\[ \int (f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {B \left (-a d +b c \right ) g \left (a^{3} d^{3} g^{3}-a^{2} b \,d^{2} g^{2} \left (-c g +5 d f \right )+a \,b^{2} d g \left (c^{2} g^{2}-5 c d f g +10 d^{2} f^{2}\right )-b^{3} \left (-c^{3} g^{3}+5 c^{2} d f \,g^{2}-10 c \,d^{2} f^{2} g +10 d^{3} f^{3}\right )\right ) x}{5 b^{4} d^{4}}-\frac {B \left (-a d +b c \right ) g^{2} \left (a^{2} d^{2} g^{2}-a b d g \left (-c g +5 d f \right )+b^{2} \left (c^{2} g^{2}-5 c d f g +10 d^{2} f^{2}\right )\right ) x^{2}}{10 b^{3} d^{3}}-\frac {B \left (-a d +b c \right ) g^{3} \left (-a d g -b c g +5 b d f \right ) x^{3}}{15 b^{2} d^{2}}-\frac {B \left (-a d +b c \right ) g^{4} x^{4}}{20 b d}-\frac {B \left (-a g +b f \right )^{5} \ln \left (b x +a \right )}{5 b^{5} g}+\frac {\left (g x +f \right )^{5} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{5 g}+\frac {B \left (-c g +d f \right )^{5} \ln \left (d x +c \right )}{5 d^{5} g} \]
command
integrate((g*x+f)^4*(A+B*log(e*(b*x+a)/(d*x+c))),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 \[ \text {Timed out} \]_______________________________________________________