\[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {e \arctanh \! \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{d^{5}}+\frac {-3 e x +4 d}{3 d^{4} x \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {1}{3 d^{2} x \left (e x +d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {8 \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{5} x} \]
command
integrate(1/x^2/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ +\infty \]