\[ \int \left (b x+c x^2\right ) \left (1+\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^n\right ) \, dx \]
Optimal antiderivative \[ \frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}+\frac {\left (\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )^{1+n}}{1+n} \]
command
integrate((c*x**2+b*x)*(1+(1/2*b*x**2+1/3*c*x**3)**n),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {3 \cdot 6^{n} b n x^{2}}{6 \cdot 6^{n} n + 6 \cdot 6^{n}} + \frac {3 \cdot 6^{n} b x^{2}}{6 \cdot 6^{n} n + 6 \cdot 6^{n}} + \frac {2 \cdot 6^{n} c n x^{3}}{6 \cdot 6^{n} n + 6 \cdot 6^{n}} + \frac {2 \cdot 6^{n} c x^{3}}{6 \cdot 6^{n} n + 6 \cdot 6^{n}} + \frac {3 b x^{2} \left (3 b x^{2} + 2 c x^{3}\right )^{n}}{6 \cdot 6^{n} n + 6 \cdot 6^{n}} + \frac {2 c x^{3} \left (3 b x^{2} + 2 c x^{3}\right )^{n}}{6 \cdot 6^{n} n + 6 \cdot 6^{n}} & \text {for}\: n \neq -1 \\\frac {b x^{2}}{2} + \frac {c x^{3}}{3} + 2 \log {\left (x \right )} + \log {\left (\frac {3 b}{2 c} + x \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________