\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2} \, dx \]
Optimal antiderivative \[ -\frac {b^{\frac {1}{3}} p \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{a^{\frac {1}{3}}}+\frac {b^{\frac {1}{3}} p \ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{2 a^{\frac {1}{3}}}-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{x}-\frac {b^{\frac {1}{3}} p \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{a^{\frac {1}{3}}} \]
command
integrate(ln(c*(b*x**3+a)**p)/x**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {\log {\left (0^{p} c \right )}}{x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3 p}{x} - \frac {\log {\left (c \left (b x^{3}\right )^{p} \right )}}{x} & \text {for}\: a = 0 \\- \frac {\log {\left (a^{p} c \right )}}{x} & \text {for}\: b = 0 \\- \frac {\log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{x} + \frac {3 b p \left (- \frac {a}{b}\right )^{\frac {2}{3}} \log {\left (4 x^{2} + 4 x \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 a} - \frac {\sqrt {3} b p \left (- \frac {a}{b}\right )^{\frac {2}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{a} - \frac {b \left (- \frac {a}{b}\right )^{\frac {2}{3}} \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{a} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________