General expression for entropy and free energy
Since G and S are taken to be zero at T = 0, H is also 0 at T = 0.  So you can calculate H by integrating the heat capacity from 0 to T:  $$H=\int_0^TC_pdT=\frac{a}{2}T^2+\frac{b}{4}T^4$$
Then you substitute H into the equation $$\frac{\partial (G/T)}{\partial T}=-\frac{H}{T^2}=-\frac{a}{2}-\frac{b}{4}T^2$$
and integrate between 0 and T.  This will give you G.  You can then solve for S from G = H - TS.  This gives you the same result as integrating directly to get S.