\[ \int (c x)^{7/2} \left (a+b x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (c x \right )^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 c}+\frac {8 a^{2} c \left (c x \right )^{\frac {5}{2}} \sqrt {b \,x^{2}+a}}{385 b}+\frac {4 a \left (c x \right )^{\frac {9}{2}} \sqrt {b \,x^{2}+a}}{55 c}-\frac {8 a^{3} c^{3} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}{231 b^{2}}+\frac {4 a^{\frac {15}{4}} c^{\frac {7}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{231 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ) b^{\frac {9}{4}} \sqrt {b \,x^{2}+a}} \]
command
integrate((c*x)^(7/2)*(b*x^2+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (20 \, \sqrt {b c} a^{4} c^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (77 \, b^{4} c^{3} x^{6} + 119 \, a b^{3} c^{3} x^{4} + 12 \, a^{2} b^{2} c^{3} x^{2} - 20 \, a^{3} b c^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{1155 \, b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b c^{3} x^{5} + a c^{3} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c x}, x\right ) \]