Maxima error with system of symbolic equations

Question

I'm having trouble with Maxima. I am trying to find the symbolical solution for a set of equations:

E1: a1*x1*x1 - a2*x1*x3 + a3*x1*x2+a4*x1 -p1;
E2: b1*x3^2-b2*x3*x1 + x3 *b3 -p2;
E3: c1*x1 - c2*x2 -c3* x3*x1 - c4*x3*x3 - c5*x3*x2 + c6*x3-p3;
algsys([E1, E2, E3], [x1, x2, x3]);

Trying to solve that set of equations won't work with maxima but it will work with Matlab. Unfortunately, I tried to check the solution returned by Matlab using Maxima. When I reduce the equation by letting out the coefficients a1...c6 it still won't compute returing the following error:

E1: x1*x1 - x1*x3 + x1*x2+x1 -p1;
E2: x3^2-x3*x1- p2;
E3: x1 - x2 - x3*x1 - x3*x3 - x3*x2 + x3-p3;
algsys([E1, E2, E3], [x1, x2, x3]);    
algsys: Couldn't reduce system to a polynomial in one variable.
     -- an error.  Entering the Maxima debugger.
    Enter ':h' for help.

Now I don't know wether the system of equations is not solveable and maxima's solver won't compute this unsolveable system and the solution returned by Matlab is a numerical estimation. Or if Matlab indeed is able to solve the system ( it returns a solution for a fifth order polynomial) and Maxima just can't solve a polynomial with such a high order.

Howevert will it compute if I substitute the symbolic coefficients with values. In this case, maxima will return the same values as Matlab.

Is there an error in the input code? Or is Matlabs symbolic solver simply better than Maxima?