\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x} \, dx \]
Optimal antiderivative \[ \frac {2 \left (233 a^{3} d^{3}+48 b^{3} e^{3}+a b \,e^{2} \left (67 b d -157 c e \right )+4 a^{2} d e \left (18 b d -37 c e \right )\right ) x \left (e x +d \right )^{\frac {3}{2}} \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}}{3465 a^{3} e^{4}}-\frac {2 \left (29 a^{2} d^{2}+8 b^{2} e^{2}+a e \left (19 b d -18 c e \right )\right ) x \left (e x +d \right )^{\frac {5}{2}} \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}}{693 a^{2} e^{4}}+\frac {2 \left (a d +b e \right ) x \left (e x +d \right )^{\frac {7}{2}} \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}}{99 a \,e^{4}}-\frac {2 \left (187 a^{4} d^{4}+64 b^{4} e^{4}+4 a \,b^{2} e^{3} \left (7 b d -69 c e \right )-4 a^{3} d^{2} e \left (2 b d +3 c e \right )+3 a^{2} e^{2} \left (3 b^{2} d^{2}-29 b c d e +50 c^{2} e^{2}\right )\right ) x \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}\, \sqrt {e x +d}}{3465 a^{4} e^{4}}+\frac {2 x^{5} \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}\, \sqrt {e x +d}}{11}+\frac {\left (128 a^{5} d^{5}+128 b^{5} e^{5}-4 a^{4} d^{3} e \left (14 b d -27 c e \right )-8 a \,b^{3} e^{4} \left (7 b d +87 c e \right )-a^{2} b \,e^{3} \left (37 b^{2} d^{2}-258 b c d e -771 c^{2} e^{2}\right )-a^{3} d \,e^{2} \left (37 b^{2} d^{2}-135 b c d e +156 c^{2} e^{2}\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}}}}{3465 a^{5} e^{5} \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 (a \,d^{2}-e \left (b d -c e \right )\right ) \left (128 a^{4} d^{4}-64 b^{4} e^{4}-4 a \,b^{2} e^{3} \left (7 b d -69 c e \right )+4 a^{3} d^{2} e \left (2 b d +3 c e \right )-3 a^{2} e^{2} \left (3 b^{2} d^{2}-29 b c d e +50 c^{2} e^{2}\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 )}}}{3465 a^{5} e^{5} \left (a \,x^{2}+b x +c \right ) \sqrt {e x +d}} \]
command
integrate(x^4*(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 (128 \, a^{6} d^{6} - 120 \, a^{5} b d^{5} e - 3 \, {\left (11 \, a^{4} b^{2} - 68 \, a^{5} c\right )} d^{4} e^{2} - {\left (20 \, a^{3} b^{3} - 87 \, a^{4} b c\right )} d^{3} e^{3} - 3 \, {\left (11 \, a^{2} b^{4} - 53 \, a^{3} b^{2} c + 34 \, a^{4} c^{2}\right )} d^{2} e^{4} - 3 \, {\left (40 \, a b^{5} - 246 \, a^{2} b^{3} c + 329 \, a^{3} b c^{2}\right )} d e^{5} + {\left (128 \, b^{6} - 888 \, a b^{4} c + 1599 \, a^{2} b^{2} c^{2} - 450 \, a^{3} c^{3}\right )} e^{6}\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 (128 \, a^{6} d^{5} e - 56 \, a^{5} b d^{4} e^{2} - {\left (37 \, a^{4} b^{2} - 108 \, a^{5} c\right )} d^{3} e^{3} - {\left (37 \, a^{3} b^{3} - 135 \, a^{4} b c\right )} d^{2} e^{4} - 2 \, {\left (28 \, a^{2} b^{4} - 129 \, a^{3} b^{2} c + 78 \, a^{4} c^{2}\right )} d e^{5} + {\left (128 \, a b^{5} - 696 \, a^{2} b^{3} c + 771 \, a^{3} b c^{2}\right )} e^{6}\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 (64 \, a^{6} d^{4} x e^{2} - {\left (315 \, a^{6} x^{5} + 35 \, a^{5} b x^{4} - 10 \, {\left (4 \, a^{4} b^{2} - 9 \, a^{5} c\right )} x^{3} + {\left (48 \, a^{3} b^{3} - 157 \, a^{4} b c\right )} x^{2} - 2 \, {\left (32 \, a^{2} b^{4} - 138 \, a^{3} b^{2} c + 75 \, a^{4} c^{2}\right )} x\right )} e^{6} - {\left (35 \, a^{6} d x^{4} + 10 \, a^{5} b d x^{3} - {\left (13 \, a^{4} b^{2} - 32 \, a^{5} c\right )} d x^{2} + 10 \, {\left (2 \, a^{3} b^{3} - 7 \, a^{4} b c\right )} d x\right )} e^{5} + {\left (40 \, a^{6} d^{2} x^{3} + 13 \, a^{5} b d^{2} x^{2} - 2 \, {\left (9 \, a^{4} b^{2} - 23 \, a^{5} c\right )} d^{2} x\right )} e^{4} - 4 \, {\left (12 \, a^{6} d^{3} x^{2} + 5 \, a^{5} b d^{3} x\right )} e^{3}\right )} \sqrt {x e + d} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right )} e^{\left (-6\right )}}{10395 \, a^{6}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {e x + d} x^{4} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}, x\right ) \]