\[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {1}{\left (b \,c^{4}+a \right ) x}-\frac {4 b \,c^{3} d \ln \left (x \right )}{\left (b \,c^{4}+a \right )^{2}}+\frac {b \,c^{3} d \ln \left (a +b \left (d x +c \right )^{4}\right )}{\left (b \,c^{4}+a \right )^{2}}-\frac {c \left (-b \,c^{4}+a \right ) d \arctan \left (\frac {\left (d x +c \right )^{2} \sqrt {b}}{\sqrt {a}}\right ) \sqrt {b}}{\left (b \,c^{4}+a \right )^{2} \sqrt {a}}-\frac {b^{\frac {1}{4}} d \ln \left (-a^{\frac {1}{4}} b^{\frac {1}{4}} \left (d x +c \right ) \sqrt {2}+\sqrt {a}+\left (d x +c \right )^{2} \sqrt {b}\right ) \left (\left (-3 b \,c^{4}+a \right ) \sqrt {a}-c^{2} \left (-b \,c^{4}+3 a \right ) \sqrt {b}\right ) \sqrt {2}}{8 a^{\frac {3}{4}} \left (b \,c^{4}+a \right )^{2}}+\frac {b^{\frac {1}{4}} d \ln \left (a^{\frac {1}{4}} b^{\frac {1}{4}} \left (d x +c \right ) \sqrt {2}+\sqrt {a}+\left (d x +c \right )^{2} \sqrt {b}\right ) \left (\left (-3 b \,c^{4}+a \right ) \sqrt {a}-c^{2} \left (-b \,c^{4}+3 a \right ) \sqrt {b}\right ) \sqrt {2}}{8 a^{\frac {3}{4}} \left (b \,c^{4}+a \right )^{2}}-\frac {b^{\frac {1}{4}} d \arctan \left (-1+\frac {b^{\frac {1}{4}} \left (d x +c \right ) \sqrt {2}}{a^{\frac {1}{4}}}\right ) \left (\left (-3 b \,c^{4}+a \right ) \sqrt {a}+c^{2} \left (-b \,c^{4}+3 a \right ) \sqrt {b}\right ) \sqrt {2}}{4 a^{\frac {3}{4}} \left (b \,c^{4}+a \right )^{2}}-\frac {b^{\frac {1}{4}} d \arctan \left (1+\frac {b^{\frac {1}{4}} \left (d x +c \right ) \sqrt {2}}{a^{\frac {1}{4}}}\right ) \left (\left (-3 b \,c^{4}+a \right ) \sqrt {a}+c^{2} \left (-b \,c^{4}+3 a \right ) \sqrt {b}\right ) \sqrt {2}}{4 a^{\frac {3}{4}} \left (b \,c^{4}+a \right )^{2}} \]
command
integrate(1/x^2/(a+b*(d*x+c)^4),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]