\[ \int \frac {(a+b x)^2 \left (A+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (4 C \left (2 a d f h -3 b \left (c f h +d e h +d f g \right )\right ) \left (a d f h -2 b \left (c f h +d e h +d f g \right )\right )+5 b d f h \left (7 A b d f h -C \left (5 b \left (c e h +c f g +d e g \right )+2 a \left (c f h +d e h +d f g \right )\right )\right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}{105 d^{3} f^{3} h^{3}}+\frac {4 C \left (2 a d f h -3 b \left (c f h +d e h +d f g \right )\right ) \left (b x +a \right ) \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}{35 d^{2} f^{2} h^{2}}+\frac {2 C \left (b x +a \right )^{2} \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}{7 d f h}-\frac {4 \left (35 a^{2} C \,d^{2} f^{2} h^{2} \left (c f h +d e h +d f g \right )-7 a b d f h \left (15 A \,d^{2} f^{2} h^{2}+C \left (8 c^{2} f^{2} h^{2}+7 c d f h \left (e h +f g \right )+d^{2} \left (8 e^{2} h^{2}+7 e f g h +8 f^{2} g^{2}\right )\right )\right )+b^{2} \left (35 A \,d^{2} f^{2} h^{2} \left (c f h +d e h +d f g \right )+2 C \left (12 c^{3} f^{3} h^{3}+10 c^{2} d \,f^{2} h^{2} \left (e h +f g \right )+c \,d^{2} f h \left (10 e^{2} h^{2}+9 e f g h +10 f^{2} g^{2}\right )+2 d^{3} \left (6 e^{3} h^{3}+5 e^{2} f g \,h^{2}+5 e \,f^{2} g^{2} h +6 f^{3} g^{3}\right )\right )\right )\right ) \EllipticE \left (\frac {\sqrt {f}\, \sqrt {d x +c}}{\sqrt {c f -d e}}, \sqrt {\frac {\left (-c f +d e \right ) h}{f \left (-c h +d g \right )}}\right ) \sqrt {c f -d e}\, \sqrt {\frac {d \left (f x +e \right )}{-c f +d e}}\, \sqrt {h x +g}}{105 d^{4} f^{\frac {7}{2}} h^{4} \sqrt {f x +e}\, \sqrt {\frac {d \left (h x +g \right )}{-c h +d g}}}+\frac {2 \left (35 a^{2} d^{2} f^{2} h^{2} \left (3 A d f \,h^{2}+C \left (c h \left (-e h +f g \right )+d g \left (e h +2 f g \right )\right )\right )-14 a b d f h \left (15 A \,d^{2} f^{2} g \,h^{2}+C \left (4 c^{2} f \,h^{2} \left (-e h +f g \right )+c d h \left (-4 e^{2} h^{2}+e f g h +3 f^{2} g^{2}\right )+d^{2} g \left (4 e^{2} h^{2}+3 e f g h +8 f^{2} g^{2}\right )\right )\right )+b^{2} \left (35 A \,d^{2} f^{2} h^{2} \left (c h \left (-e h +f g \right )+d g \left (e h +2 f g \right )\right )+C \left (24 c^{3} f^{2} h^{3} \left (-e h +f g \right )+c^{2} d f \,h^{2} \left (-23 e^{2} h^{2}+6 e f g h +17 f^{2} g^{2}\right )+2 c \,d^{2} h \left (-12 e^{3} h^{3}+3 e^{2} f g \,h^{2}+e \,f^{2} g^{2} h +8 f^{3} g^{3}\right )+d^{3} g \left (24 e^{3} h^{3}+17 e^{2} f g \,h^{2}+16 e \,f^{2} g^{2} h +48 f^{3} g^{3}\right )\right )\right )\right ) \EllipticF \left (\frac {\sqrt {f}\, \sqrt {d x +c}}{\sqrt {c f -d e}}, \sqrt {\frac {\left (-c f +d e \right ) h}{f \left (-c h +d g \right )}}\right ) \sqrt {c f -d e}\, \sqrt {\frac {d \left (f x +e \right )}{-c f +d e}}\, \sqrt {\frac {d \left (h x +g \right )}{-c h +d g}}}{105 d^{4} f^{\frac {7}{2}} h^{4} \sqrt {f x +e}\, \sqrt {h x +g}} \]
command
integrate((b*x+a)^2*(C*x^2+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (3 \, {\left (15 \, C b^{2} d^{4} f^{4} h^{4} x^{2} + 24 \, C b^{2} d^{4} f^{4} g^{2} h^{2} + 24 \, C b^{2} d^{4} f^{2} h^{4} e^{2} + {\left (23 \, C b^{2} c d^{3} - 56 \, C a b d^{4}\right )} f^{4} g h^{3} + {\left (24 \, C b^{2} c^{2} d^{2} - 56 \, C a b c d^{3} + 35 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{4} h^{4} - 6 \, {\left (3 \, C b^{2} d^{4} f^{4} g h^{3} + {\left (3 \, C b^{2} c d^{3} - 7 \, C a b d^{4}\right )} f^{4} h^{4}\right )} x - {\left (18 \, C b^{2} d^{4} f^{3} h^{4} x - 23 \, C b^{2} d^{4} f^{3} g h^{3} - {\left (23 \, C b^{2} c d^{3} - 56 \, C a b d^{4}\right )} f^{3} h^{4}\right )} e\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g} + {\left (48 \, C b^{2} d^{4} f^{4} g^{4} + 48 \, C b^{2} d^{4} h^{4} e^{4} + 16 \, {\left (C b^{2} c d^{3} - 7 \, C a b d^{4}\right )} f^{4} g^{3} h + {\left (11 \, C b^{2} c^{2} d^{2} - 42 \, C a b c d^{3} + 70 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{4} g^{2} h^{2} + {\left (16 \, C b^{2} c^{3} d - 42 \, C a b c^{2} d^{2} - 210 \, A a b d^{4} + 35 \, {\left (C a^{2} + A b^{2}\right )} c d^{3}\right )} f^{4} g h^{3} + {\left (48 \, C b^{2} c^{4} - 112 \, C a b c^{3} d - 210 \, A a b c d^{3} + 315 \, A a^{2} d^{4} + 70 \, {\left (C a^{2} + A b^{2}\right )} c^{2} d^{2}\right )} f^{4} h^{4} + 16 \, {\left (C b^{2} d^{4} f g h^{3} + {\left (C b^{2} c d^{3} - 7 \, C a b d^{4}\right )} f h^{4}\right )} e^{3} + {\left (11 \, C b^{2} d^{4} f^{2} g^{2} h^{2} + 14 \, {\left (C b^{2} c d^{3} - 3 \, C a b d^{4}\right )} f^{2} g h^{3} + {\left (11 \, C b^{2} c^{2} d^{2} - 42 \, C a b c d^{3} + 70 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{2} h^{4}\right )} e^{2} + {\left (16 \, C b^{2} d^{4} f^{3} g^{3} h + 14 \, {\left (C b^{2} c d^{3} - 3 \, C a b d^{4}\right )} f^{3} g^{2} h^{2} + 7 \, {\left (2 \, C b^{2} c^{2} d^{2} - 6 \, C a b c d^{3} + 5 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{3} g h^{3} + {\left (16 \, C b^{2} c^{3} d - 42 \, C a b c^{2} d^{2} - 210 \, A a b d^{4} + 35 \, {\left (C a^{2} + A b^{2}\right )} c d^{3}\right )} f^{3} h^{4}\right )} e\right )} \sqrt {d f h} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + c f h + d h e}{3 \, d f h}\right ) + 6 \, {\left (24 \, C b^{2} d^{4} f^{4} g^{3} h + 24 \, C b^{2} d^{4} f h^{4} e^{3} + 4 \, {\left (5 \, C b^{2} c d^{3} - 14 \, C a b d^{4}\right )} f^{4} g^{2} h^{2} + {\left (20 \, C b^{2} c^{2} d^{2} - 49 \, C a b c d^{3} + 35 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{4} g h^{3} + {\left (24 \, C b^{2} c^{3} d - 56 \, C a b c^{2} d^{2} - 105 \, A a b d^{4} + 35 \, {\left (C a^{2} + A b^{2}\right )} c d^{3}\right )} f^{4} h^{4} + 4 \, {\left (5 \, C b^{2} d^{4} f^{2} g h^{3} + {\left (5 \, C b^{2} c d^{3} - 14 \, C a b d^{4}\right )} f^{2} h^{4}\right )} e^{2} + {\left (20 \, C b^{2} d^{4} f^{3} g^{2} h^{2} + {\left (18 \, C b^{2} c d^{3} - 49 \, C a b d^{4}\right )} f^{3} g h^{3} + {\left (20 \, C b^{2} c^{2} d^{2} - 49 \, C a b c d^{3} + 35 \, {\left (C a^{2} + A b^{2}\right )} d^{4}\right )} f^{3} h^{4}\right )} e\right )} \sqrt {d f h} {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - c d f^{2} g h + c^{2} f^{2} h^{2} + d^{2} h^{2} e^{2} - {\left (d^{2} f g h + c d f h^{2}\right )} e\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, c d^{2} f^{3} g^{2} h - 3 \, c^{2} d f^{3} g h^{2} + 2 \, c^{3} f^{3} h^{3} + 2 \, d^{3} h^{3} e^{3} - 3 \, {\left (d^{3} f g h^{2} + c d^{2} f h^{3}\right )} e^{2} - 3 \, {\left (d^{3} f^{2} g^{2} h - 4 \, c d^{2} f^{2} g h^{2} + c^{2} d f^{2} h^{3}\right )} e\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + c f h + d h e}{3 \, d f h}\right )\right )\right )}}{315 \, d^{5} f^{5} h^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C b^{2} x^{4} + 2 \, C a b x^{3} + 2 \, A a b x + A a^{2} + {\left (C a^{2} + A b^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{d f h x^{3} + c e g + {\left (d f g + {\left (d e + c f\right )} h\right )} x^{2} + {\left (c e h + {\left (d e + c f\right )} g\right )} x}, x\right ) \]