\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx \]
Optimal antiderivative \[ \frac {27 e^{5} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{2 d^{4}}-\frac {8 e^{5} \left (-e x +d \right )}{d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{5 x^{5}}+\frac {e \sqrt {-e^{2} x^{2}+d^{2}}}{d \,x^{4}}-\frac {13 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{2} x^{3}}+\frac {11 e^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{3} x^{2}}-\frac {66 e^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{4} x} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^6/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {27 \, e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{2 \, d^{4}} - \frac {x^{5} {\left (\frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{3}}{x} - \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e}{x^{2}} + \frac {185 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-1\right )}}{x^{3}} - \frac {870 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-3\right )}}{x^{4}} - \frac {3670 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-5\right )}}{x^{5}} - e^{5}\right )} e^{10}}{160 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{4} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}} - \frac {\frac {1110 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{16} e^{3}}{x} - \frac {240 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{16} e}{x^{2}} + \frac {55 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{16} e^{\left (-1\right )}}{x^{3}} - \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{16} e^{\left (-3\right )}}{x^{4}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{16} e^{\left (-5\right )}}{x^{5}}}{160 \, d^{20}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________