\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e \left (e x +d \right )^{3}}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 e \left (e x +d \right )^{5}}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e \left (e x +d \right )^{7}}+\frac {\arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e}+\frac {2 \sqrt {-e^{2} x^{2}+d^{2}}}{e \left (e x +d \right )} \]
command
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) - \frac {16 \, {\left (\frac {133 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {294 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {490 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {175 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-10\right )}}{x^{5}} + 19\right )} e^{\left (-1\right )}}{105 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{7}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________