Here's an integral to try. I couldn't even get Maple to give me a closed form for it. Kriz will probably conquer it like Grant through Richmond.
$\displaystyle \int\frac{\sqrt{1+x^{2}}}{\sqrt{10-x^{3}}}dx$
My guess would be to start thusly:
$\displaystyle \int\frac{\sqrt{1+x^2}}{\sqrt{10-x^3}}dx$
$\displaystyle \int\frac{\sqrt{1+x^2}}{\sqrt{10-x^3}}\frac{\sqrt{1+x^2}}{\sqrt{1+x^2}}
dx$
$\displaystyle \int\frac{1+x^2}{\sqrt{(10-x^3)(1+x^2)}}dx$
$\displaystyle \int\frac{1}{\sqrt{(10-x^3)(1+x^2)}}dx+\int\frac{x^2}{\sqrt{(10-x^3)(1+x^2)}}dx$
How to finish I cannot tell immediately, or even whether the above is an appropriate beginning.