\[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx \]
Optimal antiderivative \[ \frac {2 \arctanh \left (\frac {\textit {\_C3}^{\frac {1}{4}} \sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}}{\sqrt {\sqrt {\textit {\_C3}}\, \textit {\_C4} -\sqrt {\textit {\_C1}}\, \textit {\_C5}}}\right ) \textit {\_C3}^{\frac {1}{4}}}{\sqrt {\sqrt {\textit {\_C3}}\, \textit {\_C4} -\sqrt {\textit {\_C1}}\, \textit {\_C5}}\, \textit {\_C7}}+\frac {2 \arctanh \left (\frac {\textit {\_C3}^{\frac {1}{4}} \sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}}{\sqrt {\sqrt {\textit {\_C3}}\, \textit {\_C4} +\sqrt {\textit {\_C1}}\, \textit {\_C5}}}\right ) \textit {\_C3}^{\frac {1}{4}}}{\sqrt {\sqrt {\textit {\_C3}}\, \textit {\_C4} +\sqrt {\textit {\_C1}}\, \textit {\_C5}}\, \textit {\_C7}}+\frac {2 \arctan \left (\frac {\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \sqrt {-\textit {\_C2} \textit {\_C7} +\textit {\_C3} \textit {\_C6}}}{\sqrt {-\textit {\_C3} \textit {\_C4} \textit {\_C6} +\textit {\_C2} \textit {\_C4} \textit {\_C7} -\textit {\_C5} \sqrt {-\textit {\_C0} \textit {\_C7} +\textit {\_C1} \textit {\_C6}}\, \sqrt {-\textit {\_C2} \textit {\_C7} +\textit {\_C3} \textit {\_C6}}}}\right ) \sqrt {-\textit {\_C2} \textit {\_C7} +\textit {\_C3} \textit {\_C6}}}{\textit {\_C7} \sqrt {-\textit {\_C3} \textit {\_C4} \textit {\_C6} +\textit {\_C2} \textit {\_C4} \textit {\_C7} -\textit {\_C5} \sqrt {-\textit {\_C0} \textit {\_C7} +\textit {\_C1} \textit {\_C6}}\, \sqrt {-\textit {\_C2} \textit {\_C7} +\textit {\_C3} \textit {\_C6}}}}+\frac {2 \arctan \left (\frac {\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \sqrt {-\textit {\_C2} \textit {\_C7} +\textit {\_C3} \textit {\_C6}}}{\sqrt {-\textit {\_C3} \textit {\_C4} \textit {\_C6} +\textit {\_C2} \textit {\_C4} \textit {\_C7} +\textit {\_C5} \sqrt {-\textit {\_C0} \textit {\_C7} +\textit {\_C1} \textit {\_C6}}\, \sqrt {-\textit {\_C2} \textit {\_C7} +\textit {\_C3} \textit {\_C6}}}}\right ) \sqrt {-\textit {\_C2} \textit {\_C7} +\textit {\_C3} \textit {\_C6}}}{\textit {\_C7} \sqrt {-\textit {\_C3} \textit {\_C4} \textit {\_C6} +\textit {\_C2} \textit {\_C4} \textit {\_C7} +\textit {\_C5} \sqrt {-\textit {\_C0} \textit {\_C7} +\textit {\_C1} \textit {\_C6}}\, \sqrt {-\textit {\_C2} \textit {\_C7} +\textit {\_C3} \textit {\_C6}}}} \]
command
integrate(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6),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 \[ \text {Timed out} \]