Answer
$$\frac{{{\theta ^3}}}{3} - \frac{{{\theta ^2}}}{2} + \theta + \frac{5}{2}\ln \left| {2\theta - 5} \right| + C $$
Work Step by Step
$$\eqalign{
& \int {\frac{{2{\theta ^3} - 7{\theta ^2} + 7\theta }}{{2\theta - 5}}} d\theta \cr
& {\text{Use long division on }}\frac{{2{\theta ^3} - 7{\theta ^2} + 7\theta }}{{2\theta - 5}} \cr
& \frac{{2{\theta ^3} - 7{\theta ^2} + 7\theta }}{{2\theta - 5}} = {\theta ^2} - \theta + 1 + \frac{5}{{2\theta - 5}} \cr
& {\text{Then}} \cr
& \int {\frac{{2{\theta ^3} - 7{\theta ^2} + 7\theta }}{{2\theta - 5}}} d\theta = \int {\left( {{\theta ^2} - \theta + 1 + \frac{5}{{2\theta - 5}}} \right)} d\theta \cr
& = \int {{\theta ^2}} d\theta - \int \theta d\theta + \int {d\theta } + \int {\frac{5}{{2\theta - 5}}} d\theta \cr
& {\text{Integrating}} \cr
& = \frac{{{\theta ^3}}}{3} - \frac{{{\theta ^2}}}{2} + \theta + \frac{5}{2}\ln \left| {2\theta - 5} \right| + C \cr} $$
