\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ -\frac {4 b n \left (e x +d \right )^{\frac {3}{2}}}{9 e^{2}}-\frac {8 b \,d^{\frac {3}{2}} n \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{3 e^{2}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 e^{2}}+\frac {8 b d n \sqrt {e x +d}}{3 e^{2}}-\frac {2 d \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e x +d}}{e^{2}} \]
command
integrate(x*(a+b*ln(c*x**n))/(e*x+d)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {- \frac {2 a d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 a \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {2 b d \left (- d \left (\frac {\log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{\sqrt {d + e x}} - \frac {2 n \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{d \sqrt {- \frac {1}{d}}}\right ) - \sqrt {d + e x} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )} - \frac {2 n \left (- e \sqrt {d + e x} - \frac {e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}}\right )}{e}\right )}{e} - \frac {2 b \left (d^{2} \left (\frac {\log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{\sqrt {d + e x}} - \frac {2 n \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{d \sqrt {- \frac {1}{d}}}\right ) - 2 d \left (- \sqrt {d + e x} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )} - \frac {2 n \left (- e \sqrt {d + e x} - \frac {e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}}\right )}{e}\right ) - \frac {\left (d + e x\right )^{\frac {3}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{3} - \frac {2 n \left (- d e \sqrt {d + e x} - \frac {d e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}} - \frac {e \left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{3 e}\right )}{e}}{e} & \text {for}\: e \neq 0 \\\frac {\frac {a x^{2}}{2} + b \left (- \frac {n x^{2}}{4} + \frac {x^{2} \log {\left (c x^{n} \right )}}{2}\right )}{\sqrt {d}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________