\[ \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a d +b c \right ) x \left (b \,x^{3}+a \right )}{6 c d \left (d \,x^{3}+c \right )^{2}}-\frac {\left (-a d +b c \right ) \left (5 a d +4 b c \right ) x}{18 c^{2} d^{2} \left (d \,x^{3}+c \right )}+\frac {\left (5 a^{2} d^{2}+2 a b c d +2 b^{2} c^{2}\right ) \ln \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )}{27 c^{\frac {8}{3}} d^{\frac {7}{3}}}-\frac {\left (5 a^{2} d^{2}+2 a b c d +2 b^{2} c^{2}\right ) \ln \left (c^{\frac {2}{3}}-c^{\frac {1}{3}} d^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right )}{54 c^{\frac {8}{3}} d^{\frac {7}{3}}}-\frac {\left (5 a^{2} d^{2}+2 a b c d +2 b^{2} c^{2}\right ) \arctan \left (\frac {\left (c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x \right ) \sqrt {3}}{3 c^{\frac {1}{3}}}\right ) \sqrt {3}}{27 c^{\frac {8}{3}} d^{\frac {7}{3}}} \]
command
int((b*x^3+a)^2/(d*x^3+c)^3,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\frac {\frac {\left (5 a^{2} d^{2}+2 a b c d -7 b^{2} c^{2}\right ) x^{4}}{18 c^{2} d}+\frac {2 \left (2 a^{2} d^{2}-a b c d -b^{2} c^{2}\right ) x}{9 d^{2} c}}{\left (d \,x^{3}+c \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (d \,\textit {\_Z}^{3}+c \right )}{\sum }\frac {\left (5 a^{2} d^{2}+2 a b c d +2 b^{2} c^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 c^{2} d^{3}}\) | \(131\) |
| default | \(\frac {\frac {\left (5 a^{2} d^{2}+2 a b c d -7 b^{2} c^{2}\right ) x^{4}}{18 c^{2} d}+\frac {2 \left (2 a^{2} d^{2}-a b c d -b^{2} c^{2}\right ) x}{9 d^{2} c}}{\left (d \,x^{3}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}+2 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right )}{9 c^{2} d^{2}}\) | \(200\) |
Maple 2021.1 output
\[ \text {hanged} \]________________________________________________________________________________________