\[ \int x \sqrt {\frac {5-7 x^2}{7+5 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {37 \arctan \! \left (\frac {\sqrt {35}\, \sqrt {\frac {-7 x^{2}+5}{5 x^{2}+7}}}{7}\right ) \sqrt {35}}{175}+\frac {\left (5 x^{2}+7\right ) \sqrt {\frac {-7 x^{2}+5}{5 x^{2}+7}}}{10} \]
command
integrate(x*((-7*x**2+5)/(5*x**2+7))**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \int x \sqrt {- \frac {7 x^{2} - 5}{5 x^{2} + 7}}\, dx \]
Sympy 1.8 under Python 3.8.8 output
\[ \begin {cases} \frac {5 \sqrt {35} \left (\frac {\sqrt {25 - 35 x^{2}} \sqrt {35 x^{2} + 49}}{125} - \frac {74 \operatorname {asin}{\left (\frac {\sqrt {74} \sqrt {25 - 35 x^{2}}}{74} \right )}}{125}\right )}{14} & \text {for}\: x > - \frac {\sqrt {35}}{7} \wedge x < \frac {\sqrt {35}}{7} \end {cases} \]