I am a regular user of Mathematica, Julia, and MATLAB but I am looking for something different. The problem I am trying to solve in Mathematica only requires (dense) linear algebra to specify but is of a size such that Mathematica takes too long (and seems to automatically halt the computation after like 18 hours, when it should finish in a week or so). I am looking for a symbolic solution and not a numeric solution (to then use a machine learning approach on the resulting equations). I was wondering if anyone knew if there are more efficient and parallel symbolic packages. Checking around it looks like Singularity or Machaly2 may be what's right for this problem?

Edit: As requested here is more detail. It's as straight-forward as it sounds. I need to solve for the symbolic solution of a system of 4x4 matrices and 1x4 vectors. I am able to succinctly write down the exact solution (attached as a photo, hard to format it right in the editor...),

but as you can see it's a little unwieldy (then it gets squared and simplified). I want to use the resulting polynomial (in terms of the matrix coefficients) in numerical computations, but I don't want to have to solve this system of equations each time the matrices change since this would increase the computational cost immensely.

So at a higher level it's really simple, and coding Mathematica to get it to try this computation is simple, but it is taking a lot of computing time (largely because Mathematica's engine is single threaded). In the simple case when the $A$ and $B$ matrices are lower triangular, I Mathematica was able to solve it after about 10 minutes. But getting rid of this assumption on even one matrix increases the complexity by a lot.

Let me lastly note that I have computing power at my disposal. I do high-performance computing, and so I have a few nodes available for a few hundred CPU cores and about a terabyte of RAM to let it churn for a month (I was hoping to change each matrix from lower triangular to full one by one to see how far I can get). However, I don't know what program/language will do this kind of "HPC Symbolic Computing". I was hoping there was some computational algebra package that has the tools to do this.