Parameterize $\mathbf r(t)=(t,2t,3t)$, 0≤t≤1. Then $d\mathbf r=(1,2,3)dt$ and $\mathbf F\cdot d\mathbf r=[6t^2+2(9t^2+2t)+36t]dt=(24t^2+40t)dt$. Integrating gives 8+20=**28**. The specified path matters because the field is not conservative.
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