3.21.82 \(\int \frac {(-2+x^3) \sqrt [3]{x+2 x^3+x^4}}{(1+x^3) (1+x^2+x^3)} \, dx\)

Optimal. Leaf size=225 \[ -\log \left (\sqrt [3]{x^4+2 x^3+x}-x\right )+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^4+2 x^3+x}-2 x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+2 x^3+x}+x}\right )+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+2 x^3+x}+x}\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^4+2 x^3+x} x+\left (x^4+2 x^3+x\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+2 x^3+x} x+\sqrt [3]{2} \left (x^4+2 x^3+x\right )^{2/3}\right )}{2^{2/3}} \]

________________________________________________________________________________________

Rubi [F]  time = 6.27, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+2 x^3+x^4}}{\left (1+x^3\right ) \left (1+x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-2 + x^3)*(x + 2*x^3 + x^4)^(1/3))/((1 + x^3)*(1 + x^2 + x^3)),x]

[Out]

(-3*(x + 2*x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + 2*x^6 + x^9)^(1/3), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*
x^2 + x^3)^(1/3)) + (3*(1 - I*Sqrt[3])*(x + 2*x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + 2*x^6 + x^9)^(1/3)
, x], x, x^(1/3)])/(2*x^(1/3)*(1 + 2*x^2 + x^3)^(1/3)) + (3*(1 + I*Sqrt[3])*(x + 2*x^3 + x^4)^(1/3)*Defer[Subs
t][Defer[Int][(1 + 2*x^6 + x^9)^(1/3), x], x, x^(1/3)])/(2*x^(1/3)*(1 + 2*x^2 + x^3)^(1/3)) + ((x + 2*x^3 + x^
4)^(1/3)*Defer[Subst][Defer[Int][(1 + 2*x^6 + x^9)^(1/3)/(1 + x), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 + x^3)^
(1/3)) - ((1 + I*Sqrt[3])*(x + 2*x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + 2*x^6 + x^9)^(1/3)/(-1 - I*Sqrt
[3] + 2*x), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 + x^3)^(1/3)) - ((1 - I*Sqrt[3])*(x + 2*x^3 + x^4)^(1/3)*Defe
r[Subst][Defer[Int][(1 + 2*x^6 + x^9)^(1/3)/(-1 + I*Sqrt[3] + 2*x), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 + x^3
)^(1/3)) - ((1 - I*Sqrt[3])^(1/3)*(1 + I*Sqrt[3])*(x + 2*x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + 2*x^6 +
x^9)^(1/3)/((1 - I*Sqrt[3])^(1/3) + (-2)^(1/3)*x), x], x, x^(1/3)])/(2*x^(1/3)*(1 + 2*x^2 + x^3)^(1/3)) - ((1
- I*Sqrt[3])*(1 + I*Sqrt[3])^(1/3)*(x + 2*x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + 2*x^6 + x^9)^(1/3)/((
1 + I*Sqrt[3])^(1/3) + (-2)^(1/3)*x), x], x, x^(1/3)])/(2*x^(1/3)*(1 + 2*x^2 + x^3)^(1/3)) - ((1 - I*Sqrt[3])^
(1/3)*(1 + I*Sqrt[3])*(x + 2*x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + 2*x^6 + x^9)^(1/3)/((1 - I*Sqrt[3])
^(1/3) - 2^(1/3)*x), x], x, x^(1/3)])/(2*x^(1/3)*(1 + 2*x^2 + x^3)^(1/3)) - ((1 - I*Sqrt[3])*(1 + I*Sqrt[3])^(
1/3)*(x + 2*x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + 2*x^6 + x^9)^(1/3)/((1 + I*Sqrt[3])^(1/3) - 2^(1/3)*
x), x], x, x^(1/3)])/(2*x^(1/3)*(1 + 2*x^2 + x^3)^(1/3)) - ((1 - I*Sqrt[3])^(1/3)*(1 + I*Sqrt[3])*(x + 2*x^3 +
x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + 2*x^6 + x^9)^(1/3)/((1 - I*Sqrt[3])^(1/3) - (-1)^(2/3)*2^(1/3)*x), x]
, x, x^(1/3)])/(2*x^(1/3)*(1 + 2*x^2 + x^3)^(1/3)) - ((1 - I*Sqrt[3])*(1 + I*Sqrt[3])^(1/3)*(x + 2*x^3 + x^4)^
(1/3)*Defer[Subst][Defer[Int][(1 + 2*x^6 + x^9)^(1/3)/((1 + I*Sqrt[3])^(1/3) - (-1)^(2/3)*2^(1/3)*x), x], x, x
^(1/3)])/(2*x^(1/3)*(1 + 2*x^2 + x^3)^(1/3)) - (6*(x + 2*x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(x^3*(1 + 2*
x^6 + x^9)^(1/3))/(1 + x^6 + x^9), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 + x^3)^(1/3)) - (9*(x + 2*x^3 + x^4)^(
1/3)*Defer[Subst][Defer[Int][(x^6*(1 + 2*x^6 + x^9)^(1/3))/(1 + x^6 + x^9), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x
^2 + x^3)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+2 x^3+x^4}}{\left (1+x^3\right ) \left (1+x^2+x^3\right )} \, dx &=\frac {\sqrt [3]{x+2 x^3+x^4} \int \frac {\sqrt [3]{x} \left (-2+x^3\right ) \sqrt [3]{1+2 x^2+x^3}}{\left (1+x^3\right ) \left (1+x^2+x^3\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1-x}+\frac {\sqrt [3]{x} (1+x) \sqrt [3]{1+2 x^2+x^3}}{1-x+x^2}+\frac {(-2-3 x) \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{1+x^2+x^3}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \int \frac {\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1-x} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\sqrt [3]{x+2 x^3+x^4} \int \frac {\sqrt [3]{x} (1+x) \sqrt [3]{1+2 x^2+x^3}}{1-x+x^2} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\sqrt [3]{x+2 x^3+x^4} \int \frac {(-2-3 x) \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{1+x^2+x^3} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1+i \sqrt {3}+2 x}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{-1-x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (-2-3 x^3\right ) \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-\sqrt [3]{1+2 x^6+x^9}-\frac {\sqrt [3]{1+2 x^6+x^9}}{-1-x^3}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {2 x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9}-\frac {3 x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1-i \sqrt {3}+2 x} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1+i \sqrt {3}+2 x} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{-1-x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (9 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{-1-i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{-1+i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [3]{1+2 x^6+x^9}}{3 (1+x)}+\frac {(-2+x) \sqrt [3]{1+2 x^6+x^9}}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (9 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2} \sqrt [3]{1+2 x^6+x^9}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{1+2 x^6+x^9}}{2 \left (-1-i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2} \sqrt [3]{1+2 x^6+x^9}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{1+2 x^6+x^9}}{2 \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\sqrt [3]{x+2 x^3+x^4} \operatorname {Subst}\left (\int \frac {(-2+x) \sqrt [3]{1+2 x^6+x^9}}{1-x+x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (9 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{-1-i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{-1+i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\sqrt [3]{x+2 x^3+x^4} \operatorname {Subst}\left (\int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{1+2 x^6+x^9}}{-1-i \sqrt {3}+2 x}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{1+2 x^6+x^9}}{-1+i \sqrt {3}+2 x}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (9 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (9 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\sqrt [3]{1-i \sqrt {3}} \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\sqrt [3]{1-i \sqrt {3}} \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\sqrt [3]{1-i \sqrt {3}} \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 2.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+2 x^3+x^4}}{\left (1+x^3\right ) \left (1+x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-2 + x^3)*(x + 2*x^3 + x^4)^(1/3))/((1 + x^3)*(1 + x^2 + x^3)),x]

[Out]

Integrate[((-2 + x^3)*(x + 2*x^3 + x^4)^(1/3))/((1 + x^3)*(1 + x^2 + x^3)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.61, size = 225, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{x^4+2 x^3+x}-x\right )+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^4+2 x^3+x}-2 x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+2 x^3+x}+x}\right )+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+2 x^3+x}+x}\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^4+2 x^3+x} x+\left (x^4+2 x^3+x\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+2 x^3+x} x+\sqrt [3]{2} \left (x^4+2 x^3+x\right )^{2/3}\right )}{2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-2 + x^3)*(x + 2*x^3 + x^4)^(1/3))/((1 + x^3)*(1 + x^2 + x^3)),x]

[Out]

-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + 2*x^3 + x^4)^(1/3))]) + 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/
3)*(x + 2*x^3 + x^4)^(1/3))] - Log[-x + (x + 2*x^3 + x^4)^(1/3)] + 2^(1/3)*Log[-2*x + 2^(2/3)*(x + 2*x^3 + x^4
)^(1/3)] + Log[x^2 + x*(x + 2*x^3 + x^4)^(1/3) + (x + 2*x^3 + x^4)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(x + 2*x^3
+ x^4)^(1/3) + 2^(1/3)*(x + 2*x^3 + x^4)^(2/3)]/2^(2/3)

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-2)*(x^4+2*x^3+x)^(1/3)/(x^3+1)/(x^3+x^2+1),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{3} + x^{2} + 1\right )} {\left (x^{3} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-2)*(x^4+2*x^3+x)^(1/3)/(x^3+1)/(x^3+x^2+1),x, algorithm="giac")

[Out]

integrate((x^4 + 2*x^3 + x)^(1/3)*(x^3 - 2)/((x^3 + x^2 + 1)*(x^3 + 1)), x)

________________________________________________________________________________________

maple [C]  time = 31.24, size = 2504, normalized size = 11.13 \begin {gather*} \text {Expression too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-2)*(x^4+2*x^3+x)^(1/3)/(x^3+1)/(x^3+x^2+1),x)

[Out]

-ln((20*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^3-40*RootOf(RootOf(_Z^3-2)^2+_Z*R
ootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^2+20*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)
^4-47*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^3-318*(x^4+2*x^3+x)^(2/3)*RootOf(Root
Of(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2+897*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*Root
Of(_Z^3-2)^2*(x^4+2*x^3+x)^(1/3)*x-746*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^2-47
*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2-904*x^3+1158*(x^4+2*x^3+x)^(2/3)+636*x*(x^4+
2*x^3+x)^(1/3)-2938*x^2-904)/(x^3+x^2+1))+RootOf(_Z^3-2)*ln((19681*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_
Z^2)^2*RootOf(_Z^3-2)^3*x^3+104831*RootOf(_Z^3-2)^4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^3-39362*
RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^2-209662*RootOf(_Z^3-2)^4*RootOf(RootOf(_
Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^2+19681*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3-
641838*(x^4+2*x^3+x)^(2/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2+104831*RootOf(_Z^3
-2)^4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)-78724*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*Ro
otOf(_Z^3-2)*x^3-419324*RootOf(_Z^3-2)^2*x^3-393620*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^
3-2)*x^2-2096620*RootOf(_Z^3-2)^2*x^2+2017896*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*(x^4+2*x^3+x)^(1
/3)*x+1283676*RootOf(_Z^3-2)*(x^4+2*x^3+x)^(1/3)*x-78724*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootO
f(_Z^3-2)+734220*(x^4+2*x^3+x)^(2/3)-419324*RootOf(_Z^3-2)^2)/(1+x)/(x^2-x+1))-ln(-(104831*RootOf(RootOf(_Z^3-
2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^3+19681*RootOf(_Z^3-2)^4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_
Z^3-2)+_Z^2)*x^3-209662*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^2-39362*RootOf(_Z
^3-2)^4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^2+104831*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_
Z^2)^2*RootOf(_Z^3-2)^3-641838*(x^4+2*x^3+x)^(2/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3
-2)^2+19681*RootOf(_Z^3-2)^4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)+628986*RootOf(RootOf(_Z^3-2)^2+_Z
*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^3+118086*RootOf(_Z^3-2)^2*x^3+1677296*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf
(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^2+314896*RootOf(_Z^3-2)^2*x^2-734220*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)
+_Z^2)*(x^4+2*x^3+x)^(1/3)*x+1283676*RootOf(_Z^3-2)*(x^4+2*x^3+x)^(1/3)*x+628986*RootOf(RootOf(_Z^3-2)^2+_Z*Ro
otOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)-2017896*(x^4+2*x^3+x)^(2/3)+118086*RootOf(_Z^3-2)^2)/(1+x)/(x^2-x+1))*RootOf
(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)-ln(-(104831*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf
(_Z^3-2)^3*x^3+19681*RootOf(_Z^3-2)^4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^3-209662*RootOf(RootOf
(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^2-39362*RootOf(_Z^3-2)^4*RootOf(RootOf(_Z^3-2)^2+_Z*Ro
otOf(_Z^3-2)+_Z^2)*x^2+104831*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3-641838*(x^4+2
*x^3+x)^(2/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2+19681*RootOf(_Z^3-2)^4*RootOf(R
ootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)+628986*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*
x^3+118086*RootOf(_Z^3-2)^2*x^3+1677296*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^2+314
896*RootOf(_Z^3-2)^2*x^2-734220*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*(x^4+2*x^3+x)^(1/3)*x+1283676*
RootOf(_Z^3-2)*(x^4+2*x^3+x)^(1/3)*x+628986*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)-201
7896*(x^4+2*x^3+x)^(2/3)+118086*RootOf(_Z^3-2)^2)/(1+x)/(x^2-x+1))*RootOf(_Z^3-2)+ln(-(11647*RootOf(RootOf(_Z^
3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^3-23294*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*
RootOf(_Z^3-2)^4*x^2+11647*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4+124118*RootOf(Ro
otOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^3-70932*(x^4+2*x^3+x)^(2/3)*RootOf(RootOf(_Z^3-2)^2+
_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2-115770*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)
^2*(x^4+2*x^3+x)^(1/3)*x+240938*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^2+124118*Ro
otOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2+76480*x^3-373404*(x^4+2*x^3+x)^(2/3)+141864*x*(
x^4+2*x^3+x)^(1/3)+168256*x^2+76480)/(x^3+x^2+1))+1/2*ln(-(11647*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^
2)^2*RootOf(_Z^3-2)^4*x^3-23294*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^2+11647*R
ootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4+124118*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^
3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^3-70932*(x^4+2*x^3+x)^(2/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*Root
Of(_Z^3-2)^2-115770*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^4+2*x^3+x)^(1/3)*x+240
938*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^2+124118*RootOf(RootOf(_Z^3-2)^2+_Z*Roo
tOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2+76480*x^3-373404*(x^4+2*x^3+x)^(2/3)+141864*x*(x^4+2*x^3+x)^(1/3)+168256*x^
2+76480)/(x^3+x^2+1))*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{3} + x^{2} + 1\right )} {\left (x^{3} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-2)*(x^4+2*x^3+x)^(1/3)/(x^3+1)/(x^3+x^2+1),x, algorithm="maxima")

[Out]

integrate((x^4 + 2*x^3 + x)^(1/3)*(x^3 - 2)/((x^3 + x^2 + 1)*(x^3 + 1)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (x^3-2\right )\,{\left (x^4+2\,x^3+x\right )}^{1/3}}{\left (x^3+1\right )\,\left (x^3+x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 2)*(x + 2*x^3 + x^4)^(1/3))/((x^3 + 1)*(x^2 + x^3 + 1)),x)

[Out]

int(((x^3 - 2)*(x + 2*x^3 + x^4)^(1/3))/((x^3 + 1)*(x^2 + x^3 + 1)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-2)*(x**4+2*x**3+x)**(1/3)/(x**3+1)/(x**3+x**2+1),x)

[Out]

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

________________________________________________________________________________________