\[ \int (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx \]
Optimal antiderivative \[ -\frac {\left (-a h +b g \right ) p r x}{2 b}-\frac {\left (-c h +d g \right ) q r x}{2 d}-\frac {p r \left (h x +g \right )^{2}}{4 h}-\frac {q r \left (h x +g \right )^{2}}{4 h}-\frac {\left (-a h +b g \right )^{2} p r \ln \left (b x +a \right )}{2 b^{2} h}-\frac {\left (-c h +d g \right )^{2} q r \ln \left (d x +c \right )}{2 d^{2} h}+\frac {\left (h x +g \right )^{2} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{2 h} \]
command
integrate((h*x+g)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{4} \, {\left (h p r + h q r - 2 \, h r \log \left (f\right ) - 2 \, h\right )} x^{2} + \frac {1}{2} \, {\left (h p r x^{2} + 2 \, g p r x\right )} \log \left (b x + a\right ) + \frac {1}{2} \, {\left (h q r x^{2} + 2 \, g q r x\right )} \log \left (d x + c\right ) - \frac {{\left (2 \, b d g p r - a d h p r + 2 \, b d g q r - b c h q r - 2 \, b d g r \log \left (f\right ) - 2 \, b d g\right )} x}{2 \, b d} + \frac {{\left (2 \, a b g p r - a^{2} h p r\right )} \log \left (-b x - a\right )}{2 \, b^{2}} + \frac {{\left (2 \, c d g q r - c^{2} h q r\right )} \log \left (d x + c\right )}{2 \, d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________