\[ \int \frac {(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 e^{3} \left (e x +d \right )}{5 d^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {e^{3} \left (23 e x +20 d \right )}{15 d^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {7 e^{3} \arctanh \! \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{d^{8}}+\frac {2 e^{3} \left (53 e x +45 d \right )}{15 d^{8} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{6} x^{3}}-\frac {e \sqrt {-e^{2} x^{2}+d^{2}}}{d^{7} x^{2}}-\frac {14 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{8} x} \]
command
integrate((e*x+d)^2/x^4/(-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
\[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left ({\left (2 \, x {\left (\frac {53 \, x e^{8}}{d^{8}} + \frac {45 \, e^{7}}{d^{7}}\right )} - \frac {235 \, e^{6}}{d^{6}}\right )} x - \frac {200 \, e^{5}}{d^{5}}\right )} x + \frac {135 \, e^{4}}{d^{4}}\right )} x + \frac {116 \, e^{3}}{d^{3}}\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} + \frac {x^{3} {\left (\frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{6}}{x} + \frac {57 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{4}}{x^{2}} + e^{8}\right )} e}{24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{8}} - \frac {7 \, e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{d^{8}} - \frac {{\left (\frac {57 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{16} e^{16}}{x} + \frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{16} e^{14}}{x^{2}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{16} e^{12}}{x^{3}}\right )} e^{\left (-15\right )}}{24 \, d^{24}} \]