\[ \int \frac {x^3 \sqrt {c+d x^3}}{4 c+d x^3} \, dx \]
Optimal antiderivative \[ \frac {x^{4} F_{1}\left (\frac {4}{3}, -\frac {1}{2}, 1, \frac {7}{3}, -\frac {d \,x^{3}}{c}, -\frac {d \,x^{3}}{4 c}\right ) \sqrt {d \,x^{3}+c}}{16 c \sqrt {1+\frac {d \,x^{3}}{c}}} \]
command
integrate(x^3*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {\sqrt {d x^{3} + c} x^{3}}{d x^{3} + 4 \, c}, x\right ) \]