This is mechanized in current CASes, e.g. the command of Maple 2019.1
solve({a*(1 - b) + a*b*a/(a + b) = u, eval(a*(1 - b) + a*b*a/(a + b), {a = b, b = a}) = v}, {a, b}, explicit);
performs a long output which can be seen here exported as a PDF file.
Addition. The command of Mathematica
Reduce[{a*(1 - b) + a*b*a/(a + b) == u, b*(1 - a) + b^2*a/(a + b) == v}, {a, b}]//ToRadicals
produces
(u == 1 - v &&
a == 1 && (b == 1/2 (v - Sqrt[v] Sqrt[4 + v]) ||
b == 1/2 (v + Sqrt[v] Sqrt[4 + v]))) || ((a == (2 + u)/4 -
1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v +
1/3 (2 u + v) + (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 -
3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3))) -
1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) -
v - (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 -
3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/(
3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(
1/3)) - (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) -
8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u -
v + 1/3 (2 u + v) + (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v -
5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
3 2^(1/3)))((2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3))))) ||
a == (2 + u)/4 -
1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v +
1/3 (2 u + v) + (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 -
3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3))) +
1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) -
v - (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 -
3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/(
3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(
1/3)) - (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) -
8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u -
v + 1/3 (2 u + v) + (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v -
5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
3 2^(1/3)))((2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3))))) ||
a == (2 + u)/4 +
1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v +
1/3 (2 u + v) + (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 -
3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3))) -
1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) -
v - (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 -
3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/(
3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(
1/3)) + (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) -
8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u -
v + 1/3 (2 u + v) + (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v -
5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
3 2^(1/3)))((2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3))))) ||
a == (2 + u)/4 +
1/2 [Sqrt](1/4 (-2 - u)^2 - 2 u - v +
1/3 (2 u + v) + (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 -
3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3))) +
1/2 [Sqrt](1/2 (-2 - u)^2 - 2 u + 1/3 (-2 u - v) -
v - (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v - 5 v^2 -
3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) - (1/(
3 2^(1/3)))((2 (2 u + v)^3 + 27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(
1/3)) + (-(-2 - u)^3 + 4 (-2 - u) (2 u + v) -
8 (u + v - u v - v^2))/(4 [Sqrt](1/4 (-2 - u)^2 - 2 u -
v + 1/3 (2 u + v) + (2^(
1/3) (6 u - 5 u^2 + 6 v - 11 u v - 3 u^2 v -
5 v^2 - 3 u v^2))/(3 (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)) + (1/(
3 2^(1/3)))((2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v -
v^2)^2 + [Sqrt](-4 (6 u - 5 u^2 + 6 v - 11 u v -
3 u^2 v - 5 v^2 - 3 u v^2)^3 + (2 (2 u + v)^3 +
27 (-2 - u)^2 (-u^2 - u v) -
72 (2 u + v) (-u^2 - u v) -
9 (-2 - u) (2 u + v) (u + v - u v - v^2) +
27 (u + v - u v - v^2)^2)^2))^(1/3)))))) && -1 +
a != 0 && b == (a - u - v)/(-1 + a) && a^2 - u - v != 0)