\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx \]
Optimal antiderivative \[ \frac {10 e^{3} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{d^{2}}-\frac {8 e^{3} \left (-e x +d \right )}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 x^{3}}+\frac {2 e \sqrt {-e^{2} x^{2}+d^{2}}}{d \,x^{2}}-\frac {23 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{2} x} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^4/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {10 \, e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{d^{2}} - \frac {x^{3} {\left (\frac {11 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e}{x} - \frac {81 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-1\right )}}{x^{2}} - \frac {477 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-3\right )}}{x^{3}} - e^{3}\right )} e^{6}}{24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}} - \frac {\frac {93 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e}{x} - \frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e^{\left (-1\right )}}{x^{2}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{\left (-3\right )}}{x^{3}}}{24 \, d^{6}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________