\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^2 \sqrt {d+e x} \, dx \]
Optimal antiderivative \[ -\frac {2 x \left (4 a^{2} d^{2}+4 b^{2} e^{2}-a e \left (2 b d -5 c e \right )-3 a e \left (a d -4 b e \right ) x \right ) \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}\, \sqrt {e x +d}}{105 a^{2} e^{2}}+\frac {2 x \left (a \,x^{2}+b x +c \right ) \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}\, \sqrt {e x +d}}{7 a}+\frac {\left (8 a^{3} d^{3}+8 b^{3} e^{3}-a^{2} d e \left (5 b d -16 c e \right )-a b \,e^{2} \left (5 b d +29 c e \right )\right ) x \EllipticE \left (\frac {\sqrt {\frac {b +2 a x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 a d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}\, \sqrt {e x +d}\, \sqrt {-\frac {a \left (a \,x^{2}+b x +c \right )}{-4 a c +b^{2}}}}{105 a^{3} e^{3} \left (a \,x^{2}+b x +c \right ) \sqrt {\frac {a \left (e x +d \right )}{2 a d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {2 \left (8 a^{2} d^{2}-4 b^{2} e^{2}-a e \left (b d -10 c e \right )\right ) \left (a \,d^{2}-e \left (b d -c e \right )\right ) x \EllipticF \left (\frac {\sqrt {\frac {b +2 a x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 a d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}\, \sqrt {-\frac {a \left (a \,x^{2}+b x +c \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {a \left (e x +d \right )}{2 a d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{105 a^{3} e^{3} \left (a \,x^{2}+b x +c \right ) \sqrt {e x +d}} \]
command
integrate(x^2*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (8 \, a^{4} d^{4} - 9 \, a^{3} b d^{3} e - 2 \, {\left (2 \, a^{2} b^{2} - 11 \, a^{3} c\right )} d^{2} e^{2} - {\left (9 \, a b^{3} - 41 \, a^{2} b c\right )} d e^{3} + {\left (8 \, b^{4} - 41 \, a b^{2} c + 30 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {a} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, a^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, a^{3}}, \frac {{\left (a d + {\left (3 \, a x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, a}\right ) + 3 \, {\left (8 \, a^{4} d^{3} e - 5 \, a^{3} b d^{2} e^{2} - {\left (5 \, a^{2} b^{2} - 16 \, a^{3} c\right )} d e^{3} + {\left (8 \, a b^{3} - 29 \, a^{2} b c\right )} e^{4}\right )} \sqrt {a} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, a^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, a^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, a^{3}}, \frac {{\left (a d + {\left (3 \, a x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, a}\right )\right ) + 3 \, {\left (4 \, a^{4} d^{2} x e^{2} - {\left (15 \, a^{4} x^{3} + 3 \, a^{3} b x^{2} - 2 \, {\left (2 \, a^{2} b^{2} - 5 \, a^{3} c\right )} x\right )} e^{4} - {\left (3 \, a^{4} d x^{2} + 2 \, a^{3} b d x\right )} e^{3}\right )} \sqrt {x e + d} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right )} e^{\left (-4\right )}}{315 \, a^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {e x + d} x^{2} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}, x\right ) \]