\[ \int (a+b x) \left (1+\left (c+a x+\frac {b x^2}{2}\right )^n\right ) \, dx \]
Optimal antiderivative \[ a x +\frac {b \,x^{2}}{2}+\frac {\left (c +a x +\frac {1}{2} b \,x^{2}\right )^{1+n}}{1+n} \]
command
integrate((b*x+a)*(1+(c+a*x+1/2*b*x**2)**n),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} a \left (x + \frac {\log {\left (x + \frac {c}{a} \right )}}{a}\right ) & \text {for}\: b = 0 \wedge n = -1 \\a \left (\frac {a n x}{a n + a} + \frac {a x \left (a x + c\right )^{n}}{a n + a} + \frac {a x}{a n + a} + \frac {c \left (a x + c\right )^{n}}{a n + a}\right ) & \text {for}\: b = 0 \\a x + \frac {b x^{2}}{2} + \log {\left (\frac {a}{b} + x - \frac {\sqrt {a^{2} - 2 b c}}{b} \right )} + \log {\left (\frac {a}{b} + x + \frac {\sqrt {a^{2} - 2 b c}}{b} \right )} & \text {for}\: n = -1 \\\frac {2 \cdot 2^{n} a b n x}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {2 \cdot 2^{n} a b x}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {2^{n} b^{2} n x^{2}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {2^{n} b^{2} x^{2}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {2 a b x \left (2 a x + b x^{2} + 2 c\right )^{n}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {b^{2} x^{2} \left (2 a x + b x^{2} + 2 c\right )^{n}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {2 b c \left (2 a x + b x^{2} + 2 c\right )^{n}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________