\[ \int \sqrt {c x} \left (a+b x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (c x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 c}+\frac {4 a \left (c x \right )^{\frac {3}{2}} \sqrt {b \,x^{2}+a}}{15 c}+\frac {8 a^{2} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}{15 \sqrt {b}\, \left (\sqrt {a}+x \sqrt {b}\right )}-\frac {8 a^{\frac {9}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {c}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ) b^{\frac {3}{4}} \sqrt {b \,x^{2}+a}}+\frac {4 a^{\frac {9}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {c}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ) b^{\frac {3}{4}} \sqrt {b \,x^{2}+a}} \]
command
integrate((c*x)^(1/2)*(b*x^2+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (12 \, \sqrt {b c} a^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (5 \, b^{2} x^{3} + 11 \, a b x\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{45 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {c x}, x\right ) \]