Integrand size = 29, antiderivative size = 85 \[ \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x+x^6}}\right )-\log \left (x+\sqrt [3]{-x+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x+x^6}+\left (-x+x^6\right )^{2/3}\right ) \]
-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^6-x)^(1/3)))-ln(x+(x^6-x)^(1/3))+1/2*ln (x^2-x*(x^6-x)^(1/3)+(x^6-x)^(2/3))
\[ \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx=\int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx \]
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.
| \(\displaystyle \int \frac {3 x^5+2}{\left (x^5+x^2-1\right ) \sqrt [3]{x^6-x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^5-1} \int -\frac {3 x^5+2}{\sqrt [3]{x} \left (-x^5-x^2+1\right ) \sqrt [3]{x^5-1}}dx}{\sqrt [3]{x^6-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^5-1} \int \frac {3 x^5+2}{\sqrt [3]{x} \left (-x^5-x^2+1\right ) \sqrt [3]{x^5-1}}dx}{\sqrt [3]{x^6-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^5-1} \int \frac {\sqrt [3]{x} \left (3 x^5+2\right )}{\left (-x^5-x^2+1\right ) \sqrt [3]{x^5-1}}d\sqrt [3]{x}}{\sqrt [3]{x^6-x}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^5-1} \int \left (\frac {\sqrt [3]{x} \left (5-3 x^2\right )}{\left (-x^5-x^2+1\right ) \sqrt [3]{x^5-1}}-\frac {3 \sqrt [3]{x}}{\sqrt [3]{x^5-1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^6-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^5-1} \left (-5 \int \frac {\sqrt [3]{x}}{\sqrt [3]{x^5-1} \left (x^5+x^2-1\right )}d\sqrt [3]{x}+3 \int \frac {x^{7/3}}{\sqrt [3]{x^5-1} \left (x^5+x^2-1\right )}d\sqrt [3]{x}-\frac {3 x^{2/3} \sqrt [3]{1-x^5} \operatorname {Hypergeometric2F1}\left (\frac {2}{15},\frac {1}{3},\frac {17}{15},x^5\right )}{2 \sqrt [3]{x^5-1}}\right )}{\sqrt [3]{x^6-x}}\) |
3.12.44.3.1 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 = 9.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93
| method | result | size |
| pseudoelliptic | \(-\ln \left (\frac {x +\left (x^{6}-x \right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {x^{2}-x \left (x^{6}-x \right )^{\frac {1}{3}}+\left (x^{6}-x \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (x^{6}-x \right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\) | \(79\) |
| trager | \(-\ln \left (-\frac {8327084306326444968 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}-10176976877377096586 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-421273930349059602349 x^{5}-64534903374029948502 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-433300799797487350553 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {2}{3}}-861051921881822746252 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {1}{3}} x -365066111281356098815 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-8327084306326444968 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-427751122084335395699 \left (x^{6}-x \right )^{\frac {2}{3}}+433300799797487350553 x \left (x^{6}-x \right )^{\frac {1}{3}}+366916003852406750433 x^{2}+10176976877377096586 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+421273930349059602349}{x^{5}+x^{2}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {54357926496652851916 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}+375243088158733195401 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}+56207819067703503534 x^{5}-421273930349059602349 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+433300799797487350553 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {2}{3}}-427751122084335395699 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {1}{3}} x -10176976877377096586 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-54357926496652851916 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-861051921881822746252 \left (x^{6}-x \right )^{\frac {2}{3}}-433300799797487350553 x \left (x^{6}-x \right )^{\frac {1}{3}}+8327084306326444968 x^{2}-375243088158733195401 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-56207819067703503534}{x^{5}+x^{2}-1}\right )\) | \(360\) |
-ln((x+(x^6-x)^(1/3))/x)+1/2*ln((x^2-x*(x^6-x)^(1/3)+(x^6-x)^(2/3))/x^2)-3 ^(1/2)*arctan(1/3*(-2*(x^6-x)^(1/3)+x)*3^(1/2)/x)
Time = 1.60 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.22 \[ \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx=-\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{6} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} - 1\right )} + 2 \, \sqrt {3} {\left (x^{6} - x\right )}^{\frac {2}{3}}}{x^{5} - 8 \, x^{2} - 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{5} + x^{2} + 3 \, {\left (x^{6} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{6} - x\right )}^{\frac {2}{3}} - 1}{x^{5} + x^{2} - 1}\right ) \]
-sqrt(3)*arctan((4*sqrt(3)*(x^6 - x)^(1/3)*x + sqrt(3)*(x^5 - 1) + 2*sqrt( 3)*(x^6 - x)^(2/3))/(x^5 - 8*x^2 - 1)) - 1/2*log((x^5 + x^2 + 3*(x^6 - x)^ (1/3)*x + 3*(x^6 - x)^(2/3) - 1)/(x^5 + x^2 - 1))
\[ \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx=\int \frac {3 x^{5} + 2}{\sqrt [3]{x \left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (x^{5} + x^{2} - 1\right )}\, dx \]
Integral((3*x**5 + 2)/((x*(x - 1)*(x**4 + x**3 + x**2 + x + 1))**(1/3)*(x* *5 + x**2 - 1)), x)
\[ \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx=\int { \frac {3 \, x^{5} + 2}{{\left (x^{6} - x\right )}^{\frac {1}{3}} {\left (x^{5} + x^{2} - 1\right )}} \,d x } \]
\[ \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx=\int { \frac {3 \, x^{5} + 2}{{\left (x^{6} - x\right )}^{\frac {1}{3}} {\left (x^{5} + x^{2} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx=\int \frac {3\,x^5+2}{{\left (x^6-x\right )}^{1/3}\,\left (x^5+x^2-1\right )} \,d x \]