\[ \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx \]
Optimal antiderivative \[ \frac {\left (g x +f \right ) \ln \left (c \left (d +\frac {e}{\left (g x +f \right )^{3}}\right )^{q}\right )}{g}+\frac {e^{\frac {1}{3}} q \ln \left (e^{\frac {1}{3}}+d^{\frac {1}{3}} \left (g x +f \right )\right )}{d^{\frac {1}{3}} g}-\frac {e^{\frac {1}{3}} q \ln \left (e^{\frac {2}{3}}-d^{\frac {1}{3}} e^{\frac {1}{3}} \left (g x +f \right )+d^{\frac {2}{3}} \left (g x +f \right )^{2}\right )}{2 d^{\frac {1}{3}} g}-\frac {e^{\frac {1}{3}} q \arctan \left (\frac {\left (e^{\frac {1}{3}}-2 d^{\frac {1}{3}} \left (g x +f \right )\right ) \sqrt {3}}{3 e^{\frac {1}{3}}}\right ) \sqrt {3}}{d^{\frac {1}{3}} g} \]
command
integrate(log(c*(d+e/(g*x+f)^3)^q),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{2} \, d g^{5} q {\left (\frac {2 \, f e^{\left (-1\right )} \log \left ({\left | d g^{3} x^{3} + 3 \, d f g^{2} x^{2} + 3 \, d f^{2} g x + d f^{3} + e \right |}\right )}{d g^{6}} - \frac {6 \, f e^{\left (-1\right )} \log \left ({\left | g x + f \right |}\right )}{d g^{6}} + \frac {{\left (2 \, \sqrt {3} \left (d^{5} g^{21}\right )^{\frac {1}{3}} \arctan \left (-\frac {d g x + d f + {\left (d^{2}\right )}^{\frac {1}{3}} e^{\frac {1}{3}}}{\sqrt {3} d g x + \sqrt {3} d f - \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{3}} e^{\frac {1}{3}}}\right ) e^{\frac {4}{3}} - \left (d^{5} g^{21}\right )^{\frac {1}{3}} e^{\frac {4}{3}} \log \left (4 \, {\left (\sqrt {3} d g x + \sqrt {3} d f - \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{3}} e^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (d g x + d f + {\left (d^{2}\right )}^{\frac {1}{3}} e^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (d^{5} g^{21}\right )^{\frac {1}{3}} e^{\frac {4}{3}} \log \left ({\left | d g x + d f + {\left (d^{2}\right )}^{\frac {1}{3}} e^{\frac {1}{3}} \right |}\right )\right )} e^{\left (-2\right )}}{d^{3} g^{13}}\right )} e + q x \log \left (d g^{3} x^{3} + 3 \, d f g^{2} x^{2} + 3 \, d f^{2} g x + d f^{3} + e\right ) - q x \log \left (g^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}\right ) + x \log \left (c\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________