\[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx \]
Optimal antiderivative \[ \frac {d \left (a d +b c \right ) x}{3 a c \left (-a d +b c \right )^{2} \left (d \,x^{3}+c \right )}+\frac {b x}{3 a \left (-a d +b c \right ) \left (b \,x^{3}+a \right ) \left (d \,x^{3}+c \right )}+\frac {2 b^{\frac {5}{3}} \left (-4 a d +b c \right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{9 a^{\frac {5}{3}} \left (-a d +b c \right )^{3}}+\frac {2 d^{\frac {5}{3}} \left (-a d +4 b c \right ) \ln \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )}{9 c^{\frac {5}{3}} \left (-a d +b c \right )^{3}}-\frac {b^{\frac {5}{3}} \left (-4 a d +b c \right ) \ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{9 a^{\frac {5}{3}} \left (-a d +b c \right )^{3}}-\frac {d^{\frac {5}{3}} \left (-a d +4 b c \right ) \ln \left (c^{\frac {2}{3}}-c^{\frac {1}{3}} d^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right )}{9 c^{\frac {5}{3}} \left (-a d +b c \right )^{3}}-\frac {2 b^{\frac {5}{3}} \left (-4 a d +b c \right ) \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{9 a^{\frac {5}{3}} \left (-a d +b c \right )^{3}}-\frac {2 d^{\frac {5}{3}} \left (-a d +4 b c \right ) \arctan \left (\frac {\left (c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x \right ) \sqrt {3}}{3 c^{\frac {1}{3}}}\right ) \sqrt {3}}{9 c^{\frac {5}{3}} \left (-a d +b c \right )^{3}} \]
command
integrate(1/(b*x^3+a)^2/(d*x^3+c)^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{4} + 2 \, \sqrt {3} {\left ({\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{3}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 2 \, \sqrt {3} {\left ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} c x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}} - \sqrt {3} d}{3 \, d}\right ) - {\left ({\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{3}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - {\left ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} + c^{2} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{3}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 2 \, {\left ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d x + c \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{9 \, {\left (a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + {\left (a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{6} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]