\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx \]
Optimal antiderivative \[ \frac {g^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 \left (-d g +e f \right ) \left (a \,g^{2}-b f g +c \,f^{2}\right ) \left (g x +f \right )^{3}}+\frac {\left (-4 a c +b^{2}\right ) g \left (-b g +2 c f \right ) \arctanh \left (\frac {b f -2 a g +\left (-b g +2 c f \right ) x}{2 \sqrt {a \,g^{2}-b f g +c \,f^{2}}\, \sqrt {c \,x^{2}+b x +a}}\right )}{16 \left (-d g +e f \right ) \left (a \,g^{2}-b f g +c \,f^{2}\right )^{\frac {5}{2}}}+\frac {\left (-4 a c +b^{2}\right ) e g \arctanh \left (\frac {b f -2 a g +\left (-b g +2 c f \right ) x}{2 \sqrt {a \,g^{2}-b f g +c \,f^{2}}\, \sqrt {c \,x^{2}+b x +a}}\right )}{8 \left (-d g +e f \right )^{2} \left (a \,g^{2}-b f g +c \,f^{2}\right )^{\frac {3}{2}}}-\frac {e^{2} \left (-b e +2 c d \right ) \arctanh \left (\frac {2 c x +b}{2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}}\right )}{2 \left (-d g +e f \right )^{4} \sqrt {c}}+\frac {e^{3} \left (-b g +2 c f \right ) \arctanh \left (\frac {2 c x +b}{2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}}\right )}{2 g \left (-d g +e f \right )^{4} \sqrt {c}}-\frac {e^{2} \arctanh \left (\frac {2 c x +b}{2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}}\right ) \sqrt {c}}{g \left (-d g +e f \right )^{3}}+\frac {e^{2} \arctanh \left (\frac {b d -2 a e +\left (-b e +2 c d \right ) x}{2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \sqrt {c \,x^{2}+b x +a}}\right ) \sqrt {a \,e^{2}-b d e +c \,d^{2}}}{\left (-d g +e f \right )^{4}}+\frac {e^{2} \left (-b g +2 c f \right ) \arctanh \left (\frac {b f -2 a g +\left (-b g +2 c f \right ) x}{2 \sqrt {a \,g^{2}-b f g +c \,f^{2}}\, \sqrt {c \,x^{2}+b x +a}}\right )}{2 g \left (-d g +e f \right )^{3} \sqrt {a \,g^{2}-b f g +c \,f^{2}}}-\frac {e^{3} \arctanh \left (\frac {b f -2 a g +\left (-b g +2 c f \right ) x}{2 \sqrt {a \,g^{2}-b f g +c \,f^{2}}\, \sqrt {c \,x^{2}+b x +a}}\right ) \sqrt {a \,g^{2}-b f g +c \,f^{2}}}{g \left (-d g +e f \right )^{4}}+\frac {e^{2} \sqrt {c \,x^{2}+b x +a}}{\left (-d g +e f \right )^{3} \left (g x +f \right )}-\frac {g \left (-b g +2 c f \right ) \left (b f -2 a g +\left (-b g +2 c f \right ) x \right ) \sqrt {c \,x^{2}+b x +a}}{8 \left (-d g +e f \right ) \left (a \,g^{2}-b f g +c \,f^{2}\right )^{2} \left (g x +f \right )^{2}}-\frac {e g \left (b f -2 a g +\left (-b g +2 c f \right ) x \right ) \sqrt {c \,x^{2}+b x +a}}{4 \left (-d g +e f \right )^{2} \left (a \,g^{2}-b f g +c \,f^{2}\right ) \left (g x +f \right )^{2}} \]
command
integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)/(g*x+f)^4,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 \[ \text {Timed out} \]_______________________________________________________