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