\[ \int (b+2 c x) \sqrt {d+e x} \sqrt {a+b x+c x^2} \, dx \]
Optimal antiderivative \[ \frac {4 \sqrt {e x +d}\, \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{7}-\frac {2 \left (8 c^{2} d^{2}+b^{2} e^{2}-c e \left (-10 a e +11 b d \right )-3 c e \left (-b e +2 c d \right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{105 c \,e^{2}}+\frac {2 \left (-b e +2 c d \right ) \left (4 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-2 a e +b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{105 c^{2} e^{3} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (16 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-5 a e +4 b d \right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{105 c^{2} e^{3} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*x+b)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (16 \, c^{4} d^{4} - 32 \, b c^{3} d^{3} e + {\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{2} e^{2} + {\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d e^{3} + {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 6 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 2 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d e^{3} + {\left (b^{3} c - 8 \, a b c^{2}\right )} e^{4}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (8 \, c^{4} d^{2} e^{2} - {\left (30 \, c^{4} x^{2} + 27 \, b c^{3} x - b^{2} c^{2} + 20 \, a c^{3}\right )} e^{4} - {\left (6 \, c^{4} d x + 11 \, b c^{3} d\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )} e^{\left (-4\right )}}{315 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {e x + d}, x\right ) \]