\[ \int \frac {\sqrt [6]{\frac {1-b x}{c+x}} \left (1+d x^2\right )}{(1+b x) (1+c x)} \, dx \]
Optimal antiderivative \[ \frac {d \left (c +x \right ) \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}}}{b c}-\frac {\left (b \,c^{2}+6 b +7 c \right ) d \arctan \left (\frac {\left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}}}{b^{\frac {1}{6}}}\right )}{3 b^{\frac {11}{6}} c^{2}}-\frac {2 \left (b +c \right )^{\frac {1}{6}} \left (c^{2}+d \right ) \arctan \left (\frac {\left (-c^{2}+1\right )^{\frac {1}{6}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}}}{\left (b +c \right )^{\frac {1}{6}}}\right )}{c^{2} \left (-b +c \right ) \left (-c^{2}+1\right )^{\frac {1}{6}}}-\frac {2^{\frac {1}{6}} \sqrt {3}\, \left (b^{2}+d \right ) \arctan \left (\frac {\left (-b^{\frac {1}{6}}+2^{\frac {5}{6}} \left (b c -1\right )^{\frac {1}{6}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}}\right ) \sqrt {3}}{3 b^{\frac {1}{6}}}\right )}{b^{\frac {11}{6}} \left (b -c \right ) \left (b c -1\right )^{\frac {1}{6}}}-\frac {2^{\frac {1}{6}} \sqrt {3}\, \left (b^{2}+d \right ) \arctan \left (\frac {\left (b^{\frac {1}{6}}+2^{\frac {5}{6}} \left (b c -1\right )^{\frac {1}{6}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}}\right ) \sqrt {3}}{3 b^{\frac {1}{6}}}\right )}{b^{\frac {11}{6}} \left (b -c \right ) \left (b c -1\right )^{\frac {1}{6}}}+\frac {\left (b \,c^{2}+6 b +7 c \right ) d \arctan \left (\frac {b^{\frac {1}{6}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}}}{-b^{\frac {1}{3}}+\left (\frac {-b x +1}{c +x}\right )^{\frac {1}{3}}}\right )}{6 b^{\frac {11}{6}} c^{2}}-\frac {\left (b +c \right )^{\frac {1}{6}} \left (c^{2}+d \right ) \arctan \left (\frac {\left (b +c \right )^{\frac {1}{6}} \left (-c^{2}+1\right )^{\frac {1}{6}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}}}{\left (b +c \right )^{\frac {1}{3}}-\left (-c^{2}+1\right )^{\frac {1}{3}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{3}}}\right )}{c^{2} \left (-b +c \right ) \left (-c^{2}+1\right )^{\frac {1}{6}}}-\frac {2 \,2^{\frac {1}{6}} \left (b^{2}+d \right ) \arctanh \left (\frac {\left (b c -1\right )^{\frac {1}{6}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}} 2^{\frac {5}{6}}}{2 b^{\frac {1}{6}}}\right )}{b^{\frac {11}{6}} \left (b -c \right ) \left (b c -1\right )^{\frac {1}{6}}}-\frac {\left (b \,c^{2}+6 b +7 c \right ) d \arctanh \left (\frac {\sqrt {3}\, b^{\frac {1}{6}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}}}{b^{\frac {1}{3}}+\left (\frac {-b x +1}{c +x}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{6 b^{\frac {11}{6}} c^{2}}-\frac {2^{\frac {1}{6}} \left (b^{2}+d \right ) \arctanh \left (\frac {2^{\frac {5}{6}} b^{\frac {1}{6}} \left (b c -1\right )^{\frac {1}{6}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}}}{2 b^{\frac {1}{3}}+2^{\frac {2}{3}} \left (b c -1\right )^{\frac {1}{3}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{3}}}\right )}{b^{\frac {11}{6}} \left (b -c \right ) \left (b c -1\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (b +c \right )^{\frac {1}{6}} \left (c^{2}+d \right ) \arctanh \left (\frac {\left (\left (b +c \right )^{\frac {1}{3}}+\left (-c^{2}+1\right )^{\frac {1}{3}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 \left (b +c \right )^{\frac {1}{6}} \left (-c^{2}+1\right )^{\frac {1}{6}} \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}}}\right )}{c^{2} \left (-b +c \right ) \left (-c^{2}+1\right )^{\frac {1}{6}}} \]
command
integrate(((-b*x+1)/(c+x))^(1/6)*(d*x^2+1)/(b*x+1)/(c*x+1),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________