\[ \int \frac {c+d x}{\sqrt {-a+b x^3}} \, dx \]
Optimal antiderivative \[ -\frac {2 d \sqrt {b \,x^{3}-a}}{b^{\frac {2}{3}} \left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right )}-\frac {2 \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}{-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}, 2 i-i \sqrt {3}\right ) \left (b^{\frac {1}{3}} c +a^{\frac {1}{3}} d \left (1+\sqrt {3}\right )\right ) \sqrt {\frac {a^{\frac {2}{3}}+a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right )^{2}}}\, \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) 3^{\frac {3}{4}}}{3 b^{\frac {2}{3}} \sqrt {b \,x^{3}-a}\, \sqrt {-\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right )}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right )^{2}}}}+\frac {3^{\frac {1}{4}} a^{\frac {1}{3}} d \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}{-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}, 2 i-i \sqrt {3}\right ) \sqrt {\frac {a^{\frac {2}{3}}+a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right )^{2}}}\, \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{b^{\frac {2}{3}} \sqrt {b \,x^{3}-a}\, \sqrt {-\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right )}{\left (-b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((d*x+c)/(b*x^3-a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (\sqrt {b} c {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right ) - \sqrt {b} d {\rm weierstrassZeta}\left (0, \frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right )\right )\right )}}{b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {d x + c}{\sqrt {b x^{3} - a}}, x\right ) \]