Evaluate the following integral : $ \int \frac{3 z^{3}-8 z+5}{\sqrt{z^{2}-4 z-7}} d z $

Question

Question: Find the value of the following integral $$ \int \frac{3 z^{3}-8 z+5}{\sqrt{z^{2}-4 z-7}} d z $$

My approach: I factorised the numerator as $(z-1)(3z^{2}+3z-5)$, then I substituted the function in denominator under the roots as $u$, but this only complicated things further, please help

Answer 1

hint :

derivative of the denominator $z^2-4z+7$ is equal to $2z-4$

now you can write $$3z^3-8z+5=l(z^2-4z-7)(2z-4) +k(z^2-4z+7)+m(2z-4)+n$$ by comparing coefficient.