<s>[INST] Evaluate $\int\frac{8x+20}{5x^2+25x+20}dx$ I tried to solve it and got $\frac{4}{5} \ln(4+5 x+x^2)+C$ as an answer, but my online homework program says it's incorrect.  What did I do wrong?
I pulled out $\frac{4}{5}$ as a constant and saw that the numerator was the derivative of the denominator. So I put the denominator in a natural log.
 [/INST] Here is a general method for integrals of this form. 
$$ I = \int \frac{mx±n}{ax^2+bx+d}dx$$  
$$I =   \int \frac{\frac{m}{2a}(2ax+b) + (n - \frac{mb}{2a})}{(x+ \frac{b}{2a})^2 +  (c -\frac{b^2}{4a^2})}dx  = \frac{m}{2a}ln|ax^2 + bx + c| + (n- \frac{mb}{2a}) \int \frac{dx}{ax^2+bx+c}dx$$
depending on the quadratic in the bottom, the second integral will be in the form of either $\frac{1}{a}\arctan(\frac{u}{a})$ + C or $ \frac{1}{2a}ln|\frac{u-a}{u+a}| + C $
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