\[ \int \frac {1}{\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {a \,f^{2} \ln \left (e x +f \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )}{d^{3} e}+\frac {a \,f^{2} \ln \left (d +e x +f \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )}{d^{3} e}-\frac {a \,f^{2}}{2 d^{2} e \left (e x +f \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )}+\frac {-1-\frac {a \,f^{2}}{d^{2}}}{2 e \left (d +e x +f \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )} \]
command
integrate(1/(d+e*x+f*(a+e^2*x^2/f^2)^(1/2))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {a f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | a f^{2} - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} d \right |}\right )}{2 \, d^{3}} + \frac {a f^{2} e^{\left (-1\right )} \log \left ({\left | -a f^{2} + 2 \, d x e + d^{2} \right |}\right )}{2 \, d^{3}} - \frac {a f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | -x e - d + \sqrt {a f^{2} + x^{2} e^{2}} \right |}\right )}{2 \, d^{3}} + \frac {a f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | -x e + \sqrt {a f^{2} + x^{2} e^{2}} \right |}\right )}{2 \, d^{3}} - \frac {\sqrt {a f^{2} + x^{2} e^{2}} {\left | f \right |} e^{\left (-1\right )}}{2 \, d^{2} f} + \frac {x}{2 \, d^{2}} + \frac {{\left (a^{2} f^{4} + 2 \, a d^{2} f^{2} + d^{4}\right )} e^{\left (-1\right )}}{4 \, {\left (a f^{2} - 2 \, d x e - d^{2}\right )} d^{3}} + \frac {{\left ({\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} a^{2} f^{4} {\left | f \right |} + 2 \, a^{2} d f^{4} {\left | f \right |} + 2 \, a d^{3} f^{2} {\left | f \right |} - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} d^{4} {\left | f \right |}\right )} e^{\left (-1\right )}}{4 \, {\left ({\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} a f^{2} + a d f^{2} - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )}^{2} d - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} d^{2}\right )} d^{3} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________