3.21.78 \(\int \frac {(-4+5 x^7) \sqrt [3]{-2 x+2 x^3-x^8}}{(2+x^7) (2-2 x^2+x^7)} \, dx\) [2078]

Optimal. Leaf size=150 \[ -\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-2 x+2 x^3-x^8}}\right )}{2^{2/3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-2 x+2 x^3-x^8}\right )}{2^{2/3}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-2 x+2 x^3-x^8}+\sqrt [3]{2} \left (-2 x+2 x^3-x^8\right )^{2/3}\right )}{2\ 2^{2/3}} \]

[Out]

-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(-x^8+2*x^3-2*x)^(1/3)))*2^(1/3)-1/2*ln(-2*x+2^(2/3)*(-x^8+2*x^3-2*x)
^(1/3))*2^(1/3)+1/4*ln(2*x^2+2^(2/3)*x*(-x^8+2*x^3-2*x)^(1/3)+2^(1/3)*(-x^8+2*x^3-2*x)^(2/3))*2^(1/3)

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Rubi [F]
time = 9.43, 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 (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-4 + 5*x^7)*(-2*x + 2*x^3 - x^8)^(1/3))/((2 + x^7)*(2 - 2*x^2 + x^7)),x]

[Out]

(-3*(-2)^(1/7)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][(-2 + 2*x^6 - x^21)^(-2/3), x], x, x^(1/3)])
/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + (3*2^(1/7)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][(-2 + 2*x^
6 - x^21)^(-2/3), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + (3*(-1)^(2/7)*2^(1/7)*(-2*x + 2*x^3 -
x^8)^(1/3)*Defer[Subst][Defer[Int][(-2 + 2*x^6 - x^21)^(-2/3), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1
/3)) - (3*(-1)^(3/7)*2^(1/7)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][(-2 + 2*x^6 - x^21)^(-2/3), x]
, x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + (3*(-1)^(4/7)*2^(1/7)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Sub
st][Defer[Int][(-2 + 2*x^6 - x^21)^(-2/3), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) - (3*(-1)^(5/7)
*2^(1/7)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][(-2 + 2*x^6 - x^21)^(-2/3), x], x, x^(1/3)])/(x^(1
/3)*(-2 + 2*x^2 - x^7)^(1/3)) + (3*(-1)^(6/7)*2^(1/7)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][(-2 +
2*x^6 - x^21)^(-2/3), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-2)^(4/21)*(-2*x + 2*x^3 - x^8)
^(1/3)*Defer[Subst][Defer[Int][1/(((-2)^(1/21) - x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2
+ 2*x^2 - x^7)^(1/3)) + (2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/((-2^(1/21) - x)*(-2 +
2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-1)^(8/21)*2^(4/21)*(-2*x + 2*x^
3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/((-((-1)^(2/21)*2^(1/21)) - x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(
1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-1)^(4/7)*2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer
[Int][1/(((-1)^(1/7)*2^(1/21) - x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1
/3)) + ((-1)^(16/21)*2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/((-((-1)^(4/21)*2^(1/21)) -
x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-1)^(20/21)*2^(4/21)*(
-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/(((-1)^(5/21)*2^(1/21) - x)*(-2 + 2*x^6 - x^21)^(2/3)), x]
, x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) - ((-1)^(1/7)*2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subs
t][Defer[Int][1/((-((-1)^(2/7)*2^(1/21)) - x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^
2 - x^7)^(1/3)) - (15*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][x^3/(-2 + 2*x^6 - x^21)^(2/3), x], x,
x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-2)^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int
][1/(((-2)^(1/21) + (-1)^(1/3)*x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/
3)) + (2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/((-2^(1/21) + (-1)^(1/3)*x)*(-2 + 2*x^6 -
x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-1)^(8/21)*2^(4/21)*(-2*x + 2*x^3 - x^8
)^(1/3)*Defer[Subst][Defer[Int][1/((-((-1)^(2/21)*2^(1/21)) + (-1)^(1/3)*x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x,
x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-1)^(4/7)*2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][D
efer[Int][1/(((-1)^(1/7)*2^(1/21) + (-1)^(1/3)*x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 +
2*x^2 - x^7)^(1/3)) + ((-1)^(16/21)*2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/((-((-1)^(4/
21)*2^(1/21)) + (-1)^(1/3)*x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3))
+ ((-1)^(20/21)*2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/(((-1)^(5/21)*2^(1/21) + (-1)^(1
/3)*x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) - ((-1)^(1/7)*2^(4/21)*
(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/((-((-1)^(2/7)*2^(1/21)) + (-1)^(1/3)*x)*(-2 + 2*x^6 - x^
21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-2)^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defe
r[Subst][Defer[Int][1/(((-2)^(1/21) - (-1)^(2/3)*x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2
+ 2*x^2 - x^7)^(1/3)) + (2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/((-2^(1/21) - (-1)^(2/3
)*x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-1)^(8/21)*2^(4/21)*(
-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/((-((-1)^(2/21)*2^(1/21)) - (-1)^(2/3)*x)*(-2 + 2*x^6 - x^
21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-1)^(4/7)*2^(4/21)*(-2*x + 2*x^3 - x^8)^(1
/3)*Defer[Subst][Defer[Int][1/(((-1)^(1/7)*2^(1/21) - (-1)^(2/3)*x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)
])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) + ((-1)^(16/21)*2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[I
nt][1/((-((-1)^(4/21)*2^(1/21)) - (-1)^(2/3)*x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*
x^2 - x^7)^(1/3)) + ((-1)^(20/21)*2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*Defer[Subst][Defer[Int][1/(((-1)^(5/21)*
2^(1/21) - (-1)^(2/3)*x)*(-2 + 2*x^6 - x^21)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(-2 + 2*x^2 - x^7)^(1/3)) - ((-
1)^(1/7)*2^(4/21)*(-2*x + 2*x^3 - x^8)^(1/3)*De...

Rubi steps

\begin {align*} \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx &=\frac {\sqrt [3]{-2 x+2 x^3-x^8} \int \frac {\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7} \left (-4+5 x^7\right )}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\sqrt [3]{-2 x+2 x^3-x^8} \int \frac {\sqrt [3]{x} \left (-4+5 x^7\right )}{\left (-2+2 x^2-x^7\right )^{2/3} \left (2+x^7\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\sqrt [3]{-2 x+2 x^3-x^8} \int \left (\frac {5 \sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3}}-\frac {14 \sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3} \left (2+x^7\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (5 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (14 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3} \left (2+x^7\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=\frac {\left (14 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \left (-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}+\sqrt [7]{-1} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-(-1)^{2/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}+(-1)^{3/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-(-1)^{4/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}+(-1)^{5/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-(-1)^{6/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-(-1)^{2/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}+(-1)^{3/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-(-1)^{4/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}+(-1)^{5/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-(-1)^{6/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-(-1)^{2/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}+(-1)^{3/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-(-1)^{4/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}+(-1)^{5/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-(-1)^{6/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [7]{2}}{\left (-\sqrt [7]{2}-x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{6/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{6/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (\frac {(-1)^{5/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{5/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}-(-1)^{2/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{4/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{4/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}+(-1)^{3/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (\frac {(-1)^{3/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{3/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}-(-1)^{4/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{2/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{2/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}+(-1)^{5/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (\frac {\sqrt [7]{-1}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {\sqrt [7]{-2}}{\left (-\sqrt [7]{2}-(-1)^{6/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{-2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-2)^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}+(-1)^{5/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{2/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{3/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{4/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{5/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{6/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3\ 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}-x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{-1} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}-(-1)^{6/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{3/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}-(-1)^{4/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{4/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}+(-1)^{3/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{5/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}-(-1)^{2/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{6/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{-2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-2)^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (\frac {\left (-\frac {1}{2}\right )^{2/21}}{3 \left (-(-1)^{2/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {\left (-\frac {1}{2}\right )^{2/21}}{3 \left (-(-1)^{2/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {\left (-\frac {1}{2}\right )^{2/21}}{3 \left (-(-1)^{2/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{2/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{3/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{4/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{5/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{6/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3\ 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (\frac {1}{3\ 2^{2/21} \left (-\sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {1}{3\ 2^{2/21} \left (-\sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {1}{3\ 2^{2/21} \left (-\sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{-1} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt [21]{-1}}{3\ 2^{2/21} \left (\sqrt [21]{-2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [21]{-1}}{3\ 2^{2/21} \left (\sqrt [21]{-2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [21]{-1}}{3\ 2^{2/21} \left (\sqrt [21]{-2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{3/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt [7]{-1}}{3\ 2^{2/21} \left (\sqrt [7]{-1} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [7]{-1}}{3\ 2^{2/21} \left (\sqrt [7]{-1} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [7]{-1}}{3\ 2^{2/21} \left (\sqrt [7]{-1} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{4/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (\frac {(-1)^{4/21}}{3\ 2^{2/21} \left (-(-1)^{4/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{4/21}}{3\ 2^{2/21} \left (-(-1)^{4/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{4/21}}{3\ 2^{2/21} \left (-(-1)^{4/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{5/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{5/21}}{3\ 2^{2/21} \left ((-1)^{5/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{5/21}}{3\ 2^{2/21} \left ((-1)^{5/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{5/21}}{3\ 2^{2/21} \left ((-1)^{5/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{6/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (\frac {(-1)^{2/7}}{3\ 2^{2/21} \left (-(-1)^{2/7} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{2/7}}{3\ 2^{2/21} \left (-(-1)^{2/7} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{2/7}}{3\ 2^{2/21} \left (-(-1)^{2/7} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{-2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-2)^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [21]{-2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-2)^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [21]{-2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-2)^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [21]{-2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{2/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{3/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{4/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{5/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{6/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{-1} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{2/7} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{-1} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{2/7} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{-1} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{2/7} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{8/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{2/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{8/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{2/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{8/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{2/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{4/7} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [7]{-1} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{4/7} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [7]{-1} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{4/7} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [7]{-1} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{16/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{4/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{16/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{4/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{16/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{4/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{20/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left ((-1)^{5/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{20/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left ((-1)^{5/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{20/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {1}{\left ((-1)^{5/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ \end {align*}

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Mathematica [A]
time = 3.52, size = 180, normalized size = 1.20 \begin {gather*} \frac {x^{2/3} \left (2-2 x^2+x^7\right )^{2/3} \left (-2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2^{2/3} \sqrt [3]{2-2 x^2+x^7}}\right )-2 \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{2-2 x^2+x^7}\right )+\log \left (-2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{2-2 x^2+x^7}-\sqrt [3]{2} \left (2-2 x^2+x^7\right )^{2/3}\right )\right )}{2\ 2^{2/3} \left (-x \left (2-2 x^2+x^7\right )\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-4 + 5*x^7)*(-2*x + 2*x^3 - x^8)^(1/3))/((2 + x^7)*(2 - 2*x^2 + x^7)),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 234.25, size = 1516, normalized size = 10.11

method	result	size
trager	\(\text {Expression too large to display}\)	\(1516\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^7-4)*(-x^8+2*x^3-2*x)^(1/3)/(x^7+2)/(x^7-2*x^2+2),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^3+2)*ln((207542028224584*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^3*
x^7+108187121643730*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^4*x^7-103771014112292*R
ootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x^7-54093560821865*RootOf(_Z^3+2)^2*x^7-6226
26084673752*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^3*x^2-324561364931190*RootOf(
RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^4*x^2+324561364931190*RootOf(RootOf(_Z^3+2)^2+2*_Z
*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*(-x^8+2*x^3-2*x)^(2/3)[phone]*RootOf(RootOf(_Z^3+2)^2+2*_Z*R
ootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^3+216374243287460*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*R
ootOf(_Z^3+2)^4-622626084673752*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*(-x^8+2*x^3-2*x)^(1/3)*x-3
24561364931190*(-x^8+2*x^3-2*x)^(1/3)*RootOf(_Z^3+2)*x-13248322594314*(-x^8+2*x^3-2*x)^(2/3)-207542028224584*R
ootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)-108187121643730*RootOf(_Z^3+2)^2)/(x^7+2))-1
/2*ln(-(108187121643730*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^3*x^7+51885507056
146*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^4*x^7-54093560821865*RootOf(RootOf(_Z^3
+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x^7-25942753528073*RootOf(_Z^3+2)^2*x^7-324561364931190*RootO
f(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^3*x^2-155656521168438*RootOf(RootOf(_Z^3+2)^2+
2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^4*x^2+162280682465595*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4
*_Z^2)*RootOf(_Z^3+2)^2*(-x^8+2*x^3-2*x)^(2/3)[phone]*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_
Z^2)^2*RootOf(_Z^3+2)^3+103771014112292*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^4+3
24561364931190*x^2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)[phone]*RootOf(_
Z^3+2)^2*x^2-13248322594314*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*(-x^8+2*x^3-2*x)^(1/3)*x-16228
0682465595*(-x^8+2*x^3-2*x)^(1/3)*RootOf(_Z^3+2)*x-155656521168438*(-x^8+2*x^3-2*x)^(2/3)-108187121643730*Root
Of(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)-51885507056146*RootOf(_Z^3+2)^2)/(x^7+2))*RootO
f(_Z^3+2)-ln(-(108187121643730*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^3*x^7+5188
5507056146*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^4*x^7-54093560821865*RootOf(Root
Of(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x^7-25942753528073*RootOf(_Z^3+2)^2*x^7-32456136493119
0*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^3*x^2-155656521168438*RootOf(RootOf(_Z^
3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^4*x^2+162280682465595*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z
^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*(-x^8+2*x^3-2*x)^(2/3)[phone]*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3
+2)+4*_Z^2)^2*RootOf(_Z^3+2)^3+103771014112292*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3
+2)^4+324561364931190*x^2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)[phone]*R
ootOf(_Z^3+2)^2*x^2-13248322594314*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*(-x^8+2*x^3-2*x)^(1/3)*
x-162280682465595*(-x^8+2*x^3-2*x)^(1/3)*RootOf(_Z^3+2)*x-155656521168438*(-x^8+2*x^3-2*x)^(2/3)-1081871216437
30*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)-51885507056146*RootOf(_Z^3+2)^2)/(x^7+2)
)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^7-4)*(-x^8+2*x^3-2*x)^(1/3)/(x^7+2)/(x^7-2*x^2+2),x, algorithm="maxima")

[Out]

integrate((-x^8 + 2*x^3 - 2*x)^(1/3)*(5*x^7 - 4)/((x^7 - 2*x^2 + 2)*(x^7 + 2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (122) = 244\).
time = 4.75, size = 390, normalized size = 2.60 \begin {gather*} \frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (-\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{15} - 18 \, x^{10} + 4 \, x^{8} + 36 \, x^{5} - 36 \, x^{3} + 4 \, x\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} - 12 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{14} - 6 \, x^{9} + 4 \, x^{7} - 12 \, x^{2} + 4\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (x^{21} - 36 \, x^{16} + 6 \, x^{14} + 180 \, x^{11} - 144 \, x^{9} + 12 \, x^{7} - 216 \, x^{6} + 360 \, x^{4} - 144 \, x^{2} + 8\right )}\right )}}{6 \, {\left (x^{21} + 6 \, x^{14} - 108 \, x^{11} + 12 \, x^{7} + 216 \, x^{6} - 216 \, x^{4} + 8\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}} {\left (x^{7} - 6 \, x^{2} + 2\right )} + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{14} - 18 \, x^{9} + 4 \, x^{7} + 36 \, x^{4} - 36 \, x^{2} + 4\right )} + 24 \, {\left (x^{8} - 3 \, x^{3} + 2 \, x\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}}}{x^{14} + 4 \, x^{7} + 4}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{7} + 2\right )} + 6 \, {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}}}{x^{7} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^7-4)*(-x^8+2*x^3-2*x)^(1/3)/(x^7+2)/(x^7-2*x^2+2),x, algorithm="fricas")

[Out]

1/6*4^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(-1/6*4^(1/6)*sqrt(3)*(6*4^(2/3)*(-1)^(2/3)*(x^15 - 18*x^10 + 4*x^8 + 36*
x^5 - 36*x^3 + 4*x)*(-x^8 + 2*x^3 - 2*x)^(1/3) - 12*(-1)^(1/3)*(x^14 - 6*x^9 + 4*x^7 - 12*x^2 + 4)*(-x^8 + 2*x
^3 - 2*x)^(2/3) - 4^(1/3)*(x^21 - 36*x^16 + 6*x^14 + 180*x^11 - 144*x^9 + 12*x^7 - 216*x^6 + 360*x^4 - 144*x^2
+ 8))/(x^21 + 6*x^14 - 108*x^11 + 12*x^7 + 216*x^6 - 216*x^4 + 8)) - 1/24*4^(2/3)*(-1)^(1/3)*log(-(6*4^(1/3)*
(-1)^(2/3)*(-x^8 + 2*x^3 - 2*x)^(2/3)*(x^7 - 6*x^2 + 2) + 4^(2/3)*(-1)^(1/3)*(x^14 - 18*x^9 + 4*x^7 + 36*x^4 -
36*x^2 + 4) + 24*(x^8 - 3*x^3 + 2*x)*(-x^8 + 2*x^3 - 2*x)^(1/3))/(x^14 + 4*x^7 + 4)) + 1/12*4^(2/3)*(-1)^(1/3
)*log(-(3*4^(2/3)*(-1)^(1/3)*(-x^8 + 2*x^3 - 2*x)^(1/3)*x + 4^(1/3)*(-1)^(2/3)*(x^7 + 2) + 6*(-x^8 + 2*x^3 - 2
*x)^(2/3))/(x^7 + 2))

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Sympy [F(-1)] Timed out
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((5*x**7-4)*(-x**8+2*x**3-2*x)**(1/3)/(x**7+2)/(x**7-2*x**2+2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^7-4)*(-x^8+2*x^3-2*x)^(1/3)/(x^7+2)/(x^7-2*x^2+2),x, algorithm="giac")

[Out]

integrate((-x^8 + 2*x^3 - 2*x)^(1/3)*(5*x^7 - 4)/((x^7 - 2*x^2 + 2)*(x^7 + 2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (5\,x^7-4\right )\,{\left (-x^8+2\,x^3-2\,x\right )}^{1/3}}{\left (x^7+2\right )\,\left (x^7-2\,x^2+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^7 - 4)*(2*x^3 - 2*x - x^8)^(1/3))/((x^7 + 2)*(x^7 - 2*x^2 + 2)),x)

[Out]

int(((5*x^7 - 4)*(2*x^3 - 2*x - x^8)^(1/3))/((x^7 + 2)*(x^7 - 2*x^2 + 2)), x)

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