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$$ F(t) = at + b $$$$W=\int_t F(t)dr, ~~~~~~~~~ ~~dr = v(t)dt$$$$W=\int_t F(t)\cdot v(t) dt$$ $$ f = \frac{dp}{dt}$$

therefore $$v(t) = \frac{1}{m}(at^2/2+bt)$$then $$W = \int_t \frac{at+b}{m}\cdot\left(\frac{at^2}{2}+bt\right)dt$$ $$W = \frac{1}{m}\int_t \frac{a^2t^3}{2}+abt^2+\frac{abt^2}{2}+b^2tdt$$ $$W =\frac{1}{m}\left(\frac{a^2t^4}{8}+\frac{abt^3}{3}+\frac{abt^3}{6}+\frac{b^2t^2}{2}\right)$$