\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {95 e^{4} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{8 d^{3}}+\frac {8 e^{4} \left (-e x +d \right )}{d^{3} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{4 x^{4}}+\frac {4 e \sqrt {-e^{2} x^{2}+d^{2}}}{3 d \,x^{3}}-\frac {31 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{8 d^{2} x^{2}}+\frac {32 e^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{3} x} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^5/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {95 \, e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{8 \, d^{3}} - \frac {x^{4} {\left (\frac {29 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{2}}{x} + \frac {864 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-2\right )}}{x^{3}} + \frac {4128 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-4\right )}}{x^{4}} - \frac {160 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2}}{x^{2}} - 3 \, e^{4}\right )} e^{8}}{192 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}} + \frac {\frac {1056 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{9} e^{2}}{x} + \frac {32 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{9} e^{\left (-2\right )}}{x^{3}} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{9} e^{\left (-4\right )}}{x^{4}} - \frac {192 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{9}}{x^{2}}}{192 \, d^{12}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________