\[ \int (d+e x) \sqrt {a+c x^4} \, dx \]
Optimal antiderivative \[ \frac {a e \arctanh \left (\frac {x^{2} \sqrt {c}}{\sqrt {c \,x^{4}+a}}\right )}{4 \sqrt {c}}+\frac {d x \sqrt {c \,x^{4}+a}}{3}+\frac {e \,x^{2} \sqrt {c \,x^{4}+a}}{4}+\frac {a^{\frac {3}{4}} d \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x^{2} \sqrt {c}\right ) \sqrt {\frac {c \,x^{4}+a}{\left (\sqrt {a}+x^{2} \sqrt {c}\right )^{2}}}}{3 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {1}{4}} \sqrt {c \,x^{4}+a}} \]
command
integrate((e*x+d)*(c*x^4+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {16 \, c^{\frac {3}{2}} d \left (-\frac {a}{c}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}, -1\right ) + 3 \, a \sqrt {c} e \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) + 2 \, \sqrt {c x^{4} + a} {\left (3 \, c x^{2} e + 4 \, c d x\right )}}{24 \, c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {c x^{4} + a} {\left (e x + d\right )}, x\right ) \]