\[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \! \left (\frac {\sqrt {1-\textit {\_C0}}\, \sqrt {-1+\textit {\_C1}}\, \sqrt {\frac {\textit {\_C3} \,x^{2}+\textit {\_C0} x +\textit {\_C4}}{\textit {\_C3} \,x^{2}+\textit {\_C1} x +\textit {\_C4}}}}{-1+\textit {\_C0}}\right ) \sqrt {-1+\textit {\_C1}}}{2 \sqrt {1-\textit {\_C0}}}-\frac {\arctan \! \left (\frac {\sqrt {-1-\textit {\_C0}}\, \sqrt {1+\textit {\_C1}}\, \sqrt {\frac {\textit {\_C3} \,x^{2}+\textit {\_C0} x +\textit {\_C4}}{\textit {\_C3} \,x^{2}+\textit {\_C1} x +\textit {\_C4}}}}{1+\textit {\_C0}}\right ) \sqrt {1+\textit {\_C1}}}{4 \sqrt {-1-\textit {\_C0}}}+\frac {3 \arctan \! \left (\frac {\sqrt {1-3 \textit {\_C0}}\, \sqrt {-1+3 \textit {\_C1}}\, \sqrt {\frac {\textit {\_C3} \,x^{2}+\textit {\_C0} x +\textit {\_C4}}{\textit {\_C3} \,x^{2}+\textit {\_C1} x +\textit {\_C4}}}}{-1+3 \textit {\_C0}}\right ) \sqrt {-1+3 \textit {\_C1}}}{4 \sqrt {1-3 \textit {\_C0}}} \]
command
int(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(3*_C3*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x)
Maple 2022.1 output
hanged
Maple 2021.1 output
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(186113818\) |