\[ \int \frac {d+e x}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {e x +d}{5 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e x +7 d}{15 d^{4} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {7 e^{2} \arctanh \! \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{2 d^{8}}+\frac {24 e x +35 d}{15 d^{6} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {7 \sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{7} x^{2}}-\frac {16 e \sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{8} x} \]
command
integrate((e*x+d)/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left (3 \, {\left (x {\left (\frac {11 \, x e^{7}}{d^{8}} + \frac {15 \, e^{6}}{d^{7}}\right )} - \frac {25 \, e^{5}}{d^{6}}\right )} x - \frac {100 \, e^{4}}{d^{5}}\right )} x + \frac {45 \, e^{3}}{d^{4}}\right )} x + \frac {58 \, e^{2}}{d^{3}}\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac {7 \, e^{2} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{2 \, d^{8}} + \frac {x^{2} {\left (\frac {4 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{4}}{x} + e^{6}\right )}}{8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{8}} - \frac {{\left (\frac {4 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{8} e^{8}}{x} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{8} e^{6}}{x^{2}}\right )} e^{\left (-8\right )}}{8 \, d^{16}} \]