Integrand size = 29, antiderivative size = 104 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (5-14 x^3+x^6\right )}{40 x^8}+\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{1+x^3}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
1/40*(x^3+1)^(2/3)*(x^6-14*x^3+5)/x^8+1/3*3^(1/2)*arctan(3^(1/2)*x/(-x+2*( x^3+1)^(1/3)))+1/3*ln(x+(x^3+1)^(1/3))-1/6*ln(x^2-x*(x^3+1)^(1/3)+(x^3+1)^ (2/3))
Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (5-14 x^3+x^6\right )}{40 x^8}+\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{1+x^3}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
((1 + x^3)^(2/3)*(5 - 14*x^3 + x^6))/(40*x^8) + ArcTan[(Sqrt[3]*x)/(-x + 2 *(1 + x^3)^(1/3))]/Sqrt[3] + Log[x + (1 + x^3)^(1/3)]/3 - Log[x^2 - x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6
Time = 0.55 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {7276, 2009}
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.
| \(\displaystyle \int \frac {\left (x^3+1\right )^{2/3} \left (3 x^6-1\right )}{x^9 \left (2 x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {2 \left (x^3+1\right )^{2/3}}{2 x^3+1}-\frac {\left (x^3+1\right )^{2/3}}{x^3}-\frac {\left (x^3+1\right )^{2/3}}{x^9}+\frac {2 \left (x^3+1\right )^{2/3}}{x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (2 x^3+1\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^3+1}-x\right )+\frac {\left (x^3+1\right )^{5/3}}{8 x^8}-\frac {19 \left (x^3+1\right )^{5/3}}{40 x^5}+\frac {\left (x^3+1\right )^{2/3}}{2 x^2}\) |
(1 + x^3)^(2/3)/(2*x^2) + (1 + x^3)^(5/3)/(8*x^8) - (19*(1 + x^3)^(5/3))/( 40*x^5) - ArcTan[(1 - (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[1 + 2* x^3]/6 + Log[-x - (1 + x^3)^(1/3)]/2
3.15.97.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 2.54 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00
| method | result | size |
| pseudoelliptic | \(\frac {40 \ln \left (\frac {x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{8}+3 \left (x^{6}-14 x^{3}+5\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+20 x^{8} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )-\ln \left (\frac {x^{2}-x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{120 x^{8}}\) | \(104\) |
| risch | \(\frac {x^{9}-13 x^{6}-9 x^{3}+5}{40 x^{8} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {\ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 x \left (x^{3}+1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+x^{3}+1}{2 x^{3}+1}\right )}{3}-\frac {\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{2 x^{3}+1}\right )}{3}-\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{2 x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(393\) |
| trager | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (x^{6}-14 x^{3}+5\right )}{40 x^{8}}-\frac {\ln \left (\frac {63 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}+21 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+10 x^{3}-63 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+5}{2 x^{3}+1}\right )}{3}-\ln \left (\frac {63 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}+21 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+10 x^{3}-63 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+5}{2 x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {63 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-15 x \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-2 x^{3}-63 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-48 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4}{2 x^{3}+1}\right )\) | \(489\) |
1/120*(40*ln((x+(x^3+1)^(1/3))/x)*x^8+3*(x^6-14*x^3+5)*(x^3+1)^(2/3)+20*x^ 8*(2*3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^3+1)^(1/3))/x)-ln((x^2-x*(x^3+1)^( 1/3)+(x^3+1)^(2/3))/x^2)))/x^8
Time = 0.54 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.22 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=-\frac {40 \, \sqrt {3} x^{8} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} + 1\right )}}{7 \, x^{3} - 1}\right ) - 20 \, x^{8} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{3} + 1}\right ) - 3 \, {\left (x^{6} - 14 \, x^{3} + 5\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{120 \, x^{8}} \]
-1/120*(40*sqrt(3)*x^8*arctan((4*sqrt(3)*(x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*( x^3 + 1)^(2/3)*x + sqrt(3)*(x^3 + 1))/(7*x^3 - 1)) - 20*x^8*log((2*x^3 + 3 *(x^3 + 1)^(1/3)*x^2 + 3*(x^3 + 1)^(2/3)*x + 1)/(2*x^3 + 1)) - 3*(x^6 - 14 *x^3 + 5)*(x^3 + 1)^(2/3))/x^8
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \cdot \left (3 x^{6} - 1\right )}{x^{9} \cdot \left (2 x^{3} + 1\right )}\, dx \]
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} + 1\right )} x^{9}} \,d x } \]
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} + 1\right )} x^{9}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (3\,x^6-1\right )}{x^9\,\left (2\,x^3+1\right )} \,d x \]