\[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^2} \, dx \]
Optimal antiderivative \[ \frac {b n \ln \left (x \right )}{a}-\frac {b n \ln \left (c \,x^{2}+b x +a \right )}{2 a}-\frac {\ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right )}{x}+\frac {n \arctanh \left (\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}\right ) \sqrt {-4 a c +b^{2}}}{a} \]
command
integrate(ln(d*(c*x**2+b*x+a)**n)/x**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {n}{x} - \frac {\log {\left (d \left (b x\right )^{n} \right )}}{x} & \text {for}\: a = 0 \wedge c = 0 \\- \frac {n}{x} - \frac {\log {\left (d \left (b x + c x^{2}\right )^{n} \right )}}{x} - \frac {2 c n \log {\left (b + c x \right )}}{b} + \frac {c \log {\left (d \left (b x + c x^{2}\right )^{n} \right )}}{b} & \text {for}\: a = 0 \\- \frac {\log {\left (d \left (a + b x\right )^{n} \right )}}{x} + \frac {b n \log {\left (x \right )}}{a} - \frac {b \log {\left (d \left (a + b x\right )^{n} \right )}}{a} & \text {for}\: c = 0 \\- \frac {\log {\left (d \left (a + b x + c x^{2}\right )^{n} \right )}}{x} + \frac {b n \log {\left (x \right )}}{a} - \frac {b \log {\left (d \left (a + b x + c x^{2}\right )^{n} \right )}}{2 a} + \frac {n \sqrt {- 4 a c + b^{2}} \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{a} - \frac {\sqrt {- 4 a c + b^{2}} \log {\left (d \left (a + b x + c x^{2}\right )^{n} \right )}}{2 a} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________