\[ \int \frac {(d+e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 e \left (e x +d \right )}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {e \left (13 e x +10 d \right )}{15 d^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 e \arctanh \! \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{d^{6}}+\frac {e \left (41 e x +30 d \right )}{15 d^{6} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{6} x} \]
command
integrate((e*x+d)^2/x^2/(-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
\[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left ({\left (x {\left (\frac {41 \, x e^{6}}{d^{6}} + \frac {30 \, e^{5}}{d^{5}}\right )} - \frac {95 \, e^{4}}{d^{4}}\right )} x - \frac {70 \, e^{3}}{d^{3}}\right )} x + \frac {60 \, e^{2}}{d^{2}}\right )} x + \frac {46 \, e}{d}\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac {2 \, e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{d^{6}} + \frac {x e^{3}}{2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{6}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, d^{6} x} \]