\[ \int \frac {1}{(d+e x)^4 \sqrt {d^2-e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{7 d e \left (e x +d \right )^{4}}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}}{35 d^{2} e \left (e x +d \right )^{3}}-\frac {2 \sqrt {-e^{2} x^{2}+d^{2}}}{35 d^{3} e \left (e x +d \right )^{2}}-\frac {2 \sqrt {-e^{2} x^{2}+d^{2}}}{35 d^{4} e \left (e x +d \right )} \]
command
integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (\frac {49 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {147 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {210 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {210 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-10\right )}}{x^{5}} + \frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-12\right )}}{x^{6}} + 12\right )} e^{\left (-1\right )}}{35 \, d^{4} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{7}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________