\(\int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx\) [2339]

Optimal result

Integrand size = 22, antiderivative size = 185 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right )-\text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-3 \log (x)+3 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 701, normalized size of antiderivative = 3.79, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2081, 919, 61, 6860, 93} \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=-\frac {\sqrt {3} \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x-1} x^{2/3}}-\frac {2 i \sqrt {3} \sqrt [3]{x^3-x^2} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [3]{x-1}}\right )}{\sqrt [3]{\sqrt {3}-i} \left (\sqrt {3}-3 i\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}+\frac {2 i \sqrt {3} \sqrt [3]{x^3-x^2} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [3]{x-1}}\right )}{\sqrt [3]{\sqrt {3}+i} \left (\sqrt {3}+3 i\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}-\frac {3 i \sqrt [3]{x^3-x^2} \log \left (-\sqrt [3]{x-1}+\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}}}\right )}{\sqrt [3]{\sqrt {3}-i} \left (\sqrt {3}-3 i\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}+\frac {3 i \sqrt [3]{x^3-x^2} \log \left (-\sqrt [3]{x-1}+\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}}}\right )}{\sqrt [3]{\sqrt {3}+i} \left (\sqrt {3}+3 i\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}-\frac {3 \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{2 \sqrt [3]{x-1} x^{2/3}}-\frac {\sqrt [3]{x^3-x^2} \log (x-1)}{2 \sqrt [3]{x-1} x^{2/3}}-\frac {i \sqrt [3]{x^3-x^2} \log \left (2 x-i \sqrt {3}+1\right )}{\sqrt [3]{\sqrt {3}+i} \left (\sqrt {3}+3 i\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}+\frac {i \sqrt [3]{x^3-x^2} \log \left (2 x+i \sqrt {3}+1\right )}{\sqrt [3]{\sqrt {3}-i} \left (\sqrt {3}-3 i\right )^{2/3} \sqrt [3]{x-1} x^{2/3}} \]

[In]

Int[(-x^2 + x^3)^(1/3)/(1 + x + x^2),x]

[Out]

-((Sqrt[3]*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/((-1 + x)^(1/3)*x^(2/3
))) - ((2*I)*Sqrt[3]*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*((I - Sqrt[3])/(3*I - Sqrt[3])
)^(1/3)*(-1 + x)^(1/3))])/((-I + Sqrt[3])^(1/3)*(-3*I + Sqrt[3])^(2/3)*(-1 + x)^(1/3)*x^(2/3)) + ((2*I)*Sqrt[3
]*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/3)*(-1 + x)^(1
/3))])/((I + Sqrt[3])^(1/3)*(3*I + Sqrt[3])^(2/3)*(-1 + x)^(1/3)*x^(2/3)) - ((3*I)*(-x^2 + x^3)^(1/3)*Log[-(-1
+ x)^(1/3) + x^(1/3)/((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/3)])/((-I + Sqrt[3])^(1/3)*(-3*I + Sqrt[3])^(2/3)*(-1
+ x)^(1/3)*x^(2/3)) + ((3*I)*(-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + x^(1/3)/((I + Sqrt[3])/(3*I + Sqrt[3]))
^(1/3)])/((I + Sqrt[3])^(1/3)*(3*I + Sqrt[3])^(2/3)*(-1 + x)^(1/3)*x^(2/3)) - (3*(-x^2 + x^3)^(1/3)*Log[-1 + x
^(1/3)/(-1 + x)^(1/3)])/(2*(-1 + x)^(1/3)*x^(2/3)) - ((-x^2 + x^3)^(1/3)*Log[-1 + x])/(2*(-1 + x)^(1/3)*x^(2/3
)) - (I*(-x^2 + x^3)^(1/3)*Log[1 - I*Sqrt[3] + 2*x])/((I + Sqrt[3])^(1/3)*(3*I + Sqrt[3])^(2/3)*(-1 + x)^(1/3)
*x^(2/3)) + (I*(-x^2 + x^3)^(1/3)*Log[1 + I*Sqrt[3] + 2*x])/((-I + Sqrt[3])^(1/3)*(-3*I + Sqrt[3])^(2/3)*(-1 +
x)^(1/3)*x^(2/3))

Rule 61

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] && Ne
Q[b*c - a*d, 0] && PosQ[d/b]

Rule 93

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])*q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1
/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q*(a + b*x)^(1/3) - (c +
d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 919

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Di
st[e*(g/c), Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[m, 0] &
& GtQ[n, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6860

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] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{-x^2+x^3} \int \frac {\sqrt [3]{-1+x} x^{2/3}}{1+x+x^2} \, dx}{\sqrt [3]{-1+x} x^{2/3}} \\ & = \frac {\sqrt [3]{-x^2+x^3} \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{-x^2+x^3} \int \frac {-1-2 x}{(-1+x)^{2/3} \sqrt [3]{x} \left (1+x+x^2\right )} \, dx}{\sqrt [3]{-1+x} x^{2/3}} \\ & = -\frac {\sqrt {3} \sqrt [3]{-x^2+x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {3 \sqrt [3]{-x^2+x^3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{2 \sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log (-1+x)}{2 \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{-x^2+x^3} \int \left (-\frac {2}{(-1+x)^{2/3} \sqrt [3]{x} \left (1-i \sqrt {3}+2 x\right )}-\frac {2}{(-1+x)^{2/3} \sqrt [3]{x} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx}{\sqrt [3]{-1+x} x^{2/3}} \\ & = -\frac {\sqrt {3} \sqrt [3]{-x^2+x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {3 \sqrt [3]{-x^2+x^3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{2 \sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log (-1+x)}{2 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (2 \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x} \left (1-i \sqrt {3}+2 x\right )} \, dx}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (2 \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x} \left (1+i \sqrt {3}+2 x\right )} \, dx}{\sqrt [3]{-1+x} x^{2/3}} \\ & = -\frac {\sqrt {3} \sqrt [3]{-x^2+x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {2 i \sqrt {3} \sqrt [3]{-x^2+x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [3]{-1+x}}\right )}{\sqrt [3]{-i+\sqrt {3}} \left (-3 i+\sqrt {3}\right )^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {2 i \sqrt {3} \sqrt [3]{-x^2+x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [3]{-1+x}}\right )}{\sqrt [3]{i+\sqrt {3}} \left (3 i+\sqrt {3}\right )^{2/3} \sqrt [3]{-1+x} x^{2/3}}-\frac {3 i \sqrt [3]{-x^2+x^3} \log \left (-\sqrt [3]{-1+x}+\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}}}\right )}{\sqrt [3]{-i+\sqrt {3}} \left (-3 i+\sqrt {3}\right )^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {3 i \sqrt [3]{-x^2+x^3} \log \left (-\sqrt [3]{-1+x}+\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}}}\right )}{\sqrt [3]{i+\sqrt {3}} \left (3 i+\sqrt {3}\right )^{2/3} \sqrt [3]{-1+x} x^{2/3}}-\frac {3 \sqrt [3]{-x^2+x^3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{2 \sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log (-1+x)}{2 \sqrt [3]{-1+x} x^{2/3}}-\frac {i \sqrt [3]{-x^2+x^3} \log \left (1-i \sqrt {3}+2 x\right )}{\sqrt [3]{i+\sqrt {3}} \left (3 i+\sqrt {3}\right )^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {i \sqrt [3]{-x^2+x^3} \log \left (1+i \sqrt {3}+2 x\right )}{\sqrt [3]{-i+\sqrt {3}} \left (-3 i+\sqrt {3}\right )^{2/3} \sqrt [3]{-1+x} x^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=-\frac {(-1+x)^{2/3} x^{4/3} \left (6 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )+6 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )-3 \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )+2 \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-3 \log (x)+9 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right )}{6 \left ((-1+x) x^2\right )^{2/3}} \]

[In]

Integrate[(-x^2 + x^3)^(1/3)/(1 + x + x^2),x]

[Out]

-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)

Maple [N/A] (verified)

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	\(\text {Expression too large to display}\)	\(2273\)

[In]

int((x^3-x^2)^(1/3)/(x^2+x+1),x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [C] (verification not implemented)

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 ) \]

[In]

integrate((x^3-x^2)^(1/3)/(x^2+x+1),x, algorithm="fricas")

[Out]

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)

Sympy [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=\int \frac {\sqrt [3]{x^{2} \left (x - 1\right )}}{x^{2} + x + 1}\, dx \]

[In]

integrate((x**3-x**2)**(1/3)/(x**2+x+1),x)

[Out]

Integral((x**2*(x - 1))**(1/3)/(x**2 + x + 1), x)

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=\int { \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x^{2} + x + 1} \,d x } \]

[In]

integrate((x^3-x^2)^(1/3)/(x^2+x+1),x, algorithm="maxima")

[Out]

integrate((x^3 - x^2)^(1/3)/(x^2 + x + 1), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((x^3-x^2)^(1/3)/(x^2+x+1),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Invalid _EXT in replace_ext Error: Bad Argument ValueDone

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx=\int \frac {{\left (x^3-x^2\right )}^{1/3}}{x^2+x+1} \,d x \]

[In]

int((x^3 - x^2)^(1/3)/(x + x^2 + 1),x)

[Out]

int((x^3 - x^2)^(1/3)/(x + x^2 + 1), x)