Integrand size = 22, antiderivative size = 185 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right )-\text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-3 \log (x)+3 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=-\frac {(-1+x)^{2/3} x^{4/3} \left (6 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )+6 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )-3 \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )+2 \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-3 \log (x)+9 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right )}{6 \left ((-1+x) x^2\right )^{2/3}} \]
-1/6*((-1 + x)^(2/3)*x^(4/3)*(6*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(2*(-1 + x)^(1/3) + x^(1/3))] + 6*Log[(-1 + x)^(1/3) - x^(1/3)] - 3*Log[(-1 + x)^(2 /3) + (-1 + x)^(1/3)*x^(1/3) + x^(2/3)] + 2*RootSum[3 - 3*#1^3 + #1^6 & , (-3*Log[x] + 9*Log[(-1 + x)^(1/3) - x^(1/3)*#1] + Log[x]*#1^3 - 3*Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1^3)/(-3*#1^2 + 2*#1^5) & ]))/((-1 + x)*x^2)^(2/ 3)
Result contains complex when optimal does not.
Time = 1.20 (sec) , antiderivative size = 937, normalized size of antiderivative = 5.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2467, 1202, 25, 71, 2035, 7279, 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.
| \(\displaystyle \int \frac {\sqrt [3]{x^3-x^2}}{x^2+x+1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x^2} \int \frac {\sqrt [3]{x-1} x^{2/3}}{x^2+x+1}dx}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 1202 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x^2} \left (\int -\frac {2 x+1}{(x-1)^{2/3} \sqrt [3]{x} \left (x^2+x+1\right )}dx+\int \frac {1}{(x-1)^{2/3} \sqrt [3]{x}}dx\right )}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x^2} \left (\int \frac {1}{(x-1)^{2/3} \sqrt [3]{x}}dx-\int \frac {2 x+1}{(x-1)^{2/3} \sqrt [3]{x} \left (x^2+x+1\right )}dx\right )}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x^2} \left (-\int \frac {2 x+1}{(x-1)^{2/3} \sqrt [3]{x} \left (x^2+x+1\right )}dx-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )-\frac {1}{2} \log (x-1)\right )}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x^2} \left (-3 \int \frac {\sqrt [3]{x} (2 x+1)}{(x-1)^{2/3} \left (x^2+x+1\right )}d\sqrt [3]{x}-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )-\frac {1}{2} \log (x-1)\right )}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x^2} \left (-3 \int \left (\frac {2 x^{4/3}}{(x-1)^{2/3} \left (x^2+x+1\right )}+\frac {\sqrt [3]{x}}{(x-1)^{2/3} \left (x^2+x+1\right )}\right )d\sqrt [3]{x}-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )-\frac {1}{2} \log (x-1)\right )}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x^2} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )-\frac {1}{2} \log (x-1)-3 \left (-\frac {2 \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{3 \sqrt [3]{-i+\sqrt {3}} \left (-3 i+\sqrt {3}\right )^{2/3}}+\frac {2}{3} i \left (\frac {i-\sqrt {3}}{3 i-\sqrt {3}}\right )^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )-\frac {2 \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{3 \sqrt [3]{i+\sqrt {3}} \left (3 i+\sqrt {3}\right )^{2/3}}-\frac {2}{3} i \left (\frac {i+\sqrt {3}}{3 i+\sqrt {3}}\right )^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )-\frac {\log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}}}-\sqrt [3]{x-1}\right )}{\sqrt {3} \sqrt [3]{-i+\sqrt {3}} \left (-3 i+\sqrt {3}\right )^{2/3}}+\frac {i \left (\frac {i-\sqrt {3}}{3 i-\sqrt {3}}\right )^{2/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}}}-\sqrt [3]{x-1}\right )}{\sqrt {3}}-\frac {\log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}}}-\sqrt [3]{x-1}\right )}{\sqrt {3} \sqrt [3]{i+\sqrt {3}} \left (3 i+\sqrt {3}\right )^{2/3}}-\frac {i \left (\frac {i+\sqrt {3}}{3 i+\sqrt {3}}\right )^{2/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}}}-\sqrt [3]{x-1}\right )}{\sqrt {3}}+\frac {\log \left (-2 x+i \sqrt {3}-1\right )}{3 \sqrt {3} \sqrt [3]{i+\sqrt {3}} \left (3 i+\sqrt {3}\right )^{2/3}}+\frac {i \left (\frac {i+\sqrt {3}}{3 i+\sqrt {3}}\right )^{2/3} \log \left (-2 x+i \sqrt {3}-1\right )}{3 \sqrt {3}}+\frac {\log \left (2 x+i \sqrt {3}+1\right )}{3 \sqrt {3} \sqrt [3]{-i+\sqrt {3}} \left (-3 i+\sqrt {3}\right )^{2/3}}-\frac {i \left (\frac {i-\sqrt {3}}{3 i-\sqrt {3}}\right )^{2/3} \log \left (2 x+i \sqrt {3}+1\right )}{3 \sqrt {3}}\right )\right )}{\sqrt [3]{x-1} x^{2/3}}\) |
((-x^2 + x^3)^(1/3)*(-(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))]) - (3*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/2 - Log[-1 + x]/2 - 3*(((2*I)/3)*((I - Sqrt[3])/(3*I - Sqrt[3]))^(2/3)*ArcTan[(1 + (2*x^(1/3)) /(((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/3)*(-1 + x)^(1/3)))/Sqrt[3]] - (2*Arc Tan[(1 + (2*x^(1/3))/(((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/3)*(-1 + x)^(1/3) ))/Sqrt[3]])/(3*(-I + Sqrt[3])^(1/3)*(-3*I + Sqrt[3])^(2/3)) - ((2*I)/3)*( (I + Sqrt[3])/(3*I + Sqrt[3]))^(2/3)*ArcTan[(1 + (2*x^(1/3))/(((I + Sqrt[3 ])/(3*I + Sqrt[3]))^(1/3)*(-1 + x)^(1/3)))/Sqrt[3]] - (2*ArcTan[(1 + (2*x^ (1/3))/(((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/3)*(-1 + x)^(1/3)))/Sqrt[3]])/( 3*(I + Sqrt[3])^(1/3)*(3*I + Sqrt[3])^(2/3)) + (I*((I - Sqrt[3])/(3*I - Sq rt[3]))^(2/3)*Log[-(-1 + x)^(1/3) + x^(1/3)/((I - Sqrt[3])/(3*I - Sqrt[3]) )^(1/3)])/Sqrt[3] - Log[-(-1 + x)^(1/3) + x^(1/3)/((I - Sqrt[3])/(3*I - Sq rt[3]))^(1/3)]/(Sqrt[3]*(-I + Sqrt[3])^(1/3)*(-3*I + Sqrt[3])^(2/3)) - (I* ((I + Sqrt[3])/(3*I + Sqrt[3]))^(2/3)*Log[-(-1 + x)^(1/3) + x^(1/3)/((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/3)])/Sqrt[3] - Log[-(-1 + x)^(1/3) + x^(1/3)/ ((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/3)]/(Sqrt[3]*(I + Sqrt[3])^(1/3)*(3*I + Sqrt[3])^(2/3)) + ((I/3)*((I + Sqrt[3])/(3*I + Sqrt[3]))^(2/3)*Log[-1 + I *Sqrt[3] - 2*x])/Sqrt[3] + Log[-1 + I*Sqrt[3] - 2*x]/(3*Sqrt[3]*(I + Sqrt[ 3])^(1/3)*(3*I + Sqrt[3])^(2/3)) - ((I/3)*((I - Sqrt[3])/(3*I - Sqrt[3]))^ (2/3)*Log[1 + I*Sqrt[3] + 2*x])/Sqrt[3] + Log[1 + I*Sqrt[3] + 2*x]/(3*S...
3.24.39.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*(g/c) Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Simp[1/c Int[Simp[c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x, x]*(d + e*x)^(m - 1)*((f + g*x)^(n - 1)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 0]
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]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 5.68 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.70
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-3 \textit {\_Z}^{3}+3\right )}{\sum }\frac {\left (\textit {\_R}^{3}-3\right ) \ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (2 \textit {\_R}^{3}-3\right )}\right )-\ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x}{x}\right )\) | \(130\) |
| trager | \(\text {Expression too large to display}\) | \(2273\) |
1/2*ln((((-1+x)*x^2)^(2/3)+((-1+x)*x^2)^(1/3)*x+x^2)/x^2)+3^(1/2)*arctan(1 /3*(2*((-1+x)*x^2)^(1/3)+x)*3^(1/2)/x)+sum((_R^3-3)*ln((-_R*x+((-1+x)*x^2) ^(1/3))/x)/_R^2/(2*_R^3-3),_R=RootOf(_Z^6-3*_Z^3+3))-ln((((-1+x)*x^2)^(1/3 )-x)/x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.72 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=\frac {1}{12} \cdot 6^{\frac {2}{3}} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (\frac {6^{\frac {2}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} x - i \, x\right )} - 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} + 24 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{12} \cdot 6^{\frac {2}{3}} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (\frac {6^{\frac {2}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} x - i \, x\right )} + 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} + 24 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{12} \cdot 6^{\frac {2}{3}} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (\frac {6^{\frac {2}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} x + i \, x\right )} + 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} + 24 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{12} \cdot 6^{\frac {2}{3}} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (\frac {6^{\frac {2}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} x + i \, x\right )} - 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} + 24 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \cdot 6^{\frac {2}{3}} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {2}{3}} {\left (i \, \sqrt {3} x - 3 \, x\right )} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} + 12 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \cdot 6^{\frac {2}{3}} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {2}{3}} {\left (-i \, \sqrt {3} x - 3 \, x\right )} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} + 12 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
1/12*6^(2/3)*(I*sqrt(3) - 1)^(1/3)*(sqrt(-3) - 1)*log((6^(2/3)*(sqrt(3)*(I *sqrt(-3)*x - I*x) - 3*sqrt(-3)*x + 3*x)*(I*sqrt(3) - 1)^(1/3) + 24*(x^3 - x^2)^(1/3))/x) - 1/12*6^(2/3)*(I*sqrt(3) - 1)^(1/3)*(sqrt(-3) + 1)*log((6 ^(2/3)*(sqrt(3)*(-I*sqrt(-3)*x - I*x) + 3*sqrt(-3)*x + 3*x)*(I*sqrt(3) - 1 )^(1/3) + 24*(x^3 - x^2)^(1/3))/x) - 1/12*6^(2/3)*(-I*sqrt(3) - 1)^(1/3)*( sqrt(-3) + 1)*log((6^(2/3)*(sqrt(3)*(I*sqrt(-3)*x + I*x) + 3*sqrt(-3)*x + 3*x)*(-I*sqrt(3) - 1)^(1/3) + 24*(x^3 - x^2)^(1/3))/x) + 1/12*6^(2/3)*(-I* sqrt(3) - 1)^(1/3)*(sqrt(-3) - 1)*log((6^(2/3)*(sqrt(3)*(-I*sqrt(-3)*x + I *x) - 3*sqrt(-3)*x + 3*x)*(-I*sqrt(3) - 1)^(1/3) + 24*(x^3 - x^2)^(1/3))/x ) + 1/6*6^(2/3)*(I*sqrt(3) - 1)^(1/3)*log((6^(2/3)*(I*sqrt(3)*x - 3*x)*(I* sqrt(3) - 1)^(1/3) + 12*(x^3 - x^2)^(1/3))/x) + 1/6*6^(2/3)*(-I*sqrt(3) - 1)^(1/3)*log((6^(2/3)*(-I*sqrt(3)*x - 3*x)*(-I*sqrt(3) - 1)^(1/3) + 12*(x^ 3 - x^2)^(1/3))/x) + sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - x^2) ^(1/3))/x) - log(-(x - (x^3 - x^2)^(1/3))/x) + 1/2*log((x^2 + (x^3 - x^2)^ (1/3)*x + (x^3 - x^2)^(2/3))/x^2)
Not integrable
Time = 0.41 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=\int \frac {\sqrt [3]{x^{2} \left (x - 1\right )}}{x^{2} + x + 1}\, dx \]
Not integrable
Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=\int { \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x^{2} + x + 1} \,d x } \]
Exception generated. \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument ValueDone
Not integrable
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=\int \frac {{\left (x^3-x^2\right )}^{1/3}}{x^2+x+1} \,d x \]