\[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^3} \, dx \]
Optimal antiderivative \[ \frac {6 e \,f^{2} \left (-b^{2} f^{2}+4 a \,e^{2}\right ) \ln \left (d +e x +f \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )}{\left (-b \,f^{2}+2 d e \right )^{4}}-\frac {6 e \,f^{2} \left (-b^{2} f^{2}+4 a \,e^{2}\right ) \ln \left (b \,f^{2}+2 e \left (e x +f \sqrt {a +\frac {x \left (b \,f^{2}+e^{2} x \right )}{f^{2}}}\right )\right )}{\left (-b \,f^{2}+2 d e \right )^{4}}+\frac {-a e \,f^{2}+b d \,f^{2}-d^{2} e}{\left (-b \,f^{2}+2 d e \right )^{2} \left (d +e x +f \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )^{2}}-\frac {2 f^{2} \left (-b^{2} f^{2}+4 a \,e^{2}\right )}{\left (-b \,f^{2}+2 d e \right )^{3} \left (d +e x +f \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}\right )}-\frac {2 e \,f^{2} \left (-b^{2} f^{2}+4 a \,e^{2}\right )}{\left (-b \,f^{2}+2 d e \right )^{3} \left (b \,f^{2}+2 e \left (e x +f \sqrt {a +\frac {x \left (b \,f^{2}+e^{2} x \right )}{f^{2}}}\right )\right )} \]
command
integrate(1/(d+e*x+f*(a+b*x+e^2*x^2/f^2)^(1/2))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________