20.63 Problem number 307

\[ \int \frac {1}{x^{11/2} \sqrt {a x^2+b x^5}} \, dx \]

Optimal antiderivative \[ -\frac {2 \sqrt {b \,x^{5}+a \,x^{2}}}{11 a \,x^{\frac {13}{2}}}+\frac {16 b \sqrt {b \,x^{5}+a \,x^{2}}}{55 a^{2} x^{\frac {7}{2}}}+\frac {16 b^{2} x^{\frac {3}{2}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \sqrt {\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right )^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}\, \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right ) \EllipticF \left (\sqrt {1-\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right )^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}, \frac {\sqrt {6}}{4}+\frac {\sqrt {2}}{4}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{165 \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right ) a^{\frac {7}{3}} \sqrt {b \,x^{5}+a \,x^{2}}\, \sqrt {\frac {b^{\frac {1}{3}} x \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}} \]

command

integrate(1/x^(11/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ -\frac {2 \, {\left (16 \, \sqrt {a} b^{2} x^{7} {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right ) - \sqrt {b x^{5} + a x^{2}} {\left (8 \, a b x^{3} - 5 \, a^{2}\right )} \sqrt {x}\right )}}{55 \, a^{3} x^{7}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\frac {\sqrt {b x^{5} + a x^{2}} \sqrt {x}}{b x^{11} + a x^{8}}, x\right ) \]