3.24.88 \(\int \frac {(-1+2 x^6) \sqrt [3]{x+x^7}}{(1-2 x^2+x^6) (1-x^2+x^6)} \, dx\)

Optimal. Leaf size=191 \[ -\frac {1}{2} \log \left (\sqrt [3]{x^7+x}-x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^7+x}-2 x\right )}{2^{2/3}}-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^7+x}+x}\right )+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^7+x}+x}\right )}{2^{2/3}}+\frac {1}{4} \log \left (\sqrt [3]{x^7+x} x+\left (x^7+x\right )^{2/3}+x^2\right )-\frac {\log \left (2^{2/3} \sqrt [3]{x^7+x} x+\sqrt [3]{2} \left (x^7+x\right )^{2/3}+2 x^2\right )}{2\ 2^{2/3}} \]

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Rubi [F]  time = 6.54, 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 (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + 2*x^6)*(x + x^7)^(1/3))/((1 - 2*x^2 + x^6)*(1 - x^2 + x^6)),x]

[Out]

((x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/(-1 + x), x], x, x^(2/3)])/(2*x^(1/3)*(1 + x^6)^(1/3)
) - ((1 + I*Sqrt[3])*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/(1 - I*Sqrt[3] + 2*x), x], x, x^(
2/3)])/(2*x^(1/3)*(1 + x^6)^(1/3)) - ((1 - I*Sqrt[3])*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/
(1 + I*Sqrt[3] + 2*x), x], x, x^(2/3)])/(2*x^(1/3)*(1 + x^6)^(1/3)) - ((-1 - Sqrt[5])^(2/3)*(x + x^7)^(1/3)*De
fer[Subst][Defer[Int][(1 + x^9)^(1/3)/((1 + Sqrt[5])^(1/3) - (-2)^(1/3)*x), x], x, x^(2/3)])/(2^(1/3)*Sqrt[5]*
x^(1/3)*(1 + x^6)^(1/3)) + ((-1)^(2/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((1 + Sqrt[5])^
(1/3) - (-2)^(1/3)*x), x], x, x^(2/3)])/(Sqrt[5]*(2*(1 + Sqrt[5]))^(1/3)*x^(1/3)*(1 + x^6)^(1/3)) - ((1 - Sqrt
[5])^(2/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((-1 + Sqrt[5])^(1/3) + (-2)^(1/3)*x), x],
x, x^(2/3)])/(2^(1/3)*Sqrt[5]*x^(1/3)*(1 + x^6)^(1/3)) - ((-1)^(2/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(
1 + x^9)^(1/3)/((-1 + Sqrt[5])^(1/3) + (-2)^(1/3)*x), x], x, x^(2/3)])/(Sqrt[5]*(2*(-1 + Sqrt[5]))^(1/3)*x^(1/
3)*(1 + x^6)^(1/3)) - ((-1 + Sqrt[5])^(2/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((-1 + Sqr
t[5])^(1/3) - 2^(1/3)*x), x], x, x^(2/3)])/(2^(1/3)*Sqrt[5]*x^(1/3)*(1 + x^6)^(1/3)) - ((x + x^7)^(1/3)*Defer[
Subst][Defer[Int][(1 + x^9)^(1/3)/((-1 + Sqrt[5])^(1/3) - 2^(1/3)*x), x], x, x^(2/3)])/(Sqrt[5]*(2*(-1 + Sqrt[
5]))^(1/3)*x^(1/3)*(1 + x^6)^(1/3)) - ((1 + Sqrt[5])^(2/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(
1/3)/((1 + Sqrt[5])^(1/3) + 2^(1/3)*x), x], x, x^(2/3)])/(2^(1/3)*Sqrt[5]*x^(1/3)*(1 + x^6)^(1/3)) + ((x + x^7
)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((1 + Sqrt[5])^(1/3) + 2^(1/3)*x), x], x, x^(2/3)])/(Sqrt[5]*(
2*(1 + Sqrt[5]))^(1/3)*x^(1/3)*(1 + x^6)^(1/3)) + ((-1 - Sqrt[5])^(1/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int
][(1 + x^9)^(1/3)/((-1 + Sqrt[5])^(1/3) - (-1)^(2/3)*2^(1/3)*x), x], x, x^(2/3)])/(2*Sqrt[5]*x^(1/3)*(1 + x^6)
^(1/3)) + ((-1 - Sqrt[5])^(1/3)*(5 - Sqrt[5])*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((-1 + S
qrt[5])^(1/3) - (-1)^(2/3)*2^(1/3)*x), x], x, x^(2/3)])/(10*x^(1/3)*(1 + x^6)^(1/3)) - ((1 - Sqrt[5])^(1/3)*(x
+ x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((1 + Sqrt[5])^(1/3) + (-1)^(2/3)*2^(1/3)*x), x], x, x^(
2/3)])/(2*Sqrt[5]*x^(1/3)*(1 + x^6)^(1/3)) + ((1 - Sqrt[5])^(1/3)*(5 + Sqrt[5])*(x + x^7)^(1/3)*Defer[Subst][D
efer[Int][(1 + x^9)^(1/3)/((1 + Sqrt[5])^(1/3) + (-1)^(2/3)*2^(1/3)*x), x], x, x^(2/3)])/(10*x^(1/3)*(1 + x^6)
^(1/3)) + (3*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 + x^9)^(1/3))/(1 - x^3 + x^9), x], x, x^(2/3)])/(2*
x^(1/3)*(1 + x^6)^(1/3)) - (9*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(x^7*(1 + x^9)^(1/3))/(1 - x^3 + x^9), x
], x, x^(2/3)])/(2*x^(1/3)*(1 + x^6)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx &=\frac {\sqrt [3]{x+x^7} \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^6} \left (-1+2 x^6\right )}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^{18}} \left (-1+2 x^{18}\right )}{\left (1-2 x^6+x^{18}\right ) \left (1-x^6+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9} \left (-1+2 x^9\right )}{\left (1-2 x^3+x^9\right ) \left (1-x^3+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{1+x^9}}{3 (-1+x)}+\frac {(1-x) \sqrt [3]{1+x^9}}{3 \left (1+x+x^2\right )}+\frac {x \left (-1-2 x^3\right ) \sqrt [3]{1+x^9}}{1-x^3-x^6}+\frac {x \left (1-3 x^6\right ) \sqrt [3]{1+x^9}}{1-x^3+x^9}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}\\ &=\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \frac {(1-x) \sqrt [3]{1+x^9}}{1+x+x^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \left (-1-2 x^3\right ) \sqrt [3]{1+x^9}}{1-x^3-x^6} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \left (1-3 x^6\right ) \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}\\ &=\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \left (\frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{1+x^9}}{1-i \sqrt {3}+2 x}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{1+x^9}}{1+i \sqrt {3}+2 x}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {x \sqrt [3]{1+x^9}}{-1+x^3+x^6}+\frac {2 x^4 \sqrt [3]{1+x^9}}{-1+x^3+x^6}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {x \sqrt [3]{1+x^9}}{1-x^3+x^9}-\frac {3 x^7 \sqrt [3]{1+x^9}}{1-x^3+x^9}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}\\ &=\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9}}{-1+x^3+x^6} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [3]{1+x^9}}{-1+x^3+x^6} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (9 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^7 \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{1-i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}\\ &=\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \left (-\frac {2 x \sqrt [3]{1+x^9}}{\sqrt {5} \left (-1+\sqrt {5}-2 x^3\right )}-\frac {2 x \sqrt [3]{1+x^9}}{\sqrt {5} \left (1+\sqrt {5}+2 x^3\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \left (-\frac {\left (-1+\sqrt {5}\right ) x \sqrt [3]{1+x^9}}{\sqrt {5} \left (-1+\sqrt {5}-2 x^3\right )}+\frac {\left (1+\sqrt {5}\right ) x \sqrt [3]{1+x^9}}{\sqrt {5} \left (1+\sqrt {5}+2 x^3\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (9 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^7 \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{1-i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}\\ &=\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (9 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^7 \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9}}{-1+\sqrt {5}-2 x^3} \, dx,x,x^{2/3}\right )}{\sqrt {5} \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9}}{1+\sqrt {5}+2 x^3} \, dx,x,x^{2/3}\right )}{\sqrt {5} \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{1-i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (3 \left (5-\sqrt {5}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9}}{-1+\sqrt {5}-2 x^3} \, dx,x,x^{2/3}\right )}{5 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \left (5+\sqrt {5}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9}}{1+\sqrt {5}+2 x^3} \, dx,x,x^{2/3}\right )}{5 \sqrt [3]{x} \sqrt [3]{1+x^6}}\\ &=\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (9 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^7 \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {(-1)^{2/3} \sqrt [3]{1+x^9}}{3 \sqrt [3]{2 \left (-1+\sqrt {5}\right )} \left (\sqrt [3]{-1+\sqrt {5}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1+x^9}}{3 \sqrt [3]{2 \left (-1+\sqrt {5}\right )} \left (\sqrt [3]{-1+\sqrt {5}}-\sqrt [3]{2} x\right )}-\frac {\sqrt [3]{\frac {1}{2} \left (1-\sqrt {5}\right )} \sqrt [3]{1+x^9}}{3 \left (-1+\sqrt {5}\right )^{2/3} \left (\sqrt [3]{-1+\sqrt {5}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt {5} \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1)^{2/3} \sqrt [3]{1+x^9}}{3 \sqrt [3]{2 \left (1+\sqrt {5}\right )} \left (\sqrt [3]{1+\sqrt {5}}-\sqrt [3]{-2} x\right )}-\frac {\sqrt [3]{1+x^9}}{3 \sqrt [3]{2 \left (1+\sqrt {5}\right )} \left (\sqrt [3]{1+\sqrt {5}}+\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{\frac {1}{2} \left (-1-\sqrt {5}\right )} \sqrt [3]{1+x^9}}{3 \left (1+\sqrt {5}\right )^{2/3} \left (\sqrt [3]{1+\sqrt {5}}+(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt {5} \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{1-i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (3 \left (5-\sqrt {5}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {(-1)^{2/3} \sqrt [3]{1+x^9}}{3 \sqrt [3]{2 \left (-1+\sqrt {5}\right )} \left (\sqrt [3]{-1+\sqrt {5}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1+x^9}}{3 \sqrt [3]{2 \left (-1+\sqrt {5}\right )} \left (\sqrt [3]{-1+\sqrt {5}}-\sqrt [3]{2} x\right )}-\frac {\sqrt [3]{\frac {1}{2} \left (1-\sqrt {5}\right )} \sqrt [3]{1+x^9}}{3 \left (-1+\sqrt {5}\right )^{2/3} \left (\sqrt [3]{-1+\sqrt {5}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,x^{2/3}\right )}{5 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \left (5+\sqrt {5}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1)^{2/3} \sqrt [3]{1+x^9}}{3 \sqrt [3]{2 \left (1+\sqrt {5}\right )} \left (\sqrt [3]{1+\sqrt {5}}-\sqrt [3]{-2} x\right )}-\frac {\sqrt [3]{1+x^9}}{3 \sqrt [3]{2 \left (1+\sqrt {5}\right )} \left (\sqrt [3]{1+\sqrt {5}}+\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{\frac {1}{2} \left (-1-\sqrt {5}\right )} \sqrt [3]{1+x^9}}{3 \left (1+\sqrt {5}\right )^{2/3} \left (\sqrt [3]{1+\sqrt {5}}+(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,x^{2/3}\right )}{5 \sqrt [3]{x} \sqrt [3]{1+x^6}}\\ &=\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (3 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (9 \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^7 \sqrt [3]{1+x^9}}{1-x^3+x^9} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{1-i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\sqrt [3]{-1-\sqrt {5}} \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{-1+\sqrt {5}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (\sqrt [3]{1-\sqrt {5}} \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{1+\sqrt {5}}+(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\sqrt [3]{-1-\sqrt {5}} \left (5-\sqrt {5}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{-1+\sqrt {5}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{10 \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{-1+\sqrt {5}}-\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{\sqrt {5} \sqrt [3]{2 \left (-1+\sqrt {5}\right )} \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left ((-1)^{2/3} \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{-1+\sqrt {5}}+\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{\sqrt {5} \sqrt [3]{2 \left (-1+\sqrt {5}\right )} \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (\left (5-\sqrt {5}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{-1+\sqrt {5}}-\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{5 \sqrt [3]{2 \left (-1+\sqrt {5}\right )} \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left ((-1)^{2/3} \left (5-\sqrt {5}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{-1+\sqrt {5}}+\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{5 \sqrt [3]{2 \left (-1+\sqrt {5}\right )} \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\sqrt [3]{x+x^7} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{1+\sqrt {5}}+\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{\sqrt {5} \sqrt [3]{2 \left (1+\sqrt {5}\right )} \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left ((-1)^{2/3} \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{1+\sqrt {5}}-\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{\sqrt {5} \sqrt [3]{2 \left (1+\sqrt {5}\right )} \sqrt [3]{x} \sqrt [3]{1+x^6}}+\frac {\left (\sqrt [3]{1-\sqrt {5}} \left (5+\sqrt {5}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{1+\sqrt {5}}+(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{10 \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left (\left (5+\sqrt {5}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{1+\sqrt {5}}+\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{5 \sqrt [3]{2 \left (1+\sqrt {5}\right )} \sqrt [3]{x} \sqrt [3]{1+x^6}}-\frac {\left ((-1)^{2/3} \left (5+\sqrt {5}\right ) \sqrt [3]{x+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x^9}}{\sqrt [3]{1+\sqrt {5}}-\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{5 \sqrt [3]{2 \left (1+\sqrt {5}\right )} \sqrt [3]{x} \sqrt [3]{1+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + 2*x^6)*(x + x^7)^(1/3))/((1 - 2*x^2 + x^6)*(1 - x^2 + x^6)),x]

[Out]

Integrate[((-1 + 2*x^6)*(x + x^7)^(1/3))/((1 - 2*x^2 + x^6)*(1 - x^2 + x^6)), x]

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IntegrateAlgebraic [A]  time = 2.99, size = 191, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^7}}\right )+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^7}}\right )}{2^{2/3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{x+x^7}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x+x^7}\right )}{2^{2/3}}+\frac {1}{4} \log \left (x^2+x \sqrt [3]{x+x^7}+\left (x+x^7\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^7}+\sqrt [3]{2} \left (x+x^7\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + 2*x^6)*(x + x^7)^(1/3))/((1 - 2*x^2 + x^6)*(1 - x^2 + x^6)),x]

[Out]

-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^7)^(1/3))]) + (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^7
)^(1/3))])/2^(2/3) - Log[-x + (x + x^7)^(1/3)]/2 + Log[-2*x + 2^(2/3)*(x + x^7)^(1/3)]/2^(2/3) + Log[x^2 + x*(
x + x^7)^(1/3) + (x + x^7)^(2/3)]/4 - Log[2*x^2 + 2^(2/3)*x*(x + x^7)^(1/3) + 2^(1/3)*(x + x^7)^(2/3)]/(2*2^(2
/3))

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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((2*x^6-1)*(x^7+x)^(1/3)/(x^6-2*x^2+1)/(x^6-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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{7} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - 1\right )}}{{\left (x^{6} - x^{2} + 1\right )} {\left (x^{6} - 2 \, x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6-1)*(x^7+x)^(1/3)/(x^6-2*x^2+1)/(x^6-x^2+1),x, algorithm="giac")

[Out]

integrate((x^7 + x)^(1/3)*(2*x^6 - 1)/((x^6 - x^2 + 1)*(x^6 - 2*x^2 + 1)), x)

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maple [F]  time = 2.21, size = 0, normalized size = 0.00 \[\int \frac {\left (2 x^{6}-1\right ) \left (x^{7}+x \right )^{\frac {1}{3}}}{\left (x^{6}-2 x^{2}+1\right ) \left (x^{6}-x^{2}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^6-1)*(x^7+x)^(1/3)/(x^6-2*x^2+1)/(x^6-x^2+1),x)

[Out]

int((2*x^6-1)*(x^7+x)^(1/3)/(x^6-2*x^2+1)/(x^6-x^2+1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{7} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - 1\right )}}{{\left (x^{6} - x^{2} + 1\right )} {\left (x^{6} - 2 \, x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6-1)*(x^7+x)^(1/3)/(x^6-2*x^2+1)/(x^6-x^2+1),x, algorithm="maxima")

[Out]

integrate((x^7 + x)^(1/3)*(2*x^6 - 1)/((x^6 - x^2 + 1)*(x^6 - 2*x^2 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^6-1\right )\,{\left (x^7+x\right )}^{1/3}}{\left (x^6-x^2+1\right )\,\left (x^6-2\,x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^6 - 1)*(x + x^7)^(1/3))/((x^6 - x^2 + 1)*(x^6 - 2*x^2 + 1)),x)

[Out]

int(((2*x^6 - 1)*(x + x^7)^(1/3))/((x^6 - x^2 + 1)*(x^6 - 2*x^2 + 1)), x)

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sympy [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((2*x**6-1)*(x**7+x)**(1/3)/(x**6-2*x**2+1)/(x**6-x**2+1),x)

[Out]

Timed out

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