\[ \int x \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^p \, dx \]
Optimal antiderivative \[ \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{1+p}}{2+2 p} \]
command
integrate(x*(2*c*x**2+b)*(c*x**4+b*x**2+a)**p,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {a \left (a + b x^{2} + c x^{4}\right )^{p}}{2 p + 2} + \frac {b x^{2} \left (a + b x^{2} + c x^{4}\right )^{p}}{2 p + 2} + \frac {c x^{4} \left (a + b x^{2} + c x^{4}\right )^{p}}{2 p + 2} & \text {for}\: p \neq -1 \\\frac {\log {\left (x - \frac {\sqrt {2} \sqrt {- \frac {b}{c} - \frac {\sqrt {- 4 a c + b^{2}}}{c}}}{2} \right )}}{2} + \frac {\log {\left (x + \frac {\sqrt {2} \sqrt {- \frac {b}{c} - \frac {\sqrt {- 4 a c + b^{2}}}{c}}}{2} \right )}}{2} + \frac {\log {\left (x - \frac {\sqrt {2} \sqrt {- \frac {b}{c} + \frac {\sqrt {- 4 a c + b^{2}}}{c}}}{2} \right )}}{2} + \frac {\log {\left (x + \frac {\sqrt {2} \sqrt {- \frac {b}{c} + \frac {\sqrt {- 4 a c + b^{2}}}{c}}}{2} \right )}}{2} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________