\[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {3 b c \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\left (b -c \right )^{3} \sqrt {a}}+\frac {3 b c \arctanh \left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )}{\left (b -c \right )^{3} \sqrt {a}}-\frac {2 a \sqrt {b x +a}}{\left (b -c \right )^{3} x^{2}}-\frac {\left (2 b +3 c \right ) \sqrt {b x +a}}{\left (b -c \right )^{3} x}+\frac {2 a \sqrt {c x +a}}{\left (b -c \right )^{3} x^{2}}+\frac {\left (3 b +2 c \right ) \sqrt {c x +a}}{\left (b -c \right )^{3} x} \]
command
integrate(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,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 \[ \text {Timed out} \]_______________________________________________________