\[ \int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {-2 a -2 b -c}\, \arctan \left (\frac {\sqrt {-2 a -2 b -c}\, x}{\sqrt {a}-2 x \sqrt {a}+x^{2} \sqrt {a}-\sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}}\right )}{12 a +12 b +6 c}+\frac {\arctan \left (\frac {\sqrt {a -b -c}\, x}{\sqrt {a}-x \sqrt {a}+x^{2} \sqrt {a}-\sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}}\right )}{3 \sqrt {a -b -c}}-\frac {\arctan \left (\frac {\sqrt {a +b -c}\, x}{\sqrt {a}+x \sqrt {a}+x^{2} \sqrt {a}-\sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}}\right )}{3 \sqrt {a +b -c}}-\frac {\sqrt {-2 a +2 b -c}\, \arctan \left (\frac {\sqrt {-2 a +2 b -c}\, x}{\sqrt {a}+2 x \sqrt {a}+x^{2} \sqrt {a}-\sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}}\right )}{12 a -12 b +6 c} \]
command
integrate(x^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),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 \[ \text {Timed out} \]