\[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-c f +d e \right ) \arctan \left (\frac {\left (2 d x +c \right ) \sqrt {3}\, \sqrt {c}}{\sqrt {4 d^{3} x^{3}+c^{3}}}\right ) \sqrt {3}}{9 c^{\frac {3}{2}} d^{2}}+\frac {2^{\frac {1}{3}} \left (c f +2 d e \right ) \left (c +2^{\frac {2}{3}} d x \right ) \EllipticF \left (\frac {2^{\frac {2}{3}} d x +c \left (1-\sqrt {3}\right )}{2^{\frac {2}{3}} d x +c \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c^{2}-2^{\frac {2}{3}} c d x +2 \,2^{\frac {1}{3}} d^{2} x^{2}}{\left (2^{\frac {2}{3}} d x +c \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{9 c \,d^{2} \sqrt {4 d^{3} x^{3}+c^{3}}\, \sqrt {\frac {c \left (c +2^{\frac {2}{3}} d x \right )}{\left (2^{\frac {2}{3}} d x +c \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((f*x+e)/(d*x+c)/(4*d^3*x^3+c^3)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {\sqrt {3} {\left (c d^{2} f - d^{3} e\right )} \sqrt {-c} \log \left (\frac {2 \, d^{6} x^{6} - 36 \, c d^{5} x^{5} - 18 \, c^{2} d^{4} x^{4} + 28 \, c^{3} d^{3} x^{3} + 18 \, c^{4} d^{2} x^{2} - c^{6} + \sqrt {3} {\left (4 \, d^{4} x^{4} - 10 \, c d^{3} x^{3} - 18 \, c^{2} d^{2} x^{2} - 8 \, c^{3} d x - c^{4}\right )} \sqrt {4 \, d^{3} x^{3} + c^{3}} \sqrt {-c}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right ) + 6 \, \sqrt {d^{3}} {\left (c^{2} f + 2 \, c d e\right )} {\rm weierstrassPInverse}\left (0, -\frac {c^{3}}{d^{3}}, x\right )}{18 \, c^{2} d^{4}}, \frac {\sqrt {3} {\left (c d^{2} f - d^{3} e\right )} \sqrt {c} \arctan \left (\frac {\sqrt {3} \sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (2 \, d^{3} x^{3} - 6 \, c d^{2} x^{2} - 6 \, c^{2} d x - c^{3}\right )} \sqrt {c}}{3 \, {\left (8 \, c d^{4} x^{4} + 4 \, c^{2} d^{3} x^{3} + 2 \, c^{4} d x + c^{5}\right )}}\right ) + 3 \, \sqrt {d^{3}} {\left (c^{2} f + 2 \, c d e\right )} {\rm weierstrassPInverse}\left (0, -\frac {c^{3}}{d^{3}}, x\right )}{9 \, c^{2} d^{4}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (f x + e\right )}}{4 \, d^{4} x^{4} + 4 \, c d^{3} x^{3} + c^{3} d x + c^{4}}, x\right ) \]