\[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-d g +e f \right )^{2} \left (3 a e \,g^{2}+c f \left (-2 d g +5 e f \right )\right ) \left (g x +f \right )^{\frac {3}{2}}}{3 g^{6}}-\frac {2 \left (-d g +e f \right ) \left (3 a \,e^{2} g^{2}+c \left (d^{2} g^{2}-8 d e f g +10 e^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {5}{2}}}{5 g^{6}}+\frac {2 e \left (a \,e^{2} g^{2}+c \left (3 d^{2} g^{2}-12 d e f g +10 e^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7 g^{6}}-\frac {2 c \,e^{2} \left (-3 d g +5 e f \right ) \left (g x +f \right )^{\frac {9}{2}}}{9 g^{6}}+\frac {2 c \,e^{3} \left (g x +f \right )^{\frac {11}{2}}}{11 g^{6}}-\frac {2 \left (-d g +e f \right )^{3} \left (a \,g^{2}+c \,f^{2}\right ) \sqrt {g x +f}}{g^{6}} \]
command
integrate((e*x+d)**3*(c*x**2+a)/(g*x+f)**(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 \[ \text {Timed out} \]_____________________________________________________