\[ \int \frac {(a+b x)^n \left (c+d x^3\right )^3}{x} \, dx \]
Optimal antiderivative \[ \frac {a^{2} d \left (a^{6} d^{2}-3 a^{3} b^{3} c d +3 b^{6} c^{2}\right ) \left (b x +a \right )^{1+n}}{b^{9} \left (1+n \right )}-\frac {a d \left (8 a^{6} d^{2}-15 a^{3} b^{3} c d +6 b^{6} c^{2}\right ) \left (b x +a \right )^{2+n}}{b^{9} \left (2+n \right )}+\frac {d \left (28 a^{6} d^{2}-30 a^{3} b^{3} c d +3 b^{6} c^{2}\right ) \left (b x +a \right )^{3+n}}{b^{9} \left (3+n \right )}+\frac {2 a^{2} d^{2} \left (-28 a^{3} d +15 b^{3} c \right ) \left (b x +a \right )^{4+n}}{b^{9} \left (4+n \right )}-\frac {5 a \,d^{2} \left (-14 a^{3} d +3 b^{3} c \right ) \left (b x +a \right )^{5+n}}{b^{9} \left (5+n \right )}+\frac {d^{2} \left (-56 a^{3} d +3 b^{3} c \right ) \left (b x +a \right )^{6+n}}{b^{9} \left (6+n \right )}+\frac {28 a^{2} d^{3} \left (b x +a \right )^{7+n}}{b^{9} \left (7+n \right )}-\frac {8 a \,d^{3} \left (b x +a \right )^{8+n}}{b^{9} \left (8+n \right )}+\frac {d^{3} \left (b x +a \right )^{9+n}}{b^{9} \left (9+n \right )}-\frac {c^{3} \left (b x +a \right )^{1+n} \hypergeom \left (\left [1, 1+n \right ], \left [2+n \right ], 1+\frac {b x}{a}\right )}{a \left (1+n \right )} \]
command
integrate((b*x+a)**n*(d*x**3+c)**3/x,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 \[ \text {Timed out} \]_____________________________________________________