\(\int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx\) [186]

Optimal result

Integrand size = 26, antiderivative size = 610 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=-\frac {2^{2/3} \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {2^{2/3} \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {c^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {c^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n} \] Output:

-2^(2/3)*3^(1/2)*c^(2/3)*arctan(1/3*(1-2*2^(1/3)*c^(1/3)*x^(1/3*n)/(b-(-4* 
a*c+b^2)^(1/2))^(1/3))*3^(1/2))/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^ 
(2/3)/n+2^(2/3)*3^(1/2)*c^(2/3)*arctan(1/3*(1-2*2^(1/3)*c^(1/3)*x^(1/3*n)/ 
(b+(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^ 
(1/2))^(2/3)/n+2^(2/3)*c^(2/3)*ln((b-(-4*a*c+b^2)^(1/2))^(1/3)+2^(1/3)*c^( 
1/3)*x^(1/3*n))/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(2/3)/n-2^(2/3)* 
c^(2/3)*ln((b+(-4*a*c+b^2)^(1/2))^(1/3)+2^(1/3)*c^(1/3)*x^(1/3*n))/(-4*a*c 
+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(2/3)/n-1/2*c^(2/3)*ln((b-(-4*a*c+b^2)^ 
(1/2))^(2/3)-2^(1/3)*c^(1/3)*(b-(-4*a*c+b^2)^(1/2))^(1/3)*x^(1/3*n)+2^(2/3 
)*c^(2/3)*x^(2/3*n))*2^(2/3)/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(2/ 
3)/n+1/2*c^(2/3)*ln((b+(-4*a*c+b^2)^(1/2))^(2/3)-2^(1/3)*c^(1/3)*(b+(-4*a* 
c+b^2)^(1/2))^(1/3)*x^(1/3*n)+2^(2/3)*c^(2/3)*x^(2/3*n))*2^(2/3)/(-4*a*c+b 
^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(2/3)/n
 

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 526, normalized size of antiderivative = 0.86 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\frac {c^{2/3} \left (-2 \sqrt {3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )+2 \sqrt {3} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )+2 \left (b+\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )-2 \left (b-\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )-\left (b+\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )+\left (b-\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n} \] Input:

Integrate[x^(-1 + n/3)/(a + b*x^n + c*x^(2*n)),x]
 

Output:

(c^(2/3)*(-2*Sqrt[3]*(b + Sqrt[b^2 - 4*a*c])^(2/3)*ArcTan[(1 - (2*2^(1/3)* 
c^(1/3)*x^(n/3))/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]] + 2*Sqrt[3]*(b - 
Sqrt[b^2 - 4*a*c])^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x^(n/3))/(b + Sqrt 
[b^2 - 4*a*c])^(1/3))/Sqrt[3]] + 2*(b + Sqrt[b^2 - 4*a*c])^(2/3)*Log[(b - 
Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x^(n/3)] - 2*(b - Sqrt[b^2 - 4* 
a*c])^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x^(n/3)] - 
 (b + Sqrt[b^2 - 4*a*c])^(2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3) 
*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)*x^(n/3) + 2^(2/3)*c^(2/3)*x^((2*n)/ 
3)] + (b - Sqrt[b^2 - 4*a*c])^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^ 
(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x^(n/3) + 2^(2/3)*c^(2/3)*x^(( 
2*n)/3)]))/(2^(1/3)*Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])^(2/3)*(b + S 
qrt[b^2 - 4*a*c])^(2/3)*n)
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 533, normalized size of antiderivative = 0.87, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1717, 1685, 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

\(\displaystyle \frac {3 \int \frac {1}{b x^n+c x^{2 n}+a}dx^{n/3}}{n}\)

\(\Big \downarrow \) 1685

\(\displaystyle \frac {3 \left (\frac {c \int \frac {1}{c x^n+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx^{n/3}}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {1}{c x^n+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx^{n/3}}{\sqrt {b^2-4 a c}}\right )}{n}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {3 \left (\frac {c \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x^{n/3}\right )}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{c} x^{n/3}+\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx^{n/3}}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x^{n/3}\right )}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{c} x^{n/3}+\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx^{n/3}}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\right )}{n}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {3 \left (\frac {c \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x^{n/3}\right )}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x^{n/3}\right )}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\right )}{n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (\frac {c \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x^{n/3}}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x^{n/3}}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\right )}{n}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {3 \left (\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x^{n/3}\right )}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x^{n/3}\right )}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\right )}{n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 \left (\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x^{n/3}\right )}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x^{n/3}\right )}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\right )}{n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x^{n/3}}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x^{n/3}}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\right )}{n}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3 \left (\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x^{n/3}}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}+\frac {3 \int \frac {1}{-x^{2 n/3}-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x^{n/3}}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}+\frac {3 \int \frac {1}{-x^{2 n/3}-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\right )}{n}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 \left (\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x^{n/3}}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x^{n/3}}{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2 c^{2/3} x^{2 n/3}+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx^{n/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\right )}{n}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3 \left (\frac {c \left (\frac {2\ 2^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\right )}{n}\)

Input:

Int[x^(-1 + n/3)/(a + b*x^n + c*x^(2*n)),x]
 

Output:

(3*((c*((2^(2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x^(n/ 
3)])/(3*c^(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)*c^(1/3)*x^(n/3))/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sq 
rt[3]])/c^(1/3) - Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - 
 Sqrt[b^2 - 4*a*c])^(1/3)*x^(n/3) + 2^(2/3)*c^(2/3)*x^((2*n)/3)]/(4*c^(1/3 
))))/(3*(b - Sqrt[b^2 - 4*a*c])^(2/3))))/Sqrt[b^2 - 4*a*c] - (c*((2^(2/3)* 
Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x^(n/3)])/(3*c^(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)*c^(1/3)*x^(n/3))/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/c^(1/3) - 
 Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c 
])^(1/3)*x^(n/3) + 2^(2/3)*c^(2/3)*x^((2*n)/3)]/(4*c^(1/3))))/(3*(b + Sqrt 
[b^2 - 4*a*c])^(2/3))))/Sqrt[b^2 - 4*a*c]))/n
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 750

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]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 1685

Int[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q 
= Rt[b^2 - 4*a*c, 2]}, Simp[c/q   Int[1/(b/2 - q/2 + c*x^n), x], x] - Simp[ 
c/q   Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 
2*n] && NeQ[b^2 - 4*a*c, 0]
 

rule 1717

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.42 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.43

method	result	size
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{5} c^{3} n^{6}-48 a^{4} b^{2} c^{2} n^{6}+12 a^{3} b^{4} c \,n^{6}-a^{2} b^{6} n^{6}\right ) \textit {\_Z}^{6}+\left (16 a^{2} b \,c^{2} n^{3}-8 a \,b^{3} c \,n^{3}+b^{5} n^{3}\right ) \textit {\_Z}^{3}+c^{2}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{3}}+\left (-\frac {16 n^{4} b \,a^{4} c^{2}}{2 a \,c^{2}-b^{2} c}+\frac {8 n^{4} b^{3} a^{3} c}{2 a \,c^{2}-b^{2} c}-\frac {n^{4} b^{5} a^{2}}{2 a \,c^{2}-b^{2} c}\right ) \textit {\_R}^{4}+\left (\frac {4 n \,a^{2} c^{2}}{2 a \,c^{2}-b^{2} c}-\frac {5 n \,b^{2} a c}{2 a \,c^{2}-b^{2} c}+\frac {n \,b^{4}}{2 a \,c^{2}-b^{2} c}\right ) \textit {\_R} \right )\)	\(260\)

Input:

int(x^(-1+1/3*n)/(a+b*x^n+c*x^(2*n)),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

sum(_R*ln(x^(1/3*n)+(-16/(2*a*c^2-b^2*c)*n^4*b*a^4*c^2+8/(2*a*c^2-b^2*c)*n 
^4*b^3*a^3*c-1/(2*a*c^2-b^2*c)*n^4*b^5*a^2)*_R^4+(4/(2*a*c^2-b^2*c)*n*a^2* 
c^2-5/(2*a*c^2-b^2*c)*n*b^2*a*c+1/(2*a*c^2-b^2*c)*n*b^4)*_R),_R=RootOf((64 
*a^5*c^3*n^6-48*a^4*b^2*c^2*n^6+12*a^3*b^4*c*n^6-a^2*b^6*n^6)*_Z^6+(16*a^2 
*b*c^2*n^3-8*a*b^3*c*n^3+b^5*n^3)*_Z^3+c^2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2461 vs. \(2 (465) = 930\).

Time = 0.15 (sec) , antiderivative size = 2461, normalized size of antiderivative = 4.03 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\text {Too large to display} \] Input:

integrate(x^(-1+1/3*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas")
 

Output:

-1/2*(1/2)^(1/3)*(sqrt(-3) + 1)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a* 
b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3) 
*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3)*log(-(4*(b^2*c - 2*a*c^2)*x*x 
^(1/3*n - 1) + (1/2)^(1/3)*(sqrt(-3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2)*n + (b^ 
4 - 6*a*b^2*c + 8*a^2*c^2)*n - (sqrt(-3)*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b 
*c^2)*n^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*n^4)*sqrt((b^4 - 4*a*b^ 
2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n 
^6)))*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b 
^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4* 
a^3*c)*n^3))^(1/3))/x) + 1/2*(1/2)^(1/3)*(sqrt(-3) - 1)*(((a^2*b^2 - 4*a^3 
*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a 
^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3)*log(- 
(4*(b^2*c - 2*a*c^2)*x*x^(1/3*n - 1) - (1/2)^(1/3)*(sqrt(-3)*(b^4 - 6*a*b^ 
2*c + 8*a^2*c^2)*n - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*n - (sqrt(-3)*(a^2*b^5 
- 8*a^3*b^3*c + 16*a^4*b*c^2)*n^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2) 
*n^4)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6 
*b^2*c^2 - 64*a^7*c^3)*n^6)))*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^ 
2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n 
^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3))/x) - 1/2*(1/2)^(1/3)*(sqrt(-3) 
 + 1)*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a...
 

Sympy [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{3} - 1}}{a + b x^{n} + c x^{2 n}}\, dx \] Input:

integrate(x**(-1+1/3*n)/(a+b*x**n+c*x**(2*n)),x)
 

Output:

Integral(x**(n/3 - 1)/(a + b*x**n + c*x**(2*n)), x)
 

Maxima [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \] Input:

integrate(x^(-1+1/3*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")
 

Output:

integrate(x^(1/3*n - 1)/(c*x^(2*n) + b*x^n + a), x)
 

Giac [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \] Input:

integrate(x^(-1+1/3*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")
 

Output:

integrate(x^(1/3*n - 1)/(c*x^(2*n) + b*x^n + a), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{3}-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \] Input:

int(x^(n/3 - 1)/(a + b*x^n + c*x^(2*n)),x)
 

Output:

int(x^(n/3 - 1)/(a + b*x^n + c*x^(2*n)), x)
 

Reduce [F]

\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{3}}}{x^{2 n} c x +x^{n} b x +a x}d x \] Input:

int(x^(-1+1/3*n)/(a+b*x^n+c*x^(2*n)),x)
 

Output:

int(x**(n/3)/(x**(2*n)*c*x + x**n*b*x + a*x),x)