\[ \int (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx \]
Optimal antiderivative \[ -\frac {\left (-a d +b c \right )^{2} q r x}{3 d^{2}}+\frac {\left (-a d +b c \right ) q r \left (b x +a \right )^{2}}{6 b d}-\frac {p r \left (b x +a \right )^{3}}{9 b}-\frac {q r \left (b x +a \right )^{3}}{9 b}+\frac {\left (-a d +b c \right )^{3} q r \ln \left (d x +c \right )}{3 b \,d^{3}}+\frac {\left (b x +a \right )^{3} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{3 b} \]
command
integrate((b*x+a)**2*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} a^{2} x \log {\left (e \left (a^{p} c^{q} f\right )^{r} \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {a^{3} \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )}}{3 b} - \frac {a^{2} p r x}{3} + a^{2} x \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )} - \frac {a b p r x^{2}}{3} + a b x^{2} \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )} - \frac {b^{2} p r x^{3}}{9} + \frac {b^{2} x^{3} \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )}}{3} & \text {for}\: d = 0 \\a^{2} \left (\frac {c \log {\left (e \left (a^{p} f \left (c + d x\right )^{q}\right )^{r} \right )}}{d} - q r x + x \log {\left (e \left (a^{p} f \left (c + d x\right )^{q}\right )^{r} \right )}\right ) & \text {for}\: b = 0 \\- \frac {a^{3} q r \log {\left (\frac {c}{d} + x \right )}}{3 b} + \frac {a^{3} \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{3 b} + \frac {a^{2} c q r \log {\left (\frac {c}{d} + x \right )}}{d} - \frac {a^{2} p r x}{3} - a^{2} q r x + a^{2} x \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )} - \frac {a b c^{2} q r \log {\left (\frac {c}{d} + x \right )}}{d^{2}} + \frac {a b c q r x}{d} - \frac {a b p r x^{2}}{3} - \frac {a b q r x^{2}}{2} + a b x^{2} \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )} + \frac {b^{2} c^{3} q r \log {\left (\frac {c}{d} + x \right )}}{3 d^{3}} - \frac {b^{2} c^{2} q r x}{3 d^{2}} + \frac {b^{2} c q r x^{2}}{6 d} - \frac {b^{2} p r x^{3}}{9} - \frac {b^{2} q r x^{3}}{9} + \frac {b^{2} x^{3} \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{3} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________