\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (7 c^{2} d^{2}+b^{2} e^{2}-c e \left (-3 a e +7 b d \right )\right ) \left (e x +d \right )^{\frac {3}{2}}}{e^{8}}+\frac {2 \left (70 c^{4} d^{4}+b^{4} e^{4}-4 b^{2} c \,e^{3} \left (-3 a e +5 b d \right )-20 c^{3} d^{2} e \left (-3 a e +7 b d \right )+6 c^{2} e^{2} \left (a^{2} e^{2}-10 a b d e +15 b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{8}}-\frac {10 c \left (-b e +2 c d \right ) \left (7 c^{2} d^{2}+b^{2} e^{2}-c e \left (-3 a e +7 b d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{8}}+\frac {2 c^{2} \left (14 c^{2} d^{2}+3 b^{2} e^{2}-2 c e \left (-a e +7 b d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{3 e^{8}}-\frac {14 c^{3} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11 e^{8}}+\frac {4 c^{4} \left (e x +d \right )^{\frac {13}{2}}}{13 e^{8}}+\frac {2 \left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}{e^{8} \sqrt {e x +d}}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (14 c^{2} d^{2}+3 b^{2} e^{2}-2 c e \left (-a e +7 b d \right )\right ) \sqrt {e x +d}}{e^{8}} \]
command
integrate((2*c*x+b)*(c*x**2+b*x+a)**3/(e*x+d)**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {4 c^{4} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{8}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (14 b c^{3} e - 28 c^{4} d\right )}{11 e^{8}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (12 a c^{3} e^{2} + 18 b^{2} c^{2} e^{2} - 84 b c^{3} d e + 84 c^{4} d^{2}\right )}{9 e^{8}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (30 a b c^{2} e^{3} - 60 a c^{3} d e^{2} + 10 b^{3} c e^{3} - 90 b^{2} c^{2} d e^{2} + 210 b c^{3} d^{2} e - 140 c^{4} d^{3}\right )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (12 a^{2} c^{2} e^{4} + 24 a b^{2} c e^{4} - 120 a b c^{2} d e^{3} + 120 a c^{3} d^{2} e^{2} + 2 b^{4} e^{4} - 40 b^{3} c d e^{3} + 180 b^{2} c^{2} d^{2} e^{2} - 280 b c^{3} d^{3} e + 140 c^{4} d^{4}\right )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (18 a^{2} b c e^{5} - 36 a^{2} c^{2} d e^{4} + 6 a b^{3} e^{5} - 72 a b^{2} c d e^{4} + 180 a b c^{2} d^{2} e^{3} - 120 a c^{3} d^{3} e^{2} - 6 b^{4} d e^{4} + 60 b^{3} c d^{2} e^{3} - 180 b^{2} c^{2} d^{3} e^{2} + 210 b c^{3} d^{4} e - 84 c^{4} d^{5}\right )}{3 e^{8}} + \frac {\sqrt {d + e x} \left (4 a^{3} c e^{6} + 6 a^{2} b^{2} e^{6} - 36 a^{2} b c d e^{5} + 36 a^{2} c^{2} d^{2} e^{4} - 12 a b^{3} d e^{5} + 72 a b^{2} c d^{2} e^{4} - 120 a b c^{2} d^{3} e^{3} + 60 a c^{3} d^{4} e^{2} + 6 b^{4} d^{2} e^{4} - 40 b^{3} c d^{3} e^{3} + 90 b^{2} c^{2} d^{4} e^{2} - 84 b c^{3} d^{5} e + 28 c^{4} d^{6}\right )}{e^{8}} - \frac {2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3}}{e^{8} \sqrt {d + e x}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________