\[ \int x \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx \]
Optimal antiderivative \[ \frac {a^{3} B \,x^{3}}{3}+\frac {a^{3} C \,x^{4}}{4}+\frac {a^{2} \left (3 b B +a D\right ) x^{5}}{5}+\frac {a^{2} b C \,x^{6}}{2}+\frac {3 a b \left (b B +a D\right ) x^{7}}{7}+\frac {3 a \,b^{2} C \,x^{8}}{8}+\frac {b^{2} \left (b B +3 a D\right ) x^{9}}{9}+\frac {b^{3} C \,x^{10}}{10}+\frac {b^{3} D x^{11}}{11}+\frac {A \left (b \,x^{2}+a \right )^{4}}{8 b} \]
command
integrate(x*(b*x^2+a)^3*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{11} \, D b^{3} x^{11} + \frac {1}{10} \, C b^{3} x^{10} + \frac {1}{9} \, {\left (3 \, D a b^{2} + B b^{3}\right )} x^{9} + \frac {1}{8} \, {\left (3 \, C a b^{2} + A b^{3}\right )} x^{8} + \frac {3}{7} \, {\left (D a^{2} b + B a b^{2}\right )} x^{7} + \frac {1}{3} \, B a^{3} x^{3} + \frac {1}{2} \, {\left (C a^{2} b + A a b^{2}\right )} x^{6} + \frac {1}{2} \, A a^{3} x^{2} + \frac {1}{5} \, {\left (D a^{3} + 3 \, B a^{2} b\right )} x^{5} + \frac {1}{4} \, {\left (C a^{3} + 3 \, A a^{2} b\right )} x^{4} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]