Integrand size = 29, antiderivative size = 93 \[ \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x^2+x^5}}\right )-\log \left (x+\sqrt [3]{-x^2+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x^2+x^5}+\left (-x^2+x^5\right )^{2/3}\right ) \]
-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 = 7.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.29 \[ \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx=\frac {x^{2/3} \sqrt [3]{-1+x^3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{-1+x^3}}\right )-2 \log \left (\sqrt [3]{x}+\sqrt [3]{-1+x^3}\right )+\log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right )}{2 \sqrt [3]{x^2 \left (-1+x^3\right )}} \]
(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*(-1 + x^3))^(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 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x^3-1} \int -\frac {2 x^3+1}{x^{2/3} \left (-x^3-x+1\right ) \sqrt [3]{x^3-1}}dx}{\sqrt [3]{x^5-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x^3-1} \int \frac {2 x^3+1}{x^{2/3} \left (-x^3-x+1\right ) \sqrt [3]{x^3-1}}dx}{\sqrt [3]{x^5-x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^3-1} \int \frac {2 x^3+1}{\left (-x^3-x+1\right ) \sqrt [3]{x^3-1}}d\sqrt [3]{x}}{\sqrt [3]{x^5-x^2}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^3-1} \int \left (\frac {3-2 x}{\left (-x^3-x+1\right ) \sqrt [3]{x^3-1}}-\frac {2}{\sqrt [3]{x^3-1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^5-x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^3-1} \left (-3 \int \frac {1}{\sqrt [3]{x^3-1} \left (x^3+x-1\right )}d\sqrt [3]{x}+2 \int \frac {x}{\sqrt [3]{x^3-1} \left (x^3+x-1\right )}d\sqrt [3]{x}-\frac {2 \sqrt [3]{x} \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{9},\frac {1}{3},\frac {10}{9},x^3\right )}{\sqrt [3]{x^3-1}}\right )}{\sqrt [3]{x^5-x^2}}\) |
3.13.88.3.1 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 = 5.68 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(-\ln \left (\frac {x +\left (x^{5}-x^{2}\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {x^{2}-x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}+\left (x^{5}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (x^{5}-x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\) | \(87\) |
| trager | \(-\ln \left (-\frac {-1462231700 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+9984360926 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+5117810950 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+1758826818 x^{4}+17923671861 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {2}{3}}+14667651144 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {1}{3}} x +1462231700 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -6431745091 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-6512041434 \left (x^{5}-x^{2}\right )^{\frac {2}{3}}-35847343722 x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}-9984360926 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -1256304870 x^{2}-1758826818 x}{\left (x^{3}+x -1\right ) x}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {251260974 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+7105231670 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-879413409 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+29244634000 x^{4}+35847343722 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {2}{3}}+6512041434 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {1}{3}} x -251260974 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -19968721852 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-58670604576 \left (x^{5}-x^{2}\right )^{\frac {2}{3}}-71694687444 x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}-7105231670 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +11697853600 x^{2}-29244634000 x}{\left (x^{3}+x -1\right ) x}\right )}{2}\) | \(386\) |
-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 = 0.85 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.33 \[ \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{4} + x^{2} - x\right )} + 2 \, \sqrt {3} {\left (x^{5} - x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{4} - x^{2} - x\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{4} + x^{2} + 3 \, {\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} x - x + 3 \, {\left (x^{5} - x^{2}\right )}^{\frac {2}{3}}}{x^{4} + x^{2} - x}\right ) \]
-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))
Timed out. \[ \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx=\text {Timed out} \]
\[ \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx=\int { \frac {2 \, x^{3} + 1}{{\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x - 1\right )}} \,d x } \]
\[ \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx=\int { \frac {2 \, x^{3} + 1}{{\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx=\int \frac {2\,x^3+1}{{\left (x^5-x^2\right )}^{1/3}\,\left (x^3+x-1\right )} \,d x \]