\[ \int \frac {1}{x^3 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {b \left (-21 a^{2} d^{2}-6 a b c d +35 b^{2} c^{2}\right )}{12 a^{3} c^{2} \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}-\frac {1}{2 a c \,x^{2} \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}+\frac {\frac {7 a d}{4}+\frac {7 b c}{4}}{a^{2} c^{2} x \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}-\frac {5 \left (7 a^{2} d^{2}+10 a b c d +7 b^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {c}\, \sqrt {b x +a}}{\sqrt {a}\, \sqrt {d x +c}}\right )}{4 a^{\frac {9}{2}} c^{\frac {9}{2}}}+\frac {b \left (7 a^{3} d^{3}-3 a^{2} b c \,d^{2}-55 a \,b^{2} c^{2} d +35 b^{3} c^{3}\right )}{4 a^{4} c^{2} \left (-a d +b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}+\frac {d \left (-35 a^{4} d^{4}+48 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-200 a \,b^{3} c^{3} d +105 b^{4} c^{4}\right ) \sqrt {b x +a}}{12 a^{4} c^{3} \left (-a d +b c \right )^{3} \left (d x +c \right )^{\frac {3}{2}}}+\frac {d \left (a d +b c \right ) \left (105 a^{4} d^{4}-340 a^{3} b c \,d^{3}+406 a^{2} b^{2} c^{2} d^{2}-340 a \,b^{3} c^{3} d +105 b^{4} c^{4}\right ) \sqrt {b x +a}}{12 a^{4} c^{4} \left (-a d +b c \right )^{4} \sqrt {d x +c}} \]
command
integrate(1/x^3/(b*x+a)^(5/2)/(d*x+c)^(5/2),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 \[ \text {Timed out} \]_______________________________________________________