7.0 Integral of the type $\int {\frac{{a{x^2} + bx + c}}{{(p{x^2} + qx + r)}}dx,\int {\frac{{a{x^2} + bx + c}}{{\sqrt {p{x^2} + qx + r} }}dx} } ,\int {(a{x^2} + bx + c)} \sqrt {p{x^2} + qx + r} dx$

In this type of integral, the numerator ${a{x^2} + bx + c}$ is written in terms of the derivative of denominator $p{x^2} + qx + r$ as $$a{x^2} + bx + c = \lambda (p{x^2} + qx + r) + \mu \left\{ {\frac{d}{{dx}}\left( {p{x^2} + qx + r} \right)} \right\} + \gamma $$

where $\lambda ,\mu ,\gamma $ are constants and their values are found out by comparing the coefficients of like powers.

After finding the values, the given integration reduces to the integration of three independent functions which can be solved either using standard results or by substitution.