Integrand size = 27, antiderivative size = 85 \[ \int \frac {-1+2 x^3}{\left (1+x+x^3\right ) \sqrt [3]{x^2+x^5}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x^2+x^5}}\right )-\log \left (x+\sqrt [3]{x^2+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{x^2+x^5}+\left (x^2+x^5\right )^{2/3}\right ) \]
-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^5+x^2)^(1/3)))-ln(x+(x^5+x^2)^(1/3))+1/ 2*ln(x^2-x*(x^5+x^2)^(1/3)+(x^5+x^2)^(2/3))
Time = 7.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \frac {-1+2 x^3}{\left (1+x+x^3\right ) \sqrt [3]{x^2+x^5}} \, dx=\frac {x^{2/3} \sqrt [3]{1+x^3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{1+x^3}}\right )-2 \log \left (\sqrt [3]{x}+\sqrt [3]{1+x^3}\right )+\log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right )}{2 \sqrt [3]{x^2+x^5}} \]
(x^(2/3)*(1 + x^3)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) - 2* (1 + x^3)^(1/3))] - 2*Log[x^(1/3) + (1 + x^3)^(1/3)] + Log[x^(2/3) - x^(1/ 3)*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]))/(2*(x^2 + x^5)^(1/3))
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.
| \(\displaystyle \int \frac {2 x^3-1}{\left (x^3+x+1\right ) \sqrt [3]{x^5+x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x^3+1} \int -\frac {1-2 x^3}{x^{2/3} \sqrt [3]{x^3+1} \left (x^3+x+1\right )}dx}{\sqrt [3]{x^5+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x^3+1} \int \frac {1-2 x^3}{x^{2/3} \sqrt [3]{x^3+1} \left (x^3+x+1\right )}dx}{\sqrt [3]{x^5+x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^3+1} \int \frac {1-2 x^3}{\sqrt [3]{x^3+1} \left (x^3+x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^5+x^2}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^3+1} \int \left (\frac {2 x+3}{\sqrt [3]{x^3+1} \left (x^3+x+1\right )}-\frac {2}{\sqrt [3]{x^3+1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^5+x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^3+1} \left (3 \int \frac {1}{\sqrt [3]{x^3+1} \left (x^3+x+1\right )}d\sqrt [3]{x}+2 \int \frac {x}{\sqrt [3]{x^3+1} \left (x^3+x+1\right )}d\sqrt [3]{x}-2 \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (\frac {1}{9},\frac {1}{3},\frac {10}{9},-x^3\right )\right )}{\sqrt [3]{x^5+x^2}}\) |
3.12.43.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 5.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93
| method | result | size |
| pseudoelliptic | \(-\ln \left (\frac {x +\left (x^{5}+x^{2}\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {x^{2}-x \left (x^{5}+x^{2}\right )^{\frac {1}{3}}+\left (x^{5}+x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (x^{5}+x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\) | \(79\) |
| trager | \(-\ln \left (\frac {158 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}-217 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-316 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-198 x^{4}+158 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -474 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{5}+x^{2}\right )^{\frac {2}{3}}-771 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{5}+x^{2}\right )^{\frac {1}{3}} x -40 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-217 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -297 \left (x^{5}+x^{2}\right )^{\frac {2}{3}}+474 x \left (x^{5}+x^{2}\right )^{\frac {1}{3}}+99 x^{2}-198 x}{x \left (x^{3}+x +1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}+257 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-198 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+158 x^{4}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +474 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{5}+x^{2}\right )^{\frac {2}{3}}-297 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{5}+x^{2}\right )^{\frac {1}{3}} x -217 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+257 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -771 \left (x^{5}+x^{2}\right )^{\frac {2}{3}}-474 x \left (x^{5}+x^{2}\right )^{\frac {1}{3}}+158 x^{2}+158 x}{x \left (x^{3}+x +1\right )}\right )\) | \(368\) |
-ln((x+(x^5+x^2)^(1/3))/x)+1/2*ln((x^2-x*(x^5+x^2)^(1/3)+(x^5+x^2)^(2/3))/ x^2)-3^(1/2)*arctan(1/3*(-2*(x^5+x^2)^(1/3)+x)*3^(1/2)/x)
Time = 0.87 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.27 \[ \int \frac {-1+2 x^3}{\left (1+x+x^3\right ) \sqrt [3]{x^2+x^5}} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{5} + x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{4} + x^{2} + x\right )} + 2 \, \sqrt {3} {\left (x^{5} + x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{4} - x^{2} + x\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{4} + x^{2} + 3 \, {\left (x^{5} + x^{2}\right )}^{\frac {1}{3}} x + x + 3 \, {\left (x^{5} + x^{2}\right )}^{\frac {2}{3}}}{x^{4} + x^{2} + x}\right ) \]
-sqrt(3)*arctan(1/3*(2*sqrt(3)*(x^5 + x^2)^(1/3)*x + sqrt(3)*(x^4 + x^2 + x) + 2*sqrt(3)*(x^5 + x^2)^(2/3))/(x^4 - x^2 + x)) - 1/2*log((x^4 + x^2 + 3*(x^5 + x^2)^(1/3)*x + x + 3*(x^5 + x^2)^(2/3))/(x^4 + x^2 + x))
\[ \int \frac {-1+2 x^3}{\left (1+x+x^3\right ) \sqrt [3]{x^2+x^5}} \, dx=\int \frac {2 x^{3} - 1}{\sqrt [3]{x^{2} \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{3} + x + 1\right )}\, dx \]
\[ \int \frac {-1+2 x^3}{\left (1+x+x^3\right ) \sqrt [3]{x^2+x^5}} \, dx=\int { \frac {2 \, x^{3} - 1}{{\left (x^{5} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x + 1\right )}} \,d x } \]
\[ \int \frac {-1+2 x^3}{\left (1+x+x^3\right ) \sqrt [3]{x^2+x^5}} \, dx=\int { \frac {2 \, x^{3} - 1}{{\left (x^{5} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-1+2 x^3}{\left (1+x+x^3\right ) \sqrt [3]{x^2+x^5}} \, dx=\int \frac {2\,x^3-1}{{\left (x^5+x^2\right )}^{1/3}\,\left (x^3+x+1\right )} \,d x \]