\[ \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^9} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \! \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{8 x^{8}}+\frac {3 e^{\frac {3}{2}} \sqrt {e \,x^{2}+d}}{140 d^{2} x^{5}}-\frac {e^{\frac {5}{2}} \sqrt {e \,x^{2}+d}}{35 d^{3} x^{3}}+\frac {2 e^{\frac {7}{2}} \sqrt {e \,x^{2}+d}}{35 d^{4} x}-\frac {\sqrt {e}\, \sqrt {e \,x^{2}+d}}{56 d \,x^{7}} \]
command
integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^9,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Timed out} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {\log \left (-\frac {\frac {x e^{\frac {1}{2}}}{\sqrt {x^{2} e + d}} + 1}{\frac {x e^{\frac {1}{2}}}{\sqrt {x^{2} e + d}} - 1}\right )}{16 \, x^{8}} + \frac {4 \, {\left (35 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{6} d^{3} e^{3} - 21 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{4} d^{4} e^{3} + 7 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} d^{5} e^{3} - d^{6} e^{3}\right )} e}{35 \, {\left ({\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} - d\right )}^{7} d^{3}} \]