\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ -\frac {c \left (40 c^{3} d^{3}-5 b^{3} e^{3}-2 c^{2} d e \left (-12 a e +35 b d \right )+3 b c \,e^{2} \left (-5 a e +12 b d \right )\right ) x}{e^{7}}+\frac {c^{2} \left (6 a c \,e^{2}+9 b^{2} e^{2}-28 b c d e +20 c^{2} d^{2}\right ) x^{2}}{2 e^{6}}-\frac {c^{3} \left (-7 b e +8 c d \right ) x^{3}}{3 e^{5}}+\frac {c^{4} x^{4}}{2 e^{4}}+\frac {\left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}{3 e^{8} \left (e x +d \right )^{3}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (14 c^{2} d^{2}+3 b^{2} e^{2}-2 c e \left (-a e +7 b d \right )\right )}{2 e^{8} \left (e x +d \right )^{2}}+\frac {3 \left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (7 c^{2} d^{2}+b^{2} e^{2}-c e \left (-3 a e +7 b d \right )\right )}{e^{8} \left (e x +d \right )}+\frac {\left (70 c^{4} d^{4}+b^{4} e^{4}-4 b^{2} c \,e^{3} \left (-3 a e +5 b d \right )-20 c^{3} d^{2} e \left (-3 a e +7 b d \right )+6 c^{2} e^{2} \left (a^{2} e^{2}-10 a b d e +15 b^{2} d^{2}\right )\right ) \ln \! \left (e x +d \right )}{e^{8}} \]
command
integrate((2*c*x+b)*(c*x**2+b*x+a)**3/(e*x+d)**4,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {c^{4} x^{4}}{2 e^{4}} + x^{3} \left (\frac {7 b c^{3}}{3 e^{4}} - \frac {8 c^{4} d}{3 e^{5}}\right ) + x^{2} \left (\frac {3 a c^{3}}{e^{4}} + \frac {9 b^{2} c^{2}}{2 e^{4}} - \frac {14 b c^{3} d}{e^{5}} + \frac {10 c^{4} d^{2}}{e^{6}}\right ) + x \left (\frac {15 a b c^{2}}{e^{4}} - \frac {24 a c^{3} d}{e^{5}} + \frac {5 b^{3} c}{e^{4}} - \frac {36 b^{2} c^{2} d}{e^{5}} + \frac {70 b c^{3} d^{2}}{e^{6}} - \frac {40 c^{4} d^{3}}{e^{7}}\right ) + \frac {- 2 a^{3} b e^{7} - 2 a^{3} c d e^{6} - 3 a^{2} b^{2} d e^{6} - 18 a^{2} b c d^{2} e^{5} + 66 a^{2} c^{2} d^{3} e^{4} - 6 a b^{3} d^{2} e^{5} + 132 a b^{2} c d^{3} e^{4} - 390 a b c^{2} d^{4} e^{3} + 282 a c^{3} d^{5} e^{2} + 11 b^{4} d^{3} e^{4} - 130 b^{3} c d^{4} e^{3} + 423 b^{2} c^{2} d^{5} e^{2} - 518 b c^{3} d^{6} e + 214 c^{4} d^{7} + x^{2} \left (- 54 a^{2} b c e^{7} + 108 a^{2} c^{2} d e^{6} - 18 a b^{3} e^{7} + 216 a b^{2} c d e^{6} - 540 a b c^{2} d^{2} e^{5} + 360 a c^{3} d^{3} e^{4} + 18 b^{4} d e^{6} - 180 b^{3} c d^{2} e^{5} + 540 b^{2} c^{2} d^{3} e^{4} - 630 b c^{3} d^{4} e^{3} + 252 c^{4} d^{5} e^{2}\right ) + x \left (- 6 a^{3} c e^{7} - 9 a^{2} b^{2} e^{7} - 54 a^{2} b c d e^{6} + 162 a^{2} c^{2} d^{2} e^{5} - 18 a b^{3} d e^{6} + 324 a b^{2} c d^{2} e^{5} - 900 a b c^{2} d^{3} e^{4} + 630 a c^{3} d^{4} e^{3} + 27 b^{4} d^{2} e^{5} - 300 b^{3} c d^{3} e^{4} + 945 b^{2} c^{2} d^{4} e^{3} - 1134 b c^{3} d^{5} e^{2} + 462 c^{4} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} + \frac {\left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 60 a b c^{2} d e^{3} + 60 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 20 b^{3} c d e^{3} + 90 b^{2} c^{2} d^{2} e^{2} - 140 b c^{3} d^{3} e + 70 c^{4} d^{4}\right ) \log {\left (d + e x \right )}}{e^{8}} \]