<s>[INST] Closed form integral $\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$ Is there a closed form expression for the definite integral $$I=\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$$ for $a<b<c<d$? 
Mathematica 9.0 can do it for special cases using elliptic integrals, so I suspect that there may be a more general closed form solution. Some examples:
For $a=1$, $b=2$, $c=3$, $d=4$:
$$I=-2 E\left(\frac34\right) - K\left( \frac34  \right) + 32 K\left( -3  \right) - 20\, \Pi(-1,-3)$$
For $a=1$, $b=2$, $c=3$, $d=5$:
$$I=-\sqrt{6}\, E\left(\frac23\right) - \sqrt{6}\, K\left( \frac34  \right) + 25\sqrt{2}\, K\left( -2  \right) - \frac{33}{\sqrt{2}}\, \Pi\left(-\frac12,-2\right)$$
For $a=1$, $b=2$, $c=5$, $d=7$:
$$I=-\sqrt{20}\, E\left(\frac9{10}\right) - \sqrt{5}\, K\left( \frac9{10}  \right) + 49\sqrt{2}\, K\left( -9  \right) - \frac{75}{\sqrt{2}}\, \Pi\left(-\frac32,-9\right)$$
For $a=1$, $b=2$, $c=7$, $d=11$:
$$I=-\sqrt{54}\, E\left(\frac{25}{27}\right) + \frac{103\sqrt{2}}{\sqrt{27}}\, K\left( \frac{25}{27}  \right) - \frac{42\sqrt{2}}{\sqrt{27}}\, \Pi\left(-\frac59,\frac{25}{27}\right)$$
 [/INST] This integral is listed in Byrd and Friedman
\begin{equation}
I^{(2)}(a,b,c,d)=\frac{d^2}{\sqrt{(d-b)(c-a)}}\big\{2K(k)+4(b^2/a^2-1)\Pi(\alpha^2,k)+ 
\frac{\alpha^2E(k)+(k^2-\alpha^2)K(k) + (2\alpha^2k^2+2\alpha^2-\alpha^4-3k^2)\Pi(\alpha^2,k)}{(\alpha^2-1)(k^2-\alpha^2)}\bigg\}
\end{equation}
where $\alpha^2=(c-b)/(d-b)$, $k^2=\alpha^2(d-a)/(c-a)$.
</s>