\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{13}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{17 d e \left (e x +d \right )^{13}}-\frac {4 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{255 d^{2} e \left (e x +d \right )^{12}}-\frac {4 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{1105 d^{3} e \left (e x +d \right )^{11}}-\frac {8 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{12155 d^{4} e \left (e x +d \right )^{10}}-\frac {8 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{109395 d^{5} e \left (e x +d \right )^{9}} \]
command
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^13,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (\frac {37264 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {735692 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {2511580 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {11197220 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {24999520 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-10\right )}}{x^{5}} + \frac {55979300 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-12\right )}}{x^{6}} + \frac {80466100 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-14\right )}}{x^{7}} + \frac {108787250 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-16\right )}}{x^{8}} + \frac {100935120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} e^{\left (-18\right )}}{x^{9}} + \frac {87311796 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} e^{\left (-20\right )}}{x^{10}} + \frac {50788452 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{11} e^{\left (-22\right )}}{x^{11}} + \frac {28384356 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{12} e^{\left (-24\right )}}{x^{12}} + \frac {9335040 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{13} e^{\left (-26\right )}}{x^{13}} + \frac {3354780 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{14} e^{\left (-28\right )}}{x^{14}} + \frac {437580 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{15} e^{\left (-30\right )}}{x^{15}} + \frac {109395 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{16} e^{\left (-32\right )}}{x^{16}} + 8627\right )} e^{\left (-1\right )}}{109395 \, d^{5} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{17}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________