Calculate $\int_0^1 \Big( \int _0^x \sqrt{y^2+\frac{y^2}{x^2}+\frac{y^4}{x^4}} dy \Big) dx$ 
Calculate $\int_0^1 \Big( \int _0^x \sqrt{y^2+\frac{y^2}{x^2}+\frac{y^4}{x^4}} dy \Big) dx$

I have a problem with this task because I have no idea what parametrization should be used to calculate it. I tried $x=r\sin \alpha, y=r \cos \alpha$ or $s=y, t=\frac yx$ or $s=y, t=(\frac yx)^2$ but in each case I got too complicated calculations.
Does anyone have any clever ideas?

What do you mean by too complicated? Upon substitution $y=x\sqrt{z}$ the integral becomes $$\int_0^1 {\rm d}x \frac{x}{2} \int_0^1 \sqrt{x^2+z+1} \, {\rm d}z = \int_0^1 {\rm d}x \frac{x}{3} \left[ \left( x^2 + 2 \right)^{3/2} - \left(x^2+1\right)^{3/2} \right] \\
\stackrel{t=x^2}{=} \frac{1}{6} \int_0^1 {\rm d}t \left[ (t+2)^{3/2} - (t+1)^{3/2} \right] = \frac{1}{15} - \frac{8\sqrt{2}}{15} + \frac{3\sqrt{3}}{5}\, .$$
