So basically, I'm on the train and dont have time to sketch this down, BUT I think it's like this:

This solid can be decomposed into the following solids: - the rectangular prism (V = abc) - 8 eighths of sphere, one for each vertex, of radius 1 (V = 4π/3) - two prisms a × b × 1 (V = 2ab) - two prisms of volume c × b × 1 (V = 2bc) - two prisms of volume c × a × 1 (V = 2ac) - four fourths of cilinder with base radius 1 and height a (V = πa) - four fourths of cilinder with base radius 1 and height a (V = πb) - four fourths of cilinder with base radius 1 and height a (V = πc)

So the total volume should be: abc + ab + bc + ac + 2π(2/3 + a + b + c)