\[ \int \frac {1}{x (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {-\frac {4 e x}{5}+\frac {4 d}{5}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {-11 e x +5 d}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{d^{4}}+\frac {-22 e x +15 d}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(1/x/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{d^{4}} - \frac {2 \, {\left (\frac {115 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {185 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {135 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + 32\right )}}{15 \, d^{4} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________