\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^5} \, dx \]
Optimal antiderivative \[ -\frac {B \left (-a d +b c \right )}{12 \left (-a g +b f \right ) \left (-c g +d f \right ) \left (g x +f \right )^{3}}-\frac {B \left (-a d +b c \right ) \left (-a d g -b c g +2 b d f \right )}{8 \left (-a g +b f \right )^{2} \left (-c g +d f \right )^{2} \left (g x +f \right )^{2}}-\frac {B \left (-a d +b c \right ) \left (a^{2} d^{2} g^{2}-a b d g \left (-c g +3 d f \right )+b^{2} \left (c^{2} g^{2}-3 c d f g +3 d^{2} f^{2}\right )\right )}{4 \left (-a g +b f \right )^{3} \left (-c g +d f \right )^{3} \left (g x +f \right )}+\frac {b^{4} B \ln \left (b x +a \right )}{4 g \left (-a g +b f \right )^{4}}+\frac {-A -B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{4 g \left (g x +f \right )^{4}}-\frac {B \,d^{4} \ln \left (d x +c \right )}{4 g \left (-c g +d f \right )^{4}}-\frac {B \left (-a d +b c \right ) \left (-a d g -b c g +2 b d f \right ) \left (2 a b \,d^{2} f g -a^{2} d^{2} g^{2}-b^{2} \left (c^{2} g^{2}-2 c d f g +2 d^{2} f^{2}\right )\right ) \ln \left (g x +f \right )}{4 \left (-a g +b f \right )^{4} \left (-c g +d f \right )^{4}} \]
command
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(g*x+f)^5,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 \[ \text {Timed out} \]_______________________________________________________