f=Integrate[6/8*(c2-a2)^3*(s-b1)^3,{b1,0,s}]+Integrate[6/8*(c2-a2)^3*(b1-s)^3,{b1,s,1}];
s=(1-t)*a1+t*c1;
g=Integrate[f*(c2-a2),{t,0,1},{c1,0,1},{a1,0,1},{c2,a2,1}];
Integrate[g,{a2,0,1}]