\[ \int \frac {1}{\sqrt {a+b x+c x^2} \left (d+b x+c x^2\right )^4} \, dx \]
Optimal antiderivative \[ \frac {\left (b^{2}+4 c \left (a -2 d \right )\right ) \left (5 b^{4}-8 b^{2} c \left (a +4 d \right )+16 c^{2} \left (5 a^{2}-8 a d +8 d^{2}\right )\right ) \arctanh \left (\frac {\left (2 c x +b \right ) \sqrt {a -d}}{\sqrt {b^{2}-4 c d}\, \sqrt {c \,x^{2}+b x +a}}\right )}{8 \left (a -d \right )^{\frac {7}{2}} \left (b^{2}-4 c d \right )^{\frac {7}{2}}}-\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{3 \left (a -d \right ) \left (b^{2}-4 c d \right ) \left (c \,x^{2}+b x +d \right )^{3}}+\frac {5 \left (b^{2}+4 c \left (a -2 d \right )\right ) \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{12 \left (a -d \right )^{2} \left (b^{2}-4 c d \right )^{2} \left (c \,x^{2}+b x +d \right )^{2}}-\frac {\left (15 b^{4}+8 b^{2} c \left (7 a -22 d \right )+16 c^{2} \left (15 a^{2}-44 a d +44 d^{2}\right )\right ) \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{24 \left (a -d \right )^{3} \left (b^{2}-4 c d \right )^{3} \left (c \,x^{2}+b x +d \right )} \]
command
integrate(1/(c*x^2+b*x+d)^4/(c*x^2+b*x+a)^(1/2),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 \[ \text {Timed out} \]_______________________________________________________