Double-Integral over Product of three Legende-Polynomials

Question

I represent functions (for example called $f$ and $g$) by piecewise approximate it by (scaled and shifted) Legendre-Polynomials. There are intervals in $[0,1]$, where the polynomial is defined, anywhere else, it is zero. I call such a polynomial $\phi_\alpha ^i(x)$, if it is a polynomial of degree $\alpha$ on the interval specified by $i$. It holds $\int_0^1\phi_\alpha^i \cdot \phi_\alpha^i dx =1$.
So i can write $f(x) = \sum_i \sum_{\alpha=0}^p \varphi \phi_\alpha^i(x)$.
So far so good: What I come across now, is the term $$ f(x) \int_0^{1-x}g(y) dy$$ that again needs to be projected in my piecewise function space with polynomials of a maximum degree.
What I have done so far:
I can project the result into my function space by $L^2$-projection (which is the way to go for some other reason) so I come across a quadruple sum over terms of the form $$\int_0^1\phi_\alpha^i(x) \cdot \phi_\beta ^j(x) \cdot \int_0^{1-x}\phi_\gamma^k(y) \quad dy dx.$$
I already know, that it will evaluate to zero for $i\neq j$ (disjoint support of functions). Similar will the second integral be zero, if $1-x \notin k$ for all $\gamma>0$, since either orthogonality of the basis function or the support was not reached by the integral.
I could evaluate all these integrals numerically but this is unwanted since the number of intervals is rather large and/or changes. A closed form answer is appreciated.