\[ \int x (a+b x)^n \left (c+d x^2\right )^3 \, dx \]
Optimal antiderivative \[ -\frac {a \left (a^{2} d +b^{2} c \right )^{3} \left (b x +a \right )^{1+n}}{b^{8} \left (1+n \right )}+\frac {\left (a^{2} d +b^{2} c \right )^{2} \left (7 a^{2} d +b^{2} c \right ) \left (b x +a \right )^{2+n}}{b^{8} \left (2+n \right )}-\frac {3 a d \left (a^{2} d +b^{2} c \right ) \left (7 a^{2} d +3 b^{2} c \right ) \left (b x +a \right )^{3+n}}{b^{8} \left (3+n \right )}+\frac {d \left (35 a^{4} d^{2}+30 a^{2} b^{2} c d +3 b^{4} c^{2}\right ) \left (b x +a \right )^{4+n}}{b^{8} \left (4+n \right )}-\frac {5 a \,d^{2} \left (7 a^{2} d +3 b^{2} c \right ) \left (b x +a \right )^{5+n}}{b^{8} \left (5+n \right )}+\frac {3 d^{2} \left (7 a^{2} d +b^{2} c \right ) \left (b x +a \right )^{6+n}}{b^{8} \left (6+n \right )}-\frac {7 a \,d^{3} \left (b x +a \right )^{7+n}}{b^{8} \left (7+n \right )}+\frac {d^{3} \left (b x +a \right )^{8+n}}{b^{8} \left (8+n \right )} \]
command
integrate(x*(b*x+a)**n*(d*x**2+c)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________