Problem number 290

31.2 Problem number 290

$\int x \sqrt{\frac{5 - 7 x^{2}}{7 + 5 x^{2}}} d x$

Optimal antiderivative $- \frac{37 \arctan (\frac{\sqrt{35} \sqrt{\frac{- 7 x^{2} + 5}{5 x^{2} + 7}}}{7}) \sqrt{35}}{175} + \frac{(5 x^{2} + 7) \sqrt{\frac{- 7 x^{2} + 5}{5 x^{2} + 7}}}{10}$

command

integrate(x*((-7*x**2+5)/(5*x**2+7))**(1/2),x)

Sympy 1.10.1 under Python 3.10.4 output

$\int x \sqrt{- \frac{7 x^{2} - 5}{5 x^{2} + 7}} d x$

Sympy 1.8 under Python 3.8.8 output

${\begin{cases} \frac{5 \sqrt{35} (\frac{\sqrt{25 - 35 x^{2}} \sqrt{35 x^{2} + 49}}{125} - \frac{74 asin (\frac{\sqrt{74} \sqrt{25 - 35 x^{2}}}{74})}{125})}{14} & for x > - \frac{\sqrt{35}}{7} \land x < \frac{\sqrt{35}}{7} \end{cases}$

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