\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (14 c^{2} d^{2}+3 b^{2} e^{2}-2 c e \left (-a e +7 b d \right )\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e^{8}}-\frac {6 \left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (7 c^{2} d^{2}+b^{2} e^{2}-c e \left (-3 a e +7 b d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{8}}+\frac {2 \left (70 c^{4} d^{4}+b^{4} e^{4}-4 b^{2} c \,e^{3} \left (-3 a e +5 b d \right )-20 c^{3} d^{2} e \left (-3 a e +7 b d \right )+6 c^{2} e^{2} \left (a^{2} e^{2}-10 a b d e +15 b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{8}}-\frac {10 c \left (-b e +2 c d \right ) \left (7 c^{2} d^{2}+b^{2} e^{2}-c e \left (-3 a e +7 b d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9 e^{8}}+\frac {6 c^{2} \left (14 c^{2} d^{2}+3 b^{2} e^{2}-2 c e \left (-a e +7 b d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11 e^{8}}-\frac {14 c^{3} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {13}{2}}}{13 e^{8}}+\frac {4 c^{4} \left (e x +d \right )^{\frac {15}{2}}}{15 e^{8}}-\frac {2 \left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {e x +d}}{e^{8}} \]
command
integrate((2*c*x+b)*(c*x**2+b*x+a)**3/(e*x+d)**(1/2),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 \[ \text {Timed out} \]_____________________________________________________