\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)} \, dx \]
Optimal antiderivative \[ \frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 d \,x^{5}}+\frac {3 e^{5} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{8 d}-\frac {3 e^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{8 x^{2}} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^6/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {x^{5} {\left (\frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{3}}{x} + \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e}{x^{2}} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-1\right )}}{x^{3}} - \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-3\right )}}{x^{4}} - 2 \, e^{5}\right )} e^{10}}{320 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d} + \frac {3 \, e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{8 \, d} - \frac {\frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{3}}{x} + \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e}{x^{2}} - \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{\left (-1\right )}}{x^{3}} - \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} e^{\left (-3\right )}}{x^{4}} + \frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{4} e^{\left (-5\right )}}{x^{5}}}{320 \, d^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________