Integrand size = 26, antiderivative size = 699 \[ \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=-\frac {3 x^{-n/3}}{a n}-\frac {\sqrt {3} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} a^{4/3} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {\sqrt {3} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} a^{4/3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{\sqrt [3]{2} a^{4/3} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{\sqrt [3]{2} a^{4/3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}\right )}{2 \sqrt [3]{2} a^{4/3} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}\right )}{2 \sqrt [3]{2} a^{4/3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n} \]
-3/a/n/(x^(1/3*n))+1/2*ln(2^(1/3)*a^(1/3)/(x^(1/3*n))+(b-(-4*a*c+b^2)^(1/2 ))^(1/3))*(b+(2*a*c-b^2)/(-4*a*c+b^2)^(1/2))*2^(2/3)/a^(4/3)/n/(b-(-4*a*c+ b^2)^(1/2))^(2/3)-1/4*ln(2^(2/3)*a^(2/3)/(x^(2/3*n))-2^(1/3)*a^(1/3)*(b-(- 4*a*c+b^2)^(1/2))^(1/3)/(x^(1/3*n))+(b-(-4*a*c+b^2)^(1/2))^(2/3))*(b+(2*a* c-b^2)/(-4*a*c+b^2)^(1/2))*2^(2/3)/a^(4/3)/n/(b-(-4*a*c+b^2)^(1/2))^(2/3)- 1/2*arctan(1/3*(1-2*2^(1/3)*a^(1/3)/(x^(1/3*n))/(b-(-4*a*c+b^2)^(1/2))^(1/ 3))*3^(1/2))*3^(1/2)*(b+(2*a*c-b^2)/(-4*a*c+b^2)^(1/2))*2^(2/3)/a^(4/3)/n/ (b-(-4*a*c+b^2)^(1/2))^(2/3)+1/2*ln(2^(1/3)*a^(1/3)/(x^(1/3*n))+(b+(-4*a*c +b^2)^(1/2))^(1/3))*(b+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))*2^(2/3)/a^(4/3)/n/ (b+(-4*a*c+b^2)^(1/2))^(2/3)-1/4*ln(2^(2/3)*a^(2/3)/(x^(2/3*n))-2^(1/3)*a^ (1/3)*(b+(-4*a*c+b^2)^(1/2))^(1/3)/(x^(1/3*n))+(b+(-4*a*c+b^2)^(1/2))^(2/3 ))*(b+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))*2^(2/3)/a^(4/3)/n/(b+(-4*a*c+b^2)^( 1/2))^(2/3)-1/2*arctan(1/3*(1-2*2^(1/3)*a^(1/3)/(x^(1/3*n))/(b+(-4*a*c+b^2 )^(1/2))^(1/3))*3^(1/2))*3^(1/2)*(b+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))*2^(2/ 3)/a^(4/3)/n/(b+(-4*a*c+b^2)^(1/2))^(2/3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.18 \[ \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\frac {6 c x^{-n/3} \left (\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{3},1,\frac {2}{3},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{3},1,\frac {2}{3},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}\right )}{n} \]
(6*c*(Hypergeometric2F1[-1/3, 1, 2/3, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/ (b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + Hypergeometric2F1[-1/3, 1, 2/3, (-2* c*x^n)/(b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])))/(n*x ^(n/3))
Time = 0.99 (sec) , antiderivative size = 576, normalized size of antiderivative = 0.82, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1717, 1679, 1703, 1752, 750, 16, 27, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^{-\frac {n}{3}-1}}{a+b x^n+c x^{2 n}} \, dx\) |
| \(\Big \downarrow \) 1717 |
| \(\displaystyle -\frac {3 \int \frac {1}{b x^n+c x^{2 n}+a}dx^{-n/3}}{n}\) |
| \(\Big \downarrow \) 1679 |
| \(\displaystyle -\frac {3 \int \frac {x^{-2 n}}{a x^{-2 n}+b x^{-n}+c}dx^{-n/3}}{n}\) |
| \(\Big \downarrow \) 1703 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\int \frac {b x^{-n}+c}{a x^{-2 n}+b x^{-n}+c}dx^{-n/3}}{a}\right )}{n}\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{a x^{-n}+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx^{-n/3}+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \int \frac {1}{a x^{-n}+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx^{-n/3}}{a}\right )}{n}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{a} x^{-n/3}\right )}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{a} x^{-n/3}+\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx^{-n/3}}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{a} x^{-n/3}\right )}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{a} x^{-n/3}+\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx^{-n/3}}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}\right )}{n}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{a} x^{-n/3}\right )}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{a} x^{-n/3}\right )}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}\right )}{n}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{a} x^{-n/3}}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{a} x^{-n/3}}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}\right )}{n}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}+\sqrt [3]{b-\sqrt {b^2-4 a c}}\right )}{3 \sqrt [3]{a} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{a} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{a} x^{-n/3}\right )}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{4 \sqrt [3]{a}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}+\sqrt [3]{b+\sqrt {b^2-4 a c}}\right )}{3 \sqrt [3]{a} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b+\sqrt {b^2-4 a c}} \int \frac {1}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{a} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{a} x^{-n/3}\right )}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{4 \sqrt [3]{a}}\right )}{3 \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{a}\right )}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}+\sqrt [3]{b-\sqrt {b^2-4 a c}}\right )}{3 \sqrt [3]{a} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{a} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{a} x^{-n/3}\right )}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{4 \sqrt [3]{a}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}+\sqrt [3]{b+\sqrt {b^2-4 a c}}\right )}{3 \sqrt [3]{a} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b+\sqrt {b^2-4 a c}} \int \frac {1}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{a} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{a} x^{-n/3}\right )}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{4 \sqrt [3]{a}}\right )}{3 \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{a}\right )}{n}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{a} x^{-n/3}}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{a} x^{-n/3}}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}\right )}{n}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{a} x^{-n/3}}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}+\frac {3 \int \frac {1}{-x^{-2 n/3}-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{a}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{a} x^{-n/3}}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}+\frac {3 \int \frac {1}{-x^{-2 n/3}-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{a}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}\right )}{n}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{a} x^{-n/3}}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{a}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{a} x^{-n/3}}{2 a^{2/3} x^{-2 n/3}-2^{2/3} \sqrt [3]{a} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{-n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{-n/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{a}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}\right )}{n}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {3 \left (\frac {x^{-n/3}}{a}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (-\frac {\log \left (2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} x^{-n/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}\right )}{4 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{a}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \left (-\frac {\log \left (2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} x^{-n/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}\right )}{4 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{a}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{3 \sqrt [3]{a} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}\right )}{n}\) |
(-3*(1/(a*x^(n/3)) - (((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*((2^(2/3)*Log [(b - Sqrt[b^2 - 4*a*c])^(1/3) + (2^(1/3)*a^(1/3))/x^(n/3)])/(3*a^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) + (2*2^(2/3)*(-1/2*(Sqrt[3]*ArcTan[(1 - (2*2^ (1/3)*a^(1/3))/((b - Sqrt[b^2 - 4*a*c])^(1/3)*x^(n/3)))/Sqrt[3]])/a^(1/3) - Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) + (2^(2/3)*a^(2/3))/x^((2*n)/3) - (2^( 1/3)*a^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3))/x^(n/3)]/(4*a^(1/3))))/(3*(b - Sqrt[b^2 - 4*a*c])^(2/3))))/2 + ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*(( 2^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + (2^(1/3)*a^(1/3))/x^(n/3)])/(3 *a^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) + (2*2^(2/3)*(-1/2*(Sqrt[3]*ArcTan [(1 - (2*2^(1/3)*a^(1/3))/((b + Sqrt[b^2 - 4*a*c])^(1/3)*x^(n/3)))/Sqrt[3] ])/a^(1/3) - Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) + (2^(2/3)*a^(2/3))/x^((2*n )/3) - (2^(1/3)*a^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3))/x^(n/3)]/(4*a^(1/3) )))/(3*(b + Sqrt[b^2 - 4*a*c])^(2/3))))/2)/a))/n
3.6.60.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^( 2*n*p)*(c + b/x^n + a/x^(2*n))^p, x] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && LtQ[n, 0] && IntegerQ[p]
Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x _Symbol] :> Simp[d^(2*n - 1)*(d*x)^(m - 2*n + 1)*((a + b*x^n + c*x^(2*n))^( p + 1)/(c*(m + 2*n*p + 1))), x] - Simp[d^(2*n)/(c*(m + 2*n*p + 1)) Int[(d *x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x ^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && N eQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1, 0 ] && IntegerQ[p]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/(m + 1) Subst[Int[(a + b*x^Simplify[n/(m + 1)] + c*x^Simplify[ 2*(n/(m + 1))])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && ! IntegerQ[n]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.04 (sec) , antiderivative size = 534, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(-\frac {3 x^{-\frac {n}{3}}}{a n}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{7} c^{3} n^{6}-48 a^{6} b^{2} c^{2} n^{6}+12 a^{5} b^{4} c \,n^{6}-a^{4} b^{6} n^{6}\right ) \textit {\_Z}^{6}+\left (-32 a^{3} b \,c^{3} n^{3}+32 a^{2} b^{3} c^{2} n^{3}-10 a \,b^{5} c \,n^{3}+b^{7} n^{3}\right ) \textit {\_Z}^{3}+c^{4}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{3}}+\left (-\frac {64 n^{5} a^{8} c^{4}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}+\frac {112 n^{5} b^{2} a^{7} c^{3}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}-\frac {60 n^{5} b^{4} a^{6} c^{2}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}+\frac {13 n^{5} b^{6} a^{5} c}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}-\frac {n^{5} b^{8} a^{4}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}\right ) \textit {\_R}^{5}+\left (\frac {28 n^{2} b \,a^{4} c^{4}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}-\frac {63 n^{2} b^{3} a^{3} c^{3}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}+\frac {42 n^{2} b^{5} a^{2} c^{2}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}-\frac {11 n^{2} b^{7} a c}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}+\frac {n^{2} b^{9}}{2 a^{2} c^{5}-4 a \,b^{2} c^{4}+b^{4} c^{3}}\right ) \textit {\_R}^{2}\right )\right )\) | \(534\) |
-3/a/n/(x^(1/3*n))+sum(_R*ln(x^(1/3*n)+(-64/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3 )*n^5*a^8*c^4+112/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^5*b^2*a^7*c^3-60/(2*a^ 2*c^5-4*a*b^2*c^4+b^4*c^3)*n^5*b^4*a^6*c^2+13/(2*a^2*c^5-4*a*b^2*c^4+b^4*c ^3)*n^5*b^6*a^5*c-1/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^5*b^8*a^4)*_R^5+(28/ (2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^2*b*a^4*c^4-63/(2*a^2*c^5-4*a*b^2*c^4+b^ 4*c^3)*n^2*b^3*a^3*c^3+42/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^2*b^5*a^2*c^2- 11/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^2*b^7*a*c+1/(2*a^2*c^5-4*a*b^2*c^4+b^ 4*c^3)*n^2*b^9)*_R^2),_R=RootOf((64*a^7*c^3*n^6-48*a^6*b^2*c^2*n^6+12*a^5* b^4*c*n^6-a^4*b^6*n^6)*_Z^6+(-32*a^3*b*c^3*n^3+32*a^2*b^3*c^2*n^3-10*a*b^5 *c*n^3+b^7*n^3)*_Z^3+c^4))
Leaf count of result is larger than twice the leaf count of optimal. 3155 vs. \(2 (567) = 1134\).
Time = 0.37 (sec) , antiderivative size = 3155, normalized size of antiderivative = 4.51 \[ \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\text {Too large to display} \]
1/2*(2*(1/2)^(1/3)*a*n*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 2 0*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48* a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^ 3))^(1/3)*log((2*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*x*x^(-1/3*n - 1) + (1/2 )^(1/3)*((a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*n^4*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - (b^6 - 8*a*b^4*c + 18*a^2*b^2*c^2 - 8*a^3*c^3)*n)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^ 4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2 *c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3 ))/x) + 2*(1/2)^(1/3)*a*n*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c )*n^3))^(1/3)*log((2*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*x*x^(-1/3*n - 1) - (1/2)^(1/3)*((a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*n^4*sqrt((b^8 - 8*a*b^ 6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4* c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + (b^6 - 8*a*b^4*c + 18*a^2*b^2*c ^2 - 8*a^3*c^3)*n)*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a ^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^1 0*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^...
\[ \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{- \frac {n}{3} - 1}}{a + b x^{n} + c x^{2 n}}\, dx \]
\[ \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
-3/(a*n*x^(1/3*n)) - integrate((c*x^(5/3*n) + b*x^(2/3*n))/(a*c*x*x^(2*n) + a*b*x*x^n + a^2*x), x)
\[ \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
Timed out. \[ \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {1}{x^{\frac {n}{3}+1}\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \]