#bb phi(x,y,z)=xy \ bb hat i+yz \ bb hat j+xz \ bb hat k#
The path is:
#bb r_1 = langle 3,1,2 rangle + lambda langle-1,0,3 rangle, qquad d bb r_1 = langle-1,0,3 rangle d lambda #
The integral is:
#I_1 = int_(bb r_1) \ ((xy),(yz),(xz))\ cdot \ d bb r_1#
#= int_0^1 \ (((3 - lambda)(1)),((1)(2 + 3 lambda)),((3 - lambda)(2+3 lambda)))\ cdot ((-1),(0),(3) ) \ d lambda #
#= int_0^1 \ (lambda - 3) + 0 + 3(3 - lambda)(2+3 lambda) \ d lambda #
#= int_0^1 \ -9 lambda^2 + 22 lambda + 15 \ d \ lambda = 23#
#bb r_2 = langle 2,1,5 rangle + lambda langle-1,-1,-4 rangle, qquad d bb r_2 = langle-1,-1,-4 rangle d lambda #
# I_2 = int_0^1 \ (((2-lambda)(1- lambda)),((1- lambda)(5 - 4 lambda)),((5 - 4 lambda)(2 - lambda)))\ cdot ((-1),(-1),(-4) ) \ d lambda #
#= int_0^1 \ - (2 - lambda )(1 - lambda) - (1 - lambda)(5 - 5 lambda) - 4 (5 - 4 lambda)(2 - lambda) \ d lambda #
#= int_0^1 \ -22 lambda^2 + 65 lambda - 47 \ dlambda = - 131/6#
#bb r_3 = langle 3,1,2 rangle + lambda langle 2,1,1 rangle, qquad d bb r_3 = langle 2,1,1 rangle d lambda #
#I_3 = int_0^1 \ (((3 + 2 lambda)(1+ lambda)),((1+ lambda)(2 + lambda)),((2 + lambda)(3 + 2 lambda)))\ cdot ((2),(1),(1) ) \ d lambda #
#= int_0^1 d lambda \ 2 (3 + 2 lambda )(1 + lambda) + (1 + lambda)(2 + lambda) + (2 + lambda)(3+ 2 lambda) #
#= int_0^1 \ 7 lambda^2 + 20 lambda + 14
\ dlambda = 79/3#