\[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{a g+b g x} \, dx \]
Optimal antiderivative \[ -\frac {\ln \left (\frac {a d -b c}{d \left (b x +a \right )}\right ) \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )}{b g}-\frac {B \polylog \left (2, 1+\frac {-a d +b c}{d \left (b x +a \right )}\right )}{b g} \]
command
integrate((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________