\[ \int \frac {(a+b x)^4}{c+d x^3} \, dx \]
Optimal antiderivative \[ \frac {4 a \,b^{3} x}{d}+\frac {b^{4} x^{2}}{2 d}+\frac {\left (b \,c^{\frac {1}{3}} \left (-4 a^{3} d +b^{3} c \right )-d^{\frac {1}{3}} \left (-a^{4} d +4 a \,b^{3} c \right )\right ) \ln \! \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )}{3 c^{\frac {2}{3}} d^{\frac {5}{3}}}-\frac {\left (b \,c^{\frac {1}{3}} \left (-4 a^{3} d +b^{3} c \right )-d^{\frac {1}{3}} \left (-a^{4} d +4 a \,b^{3} c \right )\right ) \ln \! \left (c^{\frac {2}{3}}-c^{\frac {1}{3}} d^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right )}{6 c^{\frac {2}{3}} d^{\frac {5}{3}}}+\frac {2 a^{2} b^{2} \ln \! \left (d \,x^{3}+c \right )}{d}+\frac {\left (b^{4} c^{\frac {4}{3}}+4 a \,b^{3} c \,d^{\frac {1}{3}}-4 a^{3} b \,c^{\frac {1}{3}} d -a^{4} d^{\frac {4}{3}}\right ) \arctan \! \left (\frac {\left (c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x \right ) \sqrt {3}}{3 c^{\frac {1}{3}}}\right ) \sqrt {3}}{3 c^{\frac {2}{3}} d^{\frac {5}{3}}} \]
command
integrate((b*x+a)**4/(d*x**3+c),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {4 a b^{3} x}{d} + \frac {b^{4} x^{2}}{2 d} + \operatorname {RootSum} {\left (27 t^{3} c^{2} d^{5} - 162 t^{2} a^{2} b^{2} c^{2} d^{4} + t \left (36 a^{7} b c d^{4} + 171 a^{4} b^{4} c^{2} d^{3} + 36 a b^{7} c^{3} d^{2}\right ) - a^{12} d^{4} + 4 a^{9} b^{3} c d^{3} - 6 a^{6} b^{6} c^{2} d^{2} + 4 a^{3} b^{9} c^{3} d - b^{12} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {36 t^{2} a^{3} b c^{2} d^{4} - 9 t^{2} b^{4} c^{3} d^{3} + 3 t a^{8} c d^{4} - 168 t a^{5} b^{3} c^{2} d^{3} + 84 t a^{2} b^{6} c^{3} d^{2} + 26 a^{10} b^{2} c d^{3} + 48 a^{7} b^{5} c^{2} d^{2} - 66 a^{4} b^{8} c^{3} d - 8 a b^{11} c^{4}}{a^{12} d^{4} + 52 a^{9} b^{3} c d^{3} - 52 a^{3} b^{9} c^{3} d - b^{12} c^{4}} \right )} \right )\right )} \]