\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx \]
Optimal antiderivative \[ \frac {5 \left (-5+2 x \right )^{\frac {3}{2}} \left (1+4 x \right )^{\frac {3}{2}} \sqrt {2-3 x}}{28}+\frac {72479 \EllipticF \left (\frac {\sqrt {33}\, \sqrt {1+4 x}}{11}, \frac {\sqrt {3}}{3}\right ) \sqrt {66}\, \sqrt {5-2 x}}{4536 \sqrt {-5+2 x}}+\frac {136 \left (1+4 x \right )^{\frac {3}{2}} \sqrt {2-3 x}\, \sqrt {-5+2 x}}{105}-\frac {954811 \EllipticE \left (\frac {2 \sqrt {2-3 x}\, \sqrt {11}}{11}, \frac {i \sqrt {2}}{2}\right ) \sqrt {11}\, \sqrt {-5+2 x}}{22680 \sqrt {5-2 x}}-\frac {20911 \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}{3780} \]
command
integrate((7+5*x)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{1890} \, {\left (2700 \, x^{2} + 3717 \, x - 9695\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}, x\right ) \]