\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ \frac {28 b d n \left (e x +d \right )^{\frac {3}{2}}}{45 e^{3}}-\frac {4 b n \left (e x +d \right )^{\frac {5}{2}}}{25 e^{3}}+\frac {32 b \,d^{\frac {5}{2}} n \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{15 e^{3}}-\frac {4 d \left (e x +d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 e^{3}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{5 e^{3}}-\frac {32 b \,d^{2} n \sqrt {e x +d}}{15 e^{3}}+\frac {2 d^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e x +d}}{e^{3}} \]
command
integrate(x**2*(a+b*ln(c*x**n))/(e*x+d)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {- \frac {2 a d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {2 a \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {2 b d \left (d^{2} \left (\frac {\log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{\sqrt {d + e x}} - \frac {2 n \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{d \sqrt {- \frac {1}{d}}}\right ) - 2 d \left (- \sqrt {d + e x} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )} - \frac {2 n \left (- e \sqrt {d + e x} - \frac {e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}}\right )}{e}\right ) - \frac {\left (d + e x\right )^{\frac {3}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{3} - \frac {2 n \left (- d e \sqrt {d + e x} - \frac {d e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}} - \frac {e \left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{3 e}\right )}{e^{2}} - \frac {2 b \left (- d^{3} \left (\frac {\log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{\sqrt {d + e x}} - \frac {2 n \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{d \sqrt {- \frac {1}{d}}}\right ) + 3 d^{2} \left (- \sqrt {d + e x} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )} - \frac {2 n \left (- e \sqrt {d + e x} - \frac {e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}}\right )}{e}\right ) - 3 d \left (- \frac {\left (d + e x\right )^{\frac {3}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{3} - \frac {2 n \left (- d e \sqrt {d + e x} - \frac {d e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}} - \frac {e \left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{3 e}\right ) - \frac {\left (d + e x\right )^{\frac {5}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{5} - \frac {2 n \left (- d^{2} e \sqrt {d + e x} - \frac {d^{2} e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}} - \frac {d e \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {e \left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{5 e}\right )}{e^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {\frac {a x^{3}}{3} + b \left (- \frac {n x^{3}}{9} + \frac {x^{3} \log {\left (c x^{n} \right )}}{3}\right )}{\sqrt {d}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________