\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^9 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{64 d^{3} x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 d \,x^{8}}+\frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 d^{2} x^{7}}-\frac {e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{16 d^{3} x^{6}}+\frac {2 e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 d^{4} x^{5}}-\frac {3 e^{8} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{128 d^{4}}+\frac {3 e^{6} \sqrt {-e^{2} x^{2}+d^{2}}}{128 d^{3} x^{2}} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^9/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {x^{8} {\left (\frac {80 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{6}}{x} - \frac {112 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{2}}{x^{3}} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-2\right )}}{x^{5}} + \frac {1680 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-6\right )}}{x^{7}} + \frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4}}{x^{4}} - 35 \, e^{8}\right )} e^{16}}{71680 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{4}} - \frac {3 \, e^{8} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{128 \, d^{4}} + \frac {\frac {1680 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{28} e^{6}}{x} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{28} e^{2}}{x^{3}} - \frac {112 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{28} e^{\left (-2\right )}}{x^{5}} + \frac {80 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{28} e^{\left (-6\right )}}{x^{7}} - \frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{28} e^{\left (-8\right )}}{x^{8}} + \frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{28}}{x^{4}}}{71680 \, d^{32}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________