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

Optimal. Leaf size=185 \[ -\text {RootSum}\left [\text {$\#$1}^6-3 \text {$\#$1}^3+3\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+3 \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-3 \log (x)}{2 \text {$\#$1}^5-3 \text {$\#$1}^2}\& \right ]-\log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right ) \]

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Rubi [C]  time = 0.72, antiderivative size = 701, normalized size of antiderivative = 3.79, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2056, 905, 59, 6728, 91} \begin {gather*} -\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}}-\frac {\sqrt {3} \sqrt [3]{x^3-x^2} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 59

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, -Simp[(Sqrt
[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x
)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[
b*c - a*d, 0] && PosQ[d/b]

Rule 91

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 905

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 2056

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 6728

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*} \int \frac {\sqrt [3]{-x^2+x^3}}{1+x+x^2} \, dx &=\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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*}

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Mathematica [C]  time = 0.15, size = 136, normalized size = 0.74 \begin {gather*} \frac {\sqrt [3]{(x-1) x^2} \left (6 \sqrt [3]{x} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};1-x\right )+i \left (\sqrt {3}+3 i\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (-i+\sqrt {3}\right ) (x-1)}{\left (-3 i+\sqrt {3}\right ) x}\right )+\left (-3-i \sqrt {3}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (i+\sqrt {3}\right ) (x-1)}{\left (3 i+\sqrt {3}\right ) x}\right )\right )}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((-1 + x)*x^2)^(1/3)*(6*x^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, 1 - x] + I*(3*I + Sqrt[3])*Hypergeometric2F1
[1/3, 1, 4/3, ((-I + Sqrt[3])*(-1 + x))/((-3*I + Sqrt[3])*x)] + (-3 - I*Sqrt[3])*Hypergeometric2F1[1/3, 1, 4/3
, ((I + Sqrt[3])*(-1 + x))/((3*I + Sqrt[3])*x)]))/(2*x)

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IntegrateAlgebraic [A]  time = 0.28, size = 185, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))]) - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 +
x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 - RootSum[3 - 3*#1^3 + #1^6 & , (-3*Log[x] + 3*Log[(-x^2 + x^3)^(1/3) - x*
#1] + Log[x]*#1^3 - Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1^2 + 2*#1^5) & ]

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fricas [B]  time = 0.70, size = 908, normalized size = 4.91

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*6^(2/3)*2^(1/3)*cos(2/9*pi)*log(24*(3*6^(1/3)*2^(2/3)*x^2 - (6^(2/3)*sqrt(3)*2^(1/3)*x*sin(2/9*pi) + 3*6^(
2/3)*2^(1/3)*x*cos(2/9*pi))*(x^3 - x^2)^(1/3) + 6*(x^3 - x^2)^(2/3))/x^2) - 2/3*6^(2/3)*2^(1/3)*arctan(1/18*(7
2*x*cos(2/9*pi)*sin(2/9*pi) + sqrt(6)*(6^(1/3)*sqrt(3)*2^(2/3)*x*cos(2/9*pi) + 3*6^(1/3)*2^(2/3)*x*sin(2/9*pi)
)*sqrt((3*6^(1/3)*2^(2/3)*x^2 - (6^(2/3)*sqrt(3)*2^(1/3)*x*sin(2/9*pi) + 3*6^(2/3)*2^(1/3)*x*cos(2/9*pi))*(x^3
- x^2)^(1/3) + 6*(x^3 - x^2)^(2/3))/x^2) + 18*sqrt(3)*x - 6*(x^3 - x^2)^(1/3)*(6^(1/3)*sqrt(3)*2^(2/3)*cos(2/
9*pi) + 3*6^(1/3)*2^(2/3)*sin(2/9*pi)))/(4*x*cos(2/9*pi)^2 - 3*x))*sin(2/9*pi) - 1/3*(6^(2/3)*sqrt(3)*2^(1/3)*
cos(2/9*pi) + 6^(2/3)*2^(1/3)*sin(2/9*pi))*arctan(-1/18*(72*x*cos(2/9*pi)*sin(2/9*pi) + sqrt(6)*(6^(1/3)*sqrt(
3)*2^(2/3)*x*cos(2/9*pi) - 3*6^(1/3)*2^(2/3)*x*sin(2/9*pi))*sqrt((3*6^(1/3)*2^(2/3)*x^2 - (6^(2/3)*sqrt(3)*2^(
1/3)*x*sin(2/9*pi) - 3*6^(2/3)*2^(1/3)*x*cos(2/9*pi))*(x^3 - x^2)^(1/3) + 6*(x^3 - x^2)^(2/3))/x^2) - 18*sqrt(
3)*x - 6*(x^3 - x^2)^(1/3)*(6^(1/3)*sqrt(3)*2^(2/3)*cos(2/9*pi) - 3*6^(1/3)*2^(2/3)*sin(2/9*pi)))/(4*x*cos(2/9
*pi)^2 - 3*x)) + 1/3*(6^(2/3)*sqrt(3)*2^(1/3)*cos(2/9*pi) - 6^(2/3)*2^(1/3)*sin(2/9*pi))*arctan(1/36*(6^(5/6)*
sqrt(3)*2^(2/3)*x*sqrt((2*6^(2/3)*sqrt(3)*2^(1/3)*(x^3 - x^2)^(1/3)*x*sin(2/9*pi) + 3*6^(1/3)*2^(2/3)*x^2 + 6*
(x^3 - x^2)^(2/3))/x^2) - 36*x*sin(2/9*pi) - 6*6^(1/3)*sqrt(3)*2^(2/3)*(x^3 - x^2)^(1/3))/(x*cos(2/9*pi))) - 1
/12*(6^(2/3)*sqrt(3)*2^(1/3)*sin(2/9*pi) + 6^(2/3)*2^(1/3)*cos(2/9*pi))*log(96*(2*6^(2/3)*sqrt(3)*2^(1/3)*(x^3
- x^2)^(1/3)*x*sin(2/9*pi) + 3*6^(1/3)*2^(2/3)*x^2 + 6*(x^3 - x^2)^(2/3))/x^2) + 1/12*(6^(2/3)*sqrt(3)*2^(1/3
)*sin(2/9*pi) - 6^(2/3)*2^(1/3)*cos(2/9*pi))*log(96*(3*6^(1/3)*2^(2/3)*x^2 - (6^(2/3)*sqrt(3)*2^(1/3)*x*sin(2/
9*pi) - 3*6^(2/3)*2^(1/3)*x*cos(2/9*pi))*(x^3 - x^2)^(1/3) + 6*(x^3 - x^2)^(2/3))/x^2) + sqrt(3)*arctan(1/3*(s
qrt(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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x^{2} + x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 4.35, size = 2446, normalized size = 13.22

method	result	size
trager	\(\text {Expression too large to display}\)	\(2446\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(9*_Z^6+3*_Z^3+1)*ln((279*x^2*RootOf(9*_Z^6+3*_Z^3+1)^8-558*RootOf(9*_Z^6+3*_Z^3+1)^8*x+1206*RootOf(9*_Z
^6+3*_Z^3+1)^5*x^2-645*RootOf(9*_Z^6+3*_Z^3+1)^5*x-654*(x^3-x^2)^(1/3)*RootOf(9*_Z^6+3*_Z^3+1)^4*x+195*(x^3-x^
2)^(2/3)*RootOf(9*_Z^6+3*_Z^3+1)^3+360*RootOf(9*_Z^6+3*_Z^3+1)^2*x^2-150*RootOf(9*_Z^6+3*_Z^3+1)^2*x-241*(x^3-
x^2)^(1/3)*RootOf(9*_Z^6+3*_Z^3+1)*x+153*(x^3-x^2)^(2/3))/(3*x*RootOf(9*_Z^6+3*_Z^3+1)^3-6*RootOf(9*_Z^6+3*_Z^
3+1)^3+3*x+1)/x)+RootOf(9*_Z^6+3*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)*ln(-(306*RootOf(_Z^3+27
*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2*RootOf(9*_Z^6+3*_Z^3+1)^6*x^2-612*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^
2*RootOf(9*_Z^6+3*_Z^3+1)^6*x+582*x^2*RootOf(9*_Z^6+3*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2+
549*(x^3-x^2)^(1/3)*RootOf(9*_Z^6+3*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)*x-501*x*RootOf(9*_Z^
6+3*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2-513*(x^3-x^2)^(2/3)*RootOf(9*_Z^6+3*_Z^3+1)^3+232*
x^2*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2+537*(x^3-x^2)^(1/3)*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3
+9)*x-87*x*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2+720*(x^3-x^2)^(2/3))/(3*x*RootOf(9*_Z^6+3*_Z^3+1)^3-6
*RootOf(9*_Z^6+3*_Z^3+1)^3-2*x-3)/x)+3*RootOf(9*_Z^6+3*_Z^3+1)^4*ln((522*x^2*RootOf(9*_Z^6+3*_Z^3+1)^8-1044*Ro
otOf(9*_Z^6+3*_Z^3+1)^8*x-420*RootOf(9*_Z^6+3*_Z^3+1)^5*x^2-291*RootOf(9*_Z^6+3*_Z^3+1)^5*x+354*(x^3-x^2)^(1/3
)*RootOf(9*_Z^6+3*_Z^3+1)^4*x-57*(x^3-x^2)^(2/3)*RootOf(9*_Z^6+3*_Z^3+1)^3-102*RootOf(9*_Z^6+3*_Z^3+1)^2*x^2-1
7*RootOf(9*_Z^6+3*_Z^3+1)^2*x+179*(x^3-x^2)^(1/3)*RootOf(9*_Z^6+3*_Z^3+1)*x-99*(x^3-x^2)^(2/3))/(3*x*RootOf(9*
_Z^6+3*_Z^3+1)^3-6*RootOf(9*_Z^6+3*_Z^3+1)^3+3*x+1)/x)-ln(-(270*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2*
RootOf(9*_Z^6+3*_Z^3+1)^6*x^2-540*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2*RootOf(9*_Z^6+3*_Z^3+1)^6*x-89
7*x^2*RootOf(9*_Z^6+3*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2-207*(x^3-x^2)^(1/3)*RootOf(9*_Z^
6+3*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)*x+84*x*RootOf(9*_Z^6+3*_Z^3+1)^3*RootOf(_Z^3+27*Root
Of(9*_Z^6+3*_Z^3+1)^3+9)^2+1755*(x^3-x^2)^(2/3)*RootOf(9*_Z^6+3*_Z^3+1)^3-341*x^2*RootOf(_Z^3+27*RootOf(9*_Z^6
+3*_Z^3+1)^3+9)^2-723*(x^3-x^2)^(1/3)*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)*x+93*x*RootOf(_Z^3+27*RootOf
(9*_Z^6+3*_Z^3+1)^3+9)^2-792*(x^3-x^2)^(2/3))/(3*x*RootOf(9*_Z^6+3*_Z^3+1)^3-6*RootOf(9*_Z^6+3*_Z^3+1)^3-2*x-3
)/x)*RootOf(9*_Z^6+3*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)-1/3*ln(-(270*RootOf(_Z^3+27*RootOf(
9*_Z^6+3*_Z^3+1)^3+9)^2*RootOf(9*_Z^6+3*_Z^3+1)^6*x^2-540*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2*RootOf
(9*_Z^6+3*_Z^3+1)^6*x-897*x^2*RootOf(9*_Z^6+3*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2-207*(x^3
-x^2)^(1/3)*RootOf(9*_Z^6+3*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)*x+84*x*RootOf(9*_Z^6+3*_Z^3+
1)^3*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2+1755*(x^3-x^2)^(2/3)*RootOf(9*_Z^6+3*_Z^3+1)^3-341*x^2*Root
Of(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2-723*(x^3-x^2)^(1/3)*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)*x+93
*x*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)^2-792*(x^3-x^2)^(2/3))/(3*x*RootOf(9*_Z^6+3*_Z^3+1)^3-6*RootOf(
9*_Z^6+3*_Z^3+1)^3-2*x-3)/x)*RootOf(_Z^3+27*RootOf(9*_Z^6+3*_Z^3+1)^3+9)+3*RootOf(9*_Z^6+3*_Z^3+1)^3*ln((45*Ro
otOf(9*_Z^6+3*_Z^3+1)^6*x^2-90*RootOf(9*_Z^6+3*_Z^3+1)^6*x-72*(x^3-x^2)^(2/3)*RootOf(9*_Z^6+3*_Z^3+1)^3-72*(x^
3-x^2)^(1/3)*RootOf(9*_Z^6+3*_Z^3+1)^3*x-57*x^2*RootOf(9*_Z^6+3*_Z^3+1)^3+9*x*RootOf(9*_Z^6+3*_Z^3+1)^3-9*(x^3
-x^2)^(2/3)-9*x*(x^3-x^2)^(1/3)-4*x^2+x)/x)+ln((45*RootOf(9*_Z^6+3*_Z^3+1)^6*x^2-90*RootOf(9*_Z^6+3*_Z^3+1)^6*
x-72*(x^3-x^2)^(2/3)*RootOf(9*_Z^6+3*_Z^3+1)^3-72*(x^3-x^2)^(1/3)*RootOf(9*_Z^6+3*_Z^3+1)^3*x-57*x^2*RootOf(9*
_Z^6+3*_Z^3+1)^3+9*x*RootOf(9*_Z^6+3*_Z^3+1)^3-9*(x^3-x^2)^(2/3)-9*x*(x^3-x^2)^(1/3)-4*x^2+x)/x)-3*RootOf(9*_Z
^6+3*_Z^3+1)^3*ln((45*RootOf(9*_Z^6+3*_Z^3+1)^6*x^2-90*RootOf(9*_Z^6+3*_Z^3+1)^6*x+72*(x^3-x^2)^(2/3)*RootOf(9
*_Z^6+3*_Z^3+1)^3+72*(x^3-x^2)^(1/3)*RootOf(9*_Z^6+3*_Z^3+1)^3*x+87*x^2*RootOf(9*_Z^6+3*_Z^3+1)^3-69*x*RootOf(
9*_Z^6+3*_Z^3+1)^3+15*(x^3-x^2)^(2/3)+15*x*(x^3-x^2)^(1/3)+20*x^2-12*x)/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x^{2} + x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-x^2\right )}^{1/3}}{x^2+x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x^{2} \left (x - 1\right )}}{x^{2} + x + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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