\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3 \, dx \]
Optimal antiderivative \[ \frac {522167393 \EllipticF \left (\frac {\sqrt {33}\, \sqrt {1+4 x}}{11}, \frac {\sqrt {3}}{3}\right ) \sqrt {66}\, \sqrt {5-2 x}}{139968 \sqrt {-5+2 x}}-\frac {6489123157 \EllipticE \left (\frac {2 \sqrt {2-3 x}\, \sqrt {11}}{11}, \frac {i \sqrt {2}}{2}\right ) \sqrt {11}\, \sqrt {-5+2 x}}{699840 \sqrt {5-2 x}}-\frac {1182926269 \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}{1603800}-\frac {12243139 \left (7+5 x \right ) \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}{356400}-\frac {17561 \left (7+5 x \right )^{2} \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}{8910}-\frac {427 \left (7+5 x \right )^{3} \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}{2970}+\frac {2 \left (7+5 x \right )^{4} \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}{55} \]
command
integrate((7+5*x)^3*(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
\[ \frac {1}{641520} \, {\left (14580000 \, x^{4} + 70119000 \, x^{3} + 91429200 \, x^{2} - 106456131 \, x - 665014315\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}, x\right ) \]