\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx \]
Optimal antiderivative \[ -\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{9 d e \left (e x +d \right )^{9}} \]
command
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^9,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (\frac {36 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {126 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {84 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-12\right )}}{x^{6}} + \frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-16\right )}}{x^{8}} + 1\right )} e^{\left (-1\right )}}{9 \, d {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{9}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________