\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx \]
Optimal antiderivative \[ \frac {e^{3} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{2 d^{3}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 d \,x^{3}}+\frac {e \sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {2 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{3} x} \]
command
integrate((-e^2*x^2+d^2)^(1/2)/x^4/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {x^{3} {\left (\frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e}{x} - \frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-1\right )}}{x^{2}} - e^{3}\right )} e^{6}}{24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3}} + \frac {e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{2 \, d^{3}} - \frac {\frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{6} e}{x} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{6} e^{\left (-1\right )}}{x^{2}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{6} e^{\left (-3\right )}}{x^{3}}}{24 \, d^{9}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________