I'm reading Differential and Integral Calculus by Piskunov, and have read up to integration using substitution. I've learned how to solve integrals of the form: $$\int{\dfrac{Ax+B}{ax^2+bx+c}}dx$$ The integrals of the above form are solved by completing the square in the denominator, and decomposing the fraction into a sum such that the first fraction of the sum is of the form: $$\int{\dfrac{2ax+b}{ax^2+bx+c}}dx$$ and the second fraction in the sum is of the form: $$k\int{\dfrac{1}{ax^2+bx+c}}dx$$ By substituion, the above two terms are readily integrable.
I tried applying similar techniques to the following integral: $$\int{\dfrac{6x^4-5x^3+4x^2}{2x^2-x+1}}dx$$ But have not been able to simplify them enough. The numerator is always two degrees higher than the denominator, and the $x^2$ term doesn't yield to the substitutions that worked in the simpler forms I wrote about above.
This is what I got after some simplification: $$\dfrac{1}{4}\int{\dfrac{\left(3t^2+t-2\right)^2+71\left(t+1\right)^2}{\left(t^2+7\right)}}dt$$
$t$ is $4x-1$.
How can I solve the integral given in the question using only integration by substitution? In particular, can't use integration by parts.
