\[ \int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx \]
Optimal antiderivative \[ \frac {\EllipticE \left (\frac {\sqrt {a +2 x}}{\sqrt {a -c}}, \sqrt {1-\frac {c}{a}}\right ) \sqrt {a -c}\, \sqrt {x}\, \sqrt {\frac {-c -2 x}{a -c}}\, \sqrt {2}}{2 \sqrt {-\frac {x}{a}}\, \sqrt {c +2 x}} \]
command
integrate(x^(1/2)/(a+2*x)^(1/2)/(c+2*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{6} \, {\left (a + c\right )} {\rm weierstrassPInverse}\left (\frac {1}{3} \, a^{2} - \frac {1}{3} \, a c + \frac {1}{3} \, c^{2}, -\frac {1}{27} \, a^{3} + \frac {1}{18} \, a^{2} c + \frac {1}{18} \, a c^{2} - \frac {1}{27} \, c^{3}, \frac {1}{6} \, a + \frac {1}{6} \, c + x\right ) - {\rm weierstrassZeta}\left (\frac {1}{3} \, a^{2} - \frac {1}{3} \, a c + \frac {1}{3} \, c^{2}, -\frac {1}{27} \, a^{3} + \frac {1}{18} \, a^{2} c + \frac {1}{18} \, a c^{2} - \frac {1}{27} \, c^{3}, {\rm weierstrassPInverse}\left (\frac {1}{3} \, a^{2} - \frac {1}{3} \, a c + \frac {1}{3} \, c^{2}, -\frac {1}{27} \, a^{3} + \frac {1}{18} \, a^{2} c + \frac {1}{18} \, a c^{2} - \frac {1}{27} \, c^{3}, \frac {1}{6} \, a + \frac {1}{6} \, c + x\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a + 2 \, x} \sqrt {c + 2 \, x} \sqrt {x}}{a c + 2 \, {\left (a + c\right )} x + 4 \, x^{2}}, x\right ) \]