3.23.94 \(\int \frac {(2-2 x+2 x^2-3 x^3+3 x^4) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) (-1+2 x-2 x^2+x^3) (-1-x^3+x^5)} \, dx\)

Optimal. Leaf size=318 \[ -\log \left (\sqrt [3]{x^6-x^4-x^3-x}+x\right )+\sqrt [3]{2} \log \left (\sqrt [3]{x^6-x^4-x^3-x}+\sqrt [3]{2} x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^6-x^4-x^3-x}}{\sqrt [3]{x^6-x^4-x^3-x}-2 x}\right )-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^6-x^4-x^3-x}}{\sqrt [3]{x^6-x^4-x^3-x}-2 \sqrt [3]{2} x}\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-x^4-x^3-x} x+\left (x^6-x^4-x^3-x\right )^{2/3}\right )-\frac {\log \left (2^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{x^6-x^4-x^3-x} x+\left (x^6-x^4-x^3-x\right )^{2/3}\right )}{2^{2/3}} \]

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Rubi [F]  time = 15.70, 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 (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((2 - 2*x + 2*x^2 - 3*x^3 + 3*x^4)*(-x - x^3 - x^4 + x^6)^(1/3))/((1 + x)*(-1 + 2*x - 2*x^2 + x^3)*(-1 - x
^3 + x^5)),x]

[Out]

-(((-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/(-1 + x), x], x, x^(1/3)
])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3))) - (2*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6
- x^9 + x^15)^(1/3)/(1 + x), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + (2*(1 + I*Sqrt[3])*(-x
- x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/(-1 - I*Sqrt[3] + 2*x), x], x,
x^(1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + ((1 - I*Sqrt[3])*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst]
[Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/(1 - I*Sqrt[3] + 2*x), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x
^5)^(1/3)) + (2*(1 - I*Sqrt[3])*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(
1/3)/(-1 + I*Sqrt[3] + 2*x), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + ((1 + I*Sqrt[3])*(-x -
x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/(1 + I*Sqrt[3] + 2*x), x], x, x^(
1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + (2*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 -
x^6 - x^9 + x^15)^(1/3)/((1 - I*Sqrt[3])^(1/3) + (-2)^(1/3)*x), x], x, x^(1/3)])/((1 - I*Sqrt[3])^(2/3)*x^(1/3
)*(-1 - x^2 - x^3 + x^5)^(1/3)) + (2*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^
15)^(1/3)/((1 + I*Sqrt[3])^(1/3) + (-2)^(1/3)*x), x], x, x^(1/3)])/((1 + I*Sqrt[3])^(2/3)*x^(1/3)*(-1 - x^2 -
x^3 + x^5)^(1/3)) + (2*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/((1
- I*Sqrt[3])^(1/3) - 2^(1/3)*x), x], x, x^(1/3)])/((1 - I*Sqrt[3])^(2/3)*x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3))
+ (2*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/((1 + I*Sqrt[3])^(1/3
) - 2^(1/3)*x), x], x, x^(1/3)])/((1 + I*Sqrt[3])^(2/3)*x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + (2*(-x - x^3 -
x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/((1 - I*Sqrt[3])^(1/3) - (-1)^(2/3)*2^
(1/3)*x), x], x, x^(1/3)])/((1 - I*Sqrt[3])^(2/3)*x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + (2*(-x - x^3 - x^4 +
x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/((1 + I*Sqrt[3])^(1/3) - (-1)^(2/3)*2^(1/3)*
x), x], x, x^(1/3)])/((1 + I*Sqrt[3])^(2/3)*x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) - (9*(-x - x^3 - x^4 + x^6)^
(1/3)*Defer[Subst][Defer[Int][(x^6*(-1 - x^6 - x^9 + x^15)^(1/3))/(-1 - x^9 + x^15), x], x, x^(1/3)])/(x^(1/3)
*(-1 - x^2 - x^3 + x^5)^(1/3)) + (15*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(x^12*(-1 - x^6 - x^
9 + x^15)^(1/3))/(-1 - x^9 + x^15), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx &=\frac {\sqrt [3]{-x-x^3-x^4+x^6} \int \frac {\sqrt [3]{x} \left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-1-x^2-x^3+x^5}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (2-2 x^3+2 x^6-3 x^9+3 x^{12}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{\left (1+x^3\right ) \left (-1+2 x^3-2 x^6+x^9\right ) \left (-1-x^9+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{3 (-1+x)}-\frac {2 \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 (1+x)}+\frac {2 (-2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (1-x+x^2\right )}+\frac {(2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (1+x+x^2\right )}+\frac {\left (1-2 x^3\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-x^3+x^6}+\frac {x^6 \left (-3+5 x^6\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=-\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {(2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x+x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {(-2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-x+x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\left (1-2 x^3\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-x^3+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (-3+5 x^6\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=-\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (-\frac {2 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x^3}-\frac {2 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x^3}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}}+\frac {5 x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=-\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (9 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (15 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=-\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{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-x^6-x^9+x^{15}}}{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-x^6-x^9+x^{15}}}{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 )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{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-x^6-x^9+x^{15}}}{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-x^6-x^9+x^{15}}}{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 )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (9 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (15 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=-\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (9 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (15 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ \end {align*}

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Mathematica [F]  time = 1.93, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((2 - 2*x + 2*x^2 - 3*x^3 + 3*x^4)*(-x - x^3 - x^4 + x^6)^(1/3))/((1 + x)*(-1 + 2*x - 2*x^2 + x^3)*(
-1 - x^3 + x^5)),x]

[Out]

Integrate[((2 - 2*x + 2*x^2 - 3*x^3 + 3*x^4)*(-x - x^3 - x^4 + x^6)^(1/3))/((1 + x)*(-1 + 2*x - 2*x^2 + x^3)*(
-1 - x^3 + x^5)), x]

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IntegrateAlgebraic [A]  time = 5.10, size = 318, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{x^6-x^4-x^3-x}+x\right )+\sqrt [3]{2} \log \left (\sqrt [3]{x^6-x^4-x^3-x}+\sqrt [3]{2} x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^6-x^4-x^3-x}}{\sqrt [3]{x^6-x^4-x^3-x}-2 x}\right )-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^6-x^4-x^3-x}}{\sqrt [3]{x^6-x^4-x^3-x}-2 \sqrt [3]{2} x}\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-x^4-x^3-x} x+\left (x^6-x^4-x^3-x\right )^{2/3}\right )-\frac {\log \left (2^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{x^6-x^4-x^3-x} x+\left (x^6-x^4-x^3-x\right )^{2/3}\right )}{2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 - 2*x + 2*x^2 - 3*x^3 + 3*x^4)*(-x - x^3 - x^4 + x^6)^(1/3))/((1 + x)*(-1 + 2*x - 2*x^2
+ x^3)*(-1 - x^3 + x^5)),x]

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*(-x - x^3 - x^4 + x^6)^(1/3))/(-2*x + (-x - x^3 - x^4 + x^6)^(1/3))] - 2^(1/3)*Sqrt[3]
*ArcTan[(Sqrt[3]*(-x - x^3 - x^4 + x^6)^(1/3))/(-2*2^(1/3)*x + (-x - x^3 - x^4 + x^6)^(1/3))] - Log[x + (-x -
x^3 - x^4 + x^6)^(1/3)] + 2^(1/3)*Log[2^(1/3)*x + (-x - x^3 - x^4 + x^6)^(1/3)] + Log[x^2 - x*(-x - x^3 - x^4
+ x^6)^(1/3) + (-x - x^3 - x^4 + x^6)^(2/3)]/2 - Log[2^(2/3)*x^2 - 2^(1/3)*x*(-x - x^3 - x^4 + x^6)^(1/3) + (-
x - x^3 - x^4 + x^6)^(2/3)]/2^(2/3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2+2*x-1)/(x^5-x^3-1),x, algorithm="fr
icas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - x^{4} - x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x^{4} - 3 \, x^{3} + 2 \, x^{2} - 2 \, x + 2\right )}}{{\left (x^{5} - x^{3} - 1\right )} {\left (x^{3} - 2 \, x^{2} + 2 \, x - 1\right )} {\left (x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2+2*x-1)/(x^5-x^3-1),x, algorithm="gi
ac")

[Out]

integrate((x^6 - x^4 - x^3 - x)^(1/3)*(3*x^4 - 3*x^3 + 2*x^2 - 2*x + 2)/((x^5 - x^3 - 1)*(x^3 - 2*x^2 + 2*x -
1)*(x + 1)), x)

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maple [C]  time = 107.01, size = 3484, normalized size = 10.96 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2+2*x-1)/(x^5-x^3-1),x)

[Out]

-1/2*ln(-(17352778151361340*RootOf(_Z^3-2)^4*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*x^5+193987692
20353248*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^5-17352778151361340*RootOf(_
Z^3-2)^4*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*x^3-19398769220353248*RootOf(4*RootOf(_Z^3-2)^2+2
*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^3-99778474370327705*RootOf(_Z^3-2)^4*RootOf(4*RootOf(_Z^3-2)^2+2
*_Z*RootOf(_Z^3-2)+_Z^2)*x^2-111542923017031176*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_
Z^3-2)^3*x^2-95440279832487370*RootOf(_Z^3-2)^2*x^5-106693230711942864*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_
Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^5+101047682304680178*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(
_Z^3-2)^2*(x^6-x^4-x^3-x)^(2/3)-17352778151361340*RootOf(_Z^3-2)^4*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-
2)+_Z^2)-19398769220353248*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3+954402798324
87370*RootOf(_Z^3-2)^2*x^3+106693230711942864*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-
2)*x^3-43381945378403350*RootOf(_Z^3-2)^2*x^2-48496923050883120*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+
_Z^2)*RootOf(_Z^3-2)*x^2+404190729218720712*RootOf(_Z^3-2)*(x^6-x^4-x^3-x)^(1/3)*x+549795936176579706*RootOf(4
*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*(x^6-x^4-x^3-x)^(1/3)*x-695401143134438700*(x^6-x^4-x^3-x)^(2/3)+9
5440279832487370*RootOf(_Z^3-2)^2+106693230711942864*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootO
f(_Z^3-2))/(1+x)^2/(-1+x)/(x^2-x+1))*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)-ln(-(1735277815136134
0*RootOf(_Z^3-2)^4*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*x^5+19398769220353248*RootOf(4*RootOf(_
Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^5-17352778151361340*RootOf(_Z^3-2)^4*RootOf(4*RootOf(_
Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*x^3-19398769220353248*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2
*RootOf(_Z^3-2)^3*x^3-99778474370327705*RootOf(_Z^3-2)^4*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*x
^2-111542923017031176*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^2-9544027983248
7370*RootOf(_Z^3-2)^2*x^5-106693230711942864*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2
)*x^5+101047682304680178*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^6-x^4-x^3-x)^
(2/3)-17352778151361340*RootOf(_Z^3-2)^4*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)-19398769220353248
*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3+95440279832487370*RootOf(_Z^3-2)^2*x^3
+106693230711942864*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^3-43381945378403350*R
ootOf(_Z^3-2)^2*x^2-48496923050883120*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^2+4
04190729218720712*RootOf(_Z^3-2)*(x^6-x^4-x^3-x)^(1/3)*x+549795936176579706*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*Roo
tOf(_Z^3-2)+_Z^2)*(x^6-x^4-x^3-x)^(1/3)*x-695401143134438700*(x^6-x^4-x^3-x)^(2/3)+95440279832487370*RootOf(_Z
^3-2)^2+106693230711942864*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2))/(1+x)^2/(-1+x)/
(x^2-x+1))*RootOf(_Z^3-2)+RootOf(_Z^3-2)*ln((77595076881412992*RootOf(_Z^3-2)^4*RootOf(4*RootOf(_Z^3-2)^2+2*_Z
*RootOf(_Z^3-2)+_Z^2)*x^5+17352778151361340*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-
2)^3*x^5-77595076881412992*RootOf(_Z^3-2)^4*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*x^3-1735277815
1361340*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^3-446171692068124704*RootOf(_
Z^3-2)^4*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*x^2-99778474370327705*RootOf(4*RootOf(_Z^3-2)^2+2
*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^3*x^2+737153230373423424*RootOf(_Z^3-2)^2*x^5+164851392437932730*Roo
tOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^5+202095364609360356*RootOf(4*RootOf(_Z^3-2)
^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^6-x^4-x^3-x)^(2/3)-77595076881412992*RootOf(_Z^3-2)^4*RootOf(
4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)-17352778151361340*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_
Z^2)^2*RootOf(_Z^3-2)^3-737153230373423424*RootOf(_Z^3-2)^2*x^3-164851392437932730*RootOf(4*RootOf(_Z^3-2)^2+2
*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^3-1590699076068966336*RootOf(_Z^3-2)^2*x^2-355731952102907470*RootOf
(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^2+808381458437441424*RootOf(_Z^3-2)*(x^6-x^4-x^
3-x)^(1/3)*x-695401143134438700*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*(x^6-x^4-x^3-x)^(1/3)*x+21
99183744706318824*(x^6-x^4-x^3-x)^(2/3)-737153230373423424*RootOf(_Z^3-2)^2-164851392437932730*RootOf(4*RootOf
(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2))/(1+x)^2/(-1+x)/(x^2-x+1))-ln(-(2378228285152225516*RootOf
(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^5-2378228285152225516*RootOf(4*RootOf(_Z^3-
2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^3-13674812639625296717*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*Root
Of(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^2-24953827777837657564*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^
2)*RootOf(_Z^3-2)^2*x^5-2378228285152225516*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-
2)^4+24953827777837657564*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^3+27374521598
3433009056*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^6-x^4-x^3-x)^(2/3)[phone]
4722205696996*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^6-x^4-x^3-x)^(1/3)*x+322
446188180271501072*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^2-133266313871886890
4992*x^5+24953827777837657564*RootOf(_Z^3-2)^2*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)[phone]
18868904992*x^3-1508581754955090751760*(x^6-x^4-x^3-x)^(2/3)+1094980863933732036224*x*(x^6-x^4-x^3-x)^(1/3)+23
84765616654818040512*x^2+1332663138718868904992)/(x^5-x^3-1))+ln((-8767520649466242796*RootOf(4*RootOf(_Z^3-2)
^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^5+8767520649466242796*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(
_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^3+50413243734430896077*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^
2*RootOf(_Z^3-2)^4*x^2+136269266312298092972*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2
)^2*x^5+8767520649466242796*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4-13626926631
2298092972*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^3+136872607991716504528*Root
Of(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^6-x^4-x^3-x)^(2/3)-188572719369386343970*R
ootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^6-x^4-x^3-x)^(1/3)*x-123792352423379264
190*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^2-71346848554566765480*x^5-13626926
6312298092972*RootOf(_Z^3-2)^2*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)+71346848554566765480*x^3+13
01781309444411393992*(x^6-x^4-x^3-x)^(2/3)+547490431966866018112*x*(x^6-x^4-x^3-x)^(1/3)+52321022273348961352*
x^2+71346848554566765480)/(x^5-x^3-1))+1/4*ln((-8767520649466242796*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3
-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^5+8767520649466242796*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*Roo
tOf(_Z^3-2)^4*x^3+50413243734430896077*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*
x^2+136269266312298092972*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^5+87675206494
66242796*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4-136269266312298092972*RootOf(4
*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^3+136872607991716504528*RootOf(4*RootOf(_Z^3-2)
^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^6-x^4-x^3-x)^(2/3)-188572719369386343970*RootOf(4*RootOf(_Z^3
-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*(x^6-x^4-x^3-x)^(1/3)*x-123792352423379264190*RootOf(4*RootOf
(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^2-71346848554566765480*x^5-136269266312298092972*RootO
f(_Z^3-2)^2*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)+71346848554566765480*x^3+130178130944441139399
2*(x^6-x^4-x^3-x)^(2/3)+547490431966866018112*x*(x^6-x^4-x^3-x)^(1/3)+52321022273348961352*x^2+713468485545667
65480)/(x^5-x^3-1))*RootOf(_Z^3-2)^2*RootOf(4*RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+_Z^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - x^{4} - x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x^{4} - 3 \, x^{3} + 2 \, x^{2} - 2 \, x + 2\right )}}{{\left (x^{5} - x^{3} - 1\right )} {\left (x^{3} - 2 \, x^{2} + 2 \, x - 1\right )} {\left (x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2+2*x-1)/(x^5-x^3-1),x, algorithm="ma
xima")

[Out]

integrate((x^6 - x^4 - x^3 - x)^(1/3)*(3*x^4 - 3*x^3 + 2*x^2 - 2*x + 2)/((x^5 - x^3 - 1)*(x^3 - 2*x^2 + 2*x -
1)*(x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {{\left (x^6-x^4-x^3-x\right )}^{1/3}\,\left (3\,x^4-3\,x^3+2\,x^2-2\,x+2\right )}{\left (x+1\right )\,\left (-x^5+x^3+1\right )\,\left (x^3-2\,x^2+2\,x-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^6 - x^3 - x^4 - x)^(1/3)*(2*x^2 - 2*x - 3*x^3 + 3*x^4 + 2))/((x + 1)*(x^3 - x^5 + 1)*(2*x - 2*x^2 + x
^3 - 1)),x)

[Out]

int(-((x^6 - x^3 - x^4 - x)^(1/3)*(2*x^2 - 2*x - 3*x^3 + 3*x^4 + 2))/((x + 1)*(x^3 - x^5 + 1)*(2*x - 2*x^2 + x
^3 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x^{5} - x^{3} - x^{2} - 1\right )} \left (3 x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 2\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{5} - x^{3} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**4-3*x**3+2*x**2-2*x+2)*(x**6-x**4-x**3-x)**(1/3)/(1+x)/(x**3-2*x**2+2*x-1)/(x**5-x**3-1),x)

[Out]

Integral((x*(x**5 - x**3 - x**2 - 1))**(1/3)*(3*x**4 - 3*x**3 + 2*x**2 - 2*x + 2)/((x - 1)*(x + 1)*(x**2 - x +
1)*(x**5 - x**3 - 1)), x)

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