= Solution
Integrating $\phi_x=2xy$ gives $\phi=x^2y+h(y,z)$. Matching $\phi_y=x^2+3y^2+z^2$ then gives $h=y^3+yz^2+k(z)$. Finally $\phi_z=2yz$ forces $k$ to be constant. A <scalar potential> is
$$
\boxed{\phi(x,y,z)=x^2y+y^3+yz^2+C.}
$$
The initial and final points of the <helix> are $(b,0,0)$ and $(0,b,cL)$, where $L=5\pi/2$. The <fundamental theorem for line integrals> gives
$$
\phi(0,b,cL)-\phi(b,0,0)=b^3+bc^2L^2
=\boxed{b\left(b^2+\frac{25\pi^2c^2}{4}\right)},
$$
agreeing with the directly evaluated $I_2$.
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