\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {3 e^{4} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{8 d^{4}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{4 d \,x^{4}}+\frac {e \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{2} x^{3}}-\frac {3 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{8 d^{3} x^{2}}+\frac {2 e^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{4} x} \]
command
integrate((-e^2*x^2+d^2)^(1/2)/x^5/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {x^{4} {\left (\frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{2}}{x} + \frac {72 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-2\right )}}{x^{3}} - \frac {24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2}}{x^{2}} - 3 \, e^{4}\right )} e^{8}}{192 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4}} - \frac {3 \, e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{8 \, d^{4}} + \frac {\frac {72 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{12} e^{2}}{x} + \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{12} e^{\left (-2\right )}}{x^{3}} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{12} e^{\left (-4\right )}}{x^{4}} - \frac {24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{12}}{x^{2}}}{192 \, d^{16}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________