\[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {b e n}{6 d \,x^{\frac {3}{2}}}+\frac {b \,e^{2} n}{4 d^{2} x}-\frac {b \,e^{4} n \ln \left (x \right )}{4 d^{4}}+\frac {b \,e^{4} n \ln \left (d +e \sqrt {x}\right )}{2 d^{4}}+\frac {-a -b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )}{2 x^{2}}-\frac {b \,e^{3} n}{2 d^{3} \sqrt {x}} \]
command
integrate((a+b*ln(c*(d+e*x**(1/2))**n))/x**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {6 a d^{5} \sqrt {x}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {6 a d^{4} e x}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {6 b d^{5} \sqrt {x} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {2 b d^{4} e n x}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {6 b d^{4} e x \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} + \frac {b d^{3} e^{2} n x^{\frac {3}{2}}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {3 b d^{2} e^{3} n x^{2}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {3 b d e^{4} n x^{\frac {5}{2}} \log {\left (x \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {6 b d e^{4} n x^{\frac {5}{2}}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} + \frac {6 b d e^{4} x^{\frac {5}{2}} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} - \frac {3 b e^{5} n x^{3} \log {\left (x \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} + \frac {6 b e^{5} x^{3} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{12 d^{5} x^{\frac {5}{2}} + 12 d^{4} e x^{3}} & \text {for}\: d \neq 0 \\- \frac {a}{2 x^{2}} - \frac {b n}{8 x^{2}} - \frac {b \log {\left (c \left (e \sqrt {x}\right )^{n} \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________