\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{24 d \,x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 d \,x^{6}}+\frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 d^{2} x^{5}}-\frac {e^{6} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{16 d^{2}}+\frac {e^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{16 d \,x^{2}} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^7/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {x^{6} {\left (\frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{4}}{x} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{2}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-2\right )}}{x^{4}} + \frac {120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-4\right )}}{x^{5}} - \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3}}{x^{3}} - 5 \, e^{6}\right )} e^{12}}{1920 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{2}} - \frac {e^{6} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{16 \, d^{2}} + \frac {\frac {120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{10} e^{4}}{x} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{10} e^{2}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{10} e^{\left (-2\right )}}{x^{4}} + \frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{10} e^{\left (-4\right )}}{x^{5}} - \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{10} e^{\left (-6\right )}}{x^{6}} - \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{10}}{x^{3}}}{1920 \, d^{12}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________