\[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx \]
Optimal antiderivative \[ -\frac {\left (d \,h^{2}-e g h +f \,g^{2}\right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{5 h \left (a \,h^{2}+c \,g^{2}\right ) \left (h x +g \right )^{5}}+\frac {\left (5 a \,h^{2} \left (-e h +2 f g \right )+c g \left (3 f \,g^{2}+h \left (-7 d h +2 e g \right )\right )\right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{20 h \left (a \,h^{2}+c \,g^{2}\right )^{2} \left (h x +g \right )^{4}}-\frac {\left (20 a^{2} f \,h^{4}-c^{2} g^{2} \left (3 f \,g^{2}+h \left (-27 d h +2 e g \right )\right )-a c \,h^{2} \left (18 f \,g^{2}-h \left (-8 d h +33 e g \right )\right )\right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{60 h \left (a \,h^{2}+c \,g^{2}\right )^{3} \left (h x +g \right )^{3}}-\frac {a \,c^{2} \left (4 c^{2} d \,g^{3}+a^{2} h^{2} \left (-e h +6 f g \right )-a c g \left (f \,g^{2}-3 h \left (-d h +2 e g \right )\right )\right ) \arctanh \left (\frac {-c g x +a h}{\sqrt {a \,h^{2}+c \,g^{2}}\, \sqrt {c \,x^{2}+a}}\right )}{8 \left (a \,h^{2}+c \,g^{2}\right )^{\frac {9}{2}}}-\frac {c \left (4 c^{2} d \,g^{3}+a^{2} h^{2} \left (-e h +6 f g \right )-a c g \left (f \,g^{2}-3 h \left (-d h +2 e g \right )\right )\right ) \left (-c g x +a h \right ) \sqrt {c \,x^{2}+a}}{8 \left (a \,h^{2}+c \,g^{2}\right )^{4} \left (h x +g \right )^{2}} \]
command
integrate((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^6,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]