\[ \int \frac {1}{d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a e \,f^{2}-b d \,f^{2}+d^{2} e \right ) \ln \left (d +e x +f \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )}{\left (-b \,f^{2}+2 d e \right )^{2}}-\frac {f^{2} \left (-b^{2} f^{2}+4 a \,e^{2}\right ) \ln \left (b \,f^{2}+2 e \left (e x +f \sqrt {a +\frac {x \left (b \,f^{2}+e^{2} x \right )}{f^{2}}}\right )\right )}{2 e \left (-b \,f^{2}+2 d e \right )^{2}}-\frac {f^{2} \left (-b^{2} f^{2}+4 a \,e^{2}\right )}{2 e \left (-b \,f^{2}+2 d e \right ) \left (b \,f^{2}+2 e \left (e x +f \sqrt {a +\frac {x \left (b \,f^{2}+e^{2} x \right )}{f^{2}}}\right )\right )} \]
command
integrate(1/(d+e*x+f*(a+b*x+e^2*x^2/f^2)^(1/2)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {x e}{b f^{2} - 2 \, d e} - \frac {{\left (b d f^{2} {\left | f \right |} - a f^{2} {\left | f \right |} e - d^{2} {\left | f \right |} e\right )} \log \left ({\left | -{\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} b f^{2} + b d f^{2} - 2 \, a f^{2} e + 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} d e \right |}\right )}{b^{2} f^{5} - 4 \, b d f^{3} e + 4 \, d^{2} f e^{2}} - \frac {{\left (b d f^{2} - a f^{2} e - d^{2} e\right )} \log \left ({\left | -b f^{2} x - a f^{2} + 2 \, d x e + d^{2} \right |}\right )}{b^{2} f^{4} - 4 \, b d f^{2} e + 4 \, d^{2} e^{2}} - \frac {{\left (b^{2} f^{4} {\left | f \right |} - 2 \, b d f^{2} {\left | f \right |} e - 2 \, a f^{2} {\left | f \right |} e^{2} + 2 \, d^{2} {\left | f \right |} e^{2}\right )} \log \left ({\left | b f^{2} + 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} e \right |}\right )}{2 \, {\left (b^{2} f^{5} e - 4 \, b d f^{3} e^{2} + 4 \, d^{2} f e^{3}\right )}} + \frac {{\left (b d f^{2} {\left | f \right |} - a f^{2} {\left | f \right |} e - d^{2} {\left | f \right |} e\right )} \log \left ({\left | x e + d - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}} \right |}\right )}{b^{2} f^{5} - 4 \, b d f^{3} e + 4 \, d^{2} f e^{2}} + \frac {{\left (b^{2} f^{5} {\left | f \right |} - 4 \, b d f^{3} {\left | f \right |} e + 4 \, d^{2} f {\left | f \right |} e^{2}\right )} \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}}{b^{3} f^{8} - 6 \, b^{2} d f^{6} e + 12 \, b d^{2} f^{4} e^{2} - 8 \, d^{3} f^{2} e^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \left [\mathit {undef}, +\infty , 1\right ] \]________________________________________________________________________________________