\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {15 e^{2} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{2 d}+\frac {8 e^{2} \left (-e x +d \right )}{d \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 x^{2}}+\frac {4 e \sqrt {-e^{2} x^{2}+d^{2}}}{d x} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^3/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {15 \, e^{2} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{2 \, d} - \frac {x^{2} {\left (\frac {144 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-2\right )}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}}{x} - e^{2}\right )} e^{4}}{8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}} - \frac {\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{\left (-2\right )}}{x^{2}} - \frac {16 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d}{x}}{8 \, d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________