I have several double integrals, about $20$.
One of them, with $C1, C2, x_{iu}, x_{jl}$ being four real constants, is:
$\displaystyle \frac{2}{C_1}\frac{2}{C_2}\int_{x_{jl}}^{x_{iu}}\int_y^{x_{iu}}(x - y)(x - x_{iu})(y - x_{jl})dxdy$
I've managed to compute:
$\displaystyle \int_y^{x_{iu}}(x - y)(x - x_{iu})(y - x_{jl})dx$
$\displaystyle = \frac2{C_1}\frac2{C_2}(y - x_{jl})\left(\frac{x_{iu}^3-y^3}3 - (x_{iu} + y)\frac{x_{iu}^2 - y^2}2 + x_{iu}^2y - y^2x_{iu}\right)$
by hand, but that took me more than 1 hour and thus I won't have enough time to compute all of the $20$ integrals by hand.
That's why I'm looking for a free software that can help me trough the computations of these integrals. Note that all the functions within the $10$ double integrals are polynomial functions.
You need a Computer Algebra System. Most popular commercial ones are Mathematica and Maple.
WolframAlpha is a good tool to do such calculations online. The free version has a limit in calculation time but for your problem it is working.