-b}}\mathrm{d}u\,\frac{a-bu}{1-u}\cdot\frac{1}{\sqrt{u\left[\left(a-d\right)-\left(b-d\right)u\right]\left[\left(a-c\right)-\left(b-c\right)u\right]}}\\
&=\int_{0}^{\frac{\left(b-d\right)\left(z-a\right)}{\left(a-d\right)\left(z-b\right)}}\mathrm{d}v\,\frac{a-b\left(\frac{a-d}{b-d}\right)v}{1-\left(\frac{a-d}{b-d}\right)v}\\
&~~~~~\times\frac{1}{\sqrt{v\left(1-v\right)\left[\left(a-c\right)\left(b-d\right)-\left(b-c\right)\left(a-d\right)v\right]}};~~~\small{\left[u=\left(\frac{a-d}{b-d}\right)v\right]}\\
&=\frac{1}{\sqrt{\left(a-c\right)\left(b-d\right)}}\int_{0}^{\frac{\left(b-d\right)\left(z-a\right)}{\left(a-d\right)\left(z-b\right)}}\mathrm{d}v\,\left[\frac{a-b}{1-\left(\frac{a-d}{b-d}\right)v}+b\right]\frac{1}{\sqrt{v\left(1-v\right)\left[1-\frac{\left(a-d\right)\left(b-c\right)}{\left(a-c\right)\left(b-d\right)}v\right]}}\\
&=\frac{1}{\sqrt{\left(a-c\right)\left(b-d\right)}}\int_{0}^{\sin^{2}{\left(\varphi\right)}}\mathrm{d}v\,\left[\frac{a-b}{1-\eta\,v}+b\right]\frac{1}{\sqrt{v\left(1-v\right)\left(1-\kappa^{2}v\right)}}\\
&=\frac{2}{\sqrt{\left(a-c\right)\left(b-d\right)}}\int_{0}^{\sin{\left(\varphi\right)}}\mathrm{d}x\,\left[\frac{a-b}{1-\eta\,x^{2}}+b\right]\frac{1}{\sqrt{\left(1-x^{2}\right)\left(1-\kappa^{2}x^{2}\right)}};~~~\small{\left[v=x^{2}\right]}\\
&=\frac{2}{\sqrt{\left(a-c\right)\left(b-d\right)}}\left[\left(a-b\right)\Pi{\left(\varphi,\eta,\kappa\right)}+bF{\left(\varphi,\kappa\right)}\right],\\
\end{align}$$
where we've introduced the auxiliary parameters
$$\kappa:=\sqrt{\frac{\left(a-d\right)\left(b-c\right)}{\left(a-c\right)\left(b-d\right)}}\in\left(0,1\right),$$
$$\varphi:=\arcsin{\left(\sqrt{\frac{\left(b-d\right)\left(z-a\right)}{\left(a-d\right)\left(z-b\right)}}\right)}\in\left(0,\frac{\pi}{2}\right),$$
$$\eta:=\frac{a-d}{b-d}\in\left(1,\infty\right).$$
Note: even though $\eta>1$, we can still define $\Pi$ in the usual way without worrying about Cauchy principal values since $0<1-\eta\sin^{2}{\left(\varphi\right)}$.

