3.27.97 \(\int \frac {(2+5 x^7) \sqrt [3]{-x-x^3+x^8}}{(-1+x^7) (-1+x^2+x^7)} \, dx\)

Optimal. Leaf size=245 \[ -\log \left (\sqrt [3]{x^8-x^3-x}+x\right )+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^8-x^3-x}+2 x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^8-x^3-x}-x}\right )-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^8-x^3-x}-x}\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^8-x^3-x} x+\left (x^8-x^3-x\right )^{2/3}\right )-\frac {\log \left (-2 x^2+2^{2/3} \sqrt [3]{x^8-x^3-x} x-\sqrt [3]{2} \left (x^8-x^3-x\right )^{2/3}\right )}{2^{2/3}} \]

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Rubi [F]  time = 9.54, 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 (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

((-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 + x^21)^(1/3)/(-1 + x), x], x, x^(1/3)])/(x^(1/3)*(-
1 - x^2 + x^7)^(1/3)) - ((1 - I*Sqrt[3])*(-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 + x^21)^(1/3
)/(1 - I*Sqrt[3] + 2*x), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) - ((1 + I*Sqrt[3])*(-x - x^3 + x^8)
^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 + x^21)^(1/3)/(1 + I*Sqrt[3] + 2*x), x], x, x^(1/3)])/(x^(1/3)*(-1 -
x^2 + x^7)^(1/3)) + ((-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 + x^21)^(1/3)/(1 + x + x^2 + x^3
+ x^4 + x^5 + x^6), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) + (2*(-x - x^3 + x^8)^(1/3)*Defer[Subst
][Defer[Int][(x*(-1 - x^6 + x^21)^(1/3))/(1 + x + x^2 + x^3 + x^4 + x^5 + x^6), x], x, x^(1/3)])/(x^(1/3)*(-1
- x^2 + x^7)^(1/3)) + (3*(-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(x^2*(-1 - x^6 + x^21)^(1/3))/(1 + x +
x^2 + x^3 + x^4 + x^5 + x^6), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) + (4*(-x - x^3 + x^8)^(1/3)*D
efer[Subst][Defer[Int][(x^3*(-1 - x^6 + x^21)^(1/3))/(1 + x + x^2 + x^3 + x^4 + x^5 + x^6), x], x, x^(1/3)])/(
x^(1/3)*(-1 - x^2 + x^7)^(1/3)) - (2*(-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(x^4*(-1 - x^6 + x^21)^(1/
3))/(1 + x + x^2 + x^3 + x^4 + x^5 + x^6), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) - ((-x - x^3 + x^
8)^(1/3)*Defer[Subst][Defer[Int][(x^5*(-1 - x^6 + x^21)^(1/3))/(1 + x + x^2 + x^3 + x^4 + x^5 + x^6), x], x, x
^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) + (2*(-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 + x^21
)^(1/3)/(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3))
- (3*(-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(x*(-1 - x^6 + x^21)^(1/3))/(1 - x + x^3 - x^4 + x^6 - x^
8 + x^9 - x^11 + x^12), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) + (5*(-x - x^3 + x^8)^(1/3)*Defer[Su
bst][Defer[Int][(x^3*(-1 - x^6 + x^21)^(1/3))/(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12), x], x, x^(1
/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) + ((-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(x^4*(-1 - x^6 + x^21
)^(1/3))/(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)
) - (7*(-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(x^5*(-1 - x^6 + x^21)^(1/3))/(1 - x + x^3 - x^4 + x^6 -
x^8 + x^9 - x^11 + x^12), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) + (8*(-x - x^3 + x^8)^(1/3)*Defer
[Subst][Defer[Int][(x^6*(-1 - x^6 + x^21)^(1/3))/(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12), x], x, x
^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) - (10*(-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(x^8*(-1 - x^6
+ x^21)^(1/3))/(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)
^(1/3)) + (11*(-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(x^9*(-1 - x^6 + x^21)^(1/3))/(1 - x + x^3 - x^4
+ x^6 - x^8 + x^9 - x^11 + x^12), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) + ((-x - x^3 + x^8)^(1/3)*
Defer[Subst][Defer[Int][(x^11*(-1 - x^6 + x^21)^(1/3))/(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12), x]
, x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) - (6*(-x - x^3 + x^8)^(1/3)*Defer[Subst][Defer[Int][(x^3*(-1 -
x^6 + x^21)^(1/3))/(-1 + x^6 + x^21), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 + x^7)^(1/3)) - (21*(-x - x^3 + x^8
)^(1/3)*Defer[Subst][Defer[Int][(x^18*(-1 - x^6 + x^21)^(1/3))/(-1 + x^6 + x^21), x], x, x^(1/3)])/(x^(1/3)*(-
1 - x^2 + x^7)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx &=\frac {\sqrt [3]{-x-x^3+x^8} \int \frac {\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7} \left (2+5 x^7\right )}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}\\ &=\frac {\sqrt [3]{-x-x^3+x^8} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}{-1+x}+\frac {\sqrt [3]{x} \left (1+2 x+3 x^2+4 x^3+5 x^4-x^5\right ) \sqrt [3]{-1-x^2+x^7}}{1+x+x^2+x^3+x^4+x^5+x^6}+\frac {\sqrt [3]{x} \left (-2-7 x^5\right ) \sqrt [3]{-1-x^2+x^7}}{-1+x^2+x^7}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}\\ &=\frac {\sqrt [3]{-x-x^3+x^8} \int \frac {\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}{-1+x} \, dx}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \int \frac {\sqrt [3]{x} \left (1+2 x+3 x^2+4 x^3+5 x^4-x^5\right ) \sqrt [3]{-1-x^2+x^7}}{1+x+x^2+x^3+x^4+x^5+x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \int \frac {\sqrt [3]{x} \left (-2-7 x^5\right ) \sqrt [3]{-1-x^2+x^7}}{-1+x^2+x^7} \, dx}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}\\ &=\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1-x^6+x^{21}}}{-1+x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (1+2 x^3+3 x^6+4 x^9+5 x^{12}-x^{15}\right ) \sqrt [3]{-1-x^6+x^{21}}}{1+x^3+x^6+x^9+x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (-2-7 x^{15}\right ) \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}\\ &=\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \left (\sqrt [3]{-1-x^6+x^{21}}+\frac {\sqrt [3]{-1-x^6+x^{21}}}{-1+x^3}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \left (-\sqrt [3]{-1-x^6+x^{21}}+\frac {\left (1+2 x^3+3 x^6+4 x^9+5 x^{12}+6 x^{15}\right ) \sqrt [3]{-1-x^6+x^{21}}}{1+x^3+x^6+x^9+x^{12}+x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \left (-\frac {2 x^3 \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}}-\frac {7 x^{18} \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}\\ &=\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6+x^{21}}}{-1+x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {\left (1+2 x^3+3 x^6+4 x^9+5 x^{12}+6 x^{15}\right ) \sqrt [3]{-1-x^6+x^{21}}}{1+x^3+x^6+x^9+x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (6 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (21 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^{18} \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}\\ &=\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{-1-x^6+x^{21}}}{3 (-1+x)}+\frac {(-2-x) \sqrt [3]{-1-x^6+x^{21}}}{3 \left (1+x+x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (1+2 x+3 x^2+4 x^3-2 x^4-x^5\right ) \sqrt [3]{-1-x^6+x^{21}}}{3 \left (1+x+x^2+x^3+x^4+x^5+x^6\right )}+\frac {\left (2-3 x+5 x^3+x^4-7 x^5+8 x^6-10 x^8+11 x^9+x^{11}\right ) \sqrt [3]{-1-x^6+x^{21}}}{3 \left (1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (6 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (21 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^{18} \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}\\ &=\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6+x^{21}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \frac {(-2-x) \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \frac {\left (1+2 x+3 x^2+4 x^3-2 x^4-x^5\right ) \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \frac {\left (2-3 x+5 x^3+x^4-7 x^5+8 x^6-10 x^8+11 x^9+x^{11}\right ) \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (6 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (21 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^{18} \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}\\ &=\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6+x^{21}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \left (\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-1-x^6+x^{21}}}{1-i \sqrt {3}+2 x}+\frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{-1-x^6+x^{21}}}{1+i \sqrt {3}+2 x}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6}+\frac {2 x \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6}+\frac {3 x^2 \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6}+\frac {4 x^3 \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6}-\frac {2 x^4 \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6}-\frac {x^5 \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \left (\frac {2 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}}-\frac {3 x \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}}+\frac {5 x^3 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}}+\frac {x^4 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}}-\frac {7 x^5 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}}+\frac {8 x^6 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}}-\frac {10 x^8 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}}+\frac {11 x^9 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}}+\frac {x^{11} \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (6 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (21 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^{18} \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}\\ &=\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6+x^{21}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \frac {x^5 \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \frac {x^4 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\sqrt [3]{-x-x^3+x^8} \operatorname {Subst}\left (\int \frac {x^{11} \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (2 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (2 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (2 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (3 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (4 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1-x^6+x^{21}}}{1+x+x^2+x^3+x^4+x^5+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (5 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (6 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (7 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^5 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (8 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (10 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (11 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^9 \sqrt [3]{-1-x^6+x^{21}}}{1-x+x^3-x^4+x^6-x^8+x^9-x^{11}+x^{12}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}-\frac {\left (21 \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^{18} \sqrt [3]{-1-x^6+x^{21}}}{-1+x^6+x^{21}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6+x^{21}}}{1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3+x^8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6+x^{21}}}{1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^7}}\\ \end {align*}

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Mathematica [F]  time = 180.09, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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IntegrateAlgebraic [A]  time = 0.68, size = 245, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x-x^3+x^8}}\right )-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{-x-x^3+x^8}}\right )-\log \left (x+\sqrt [3]{-x-x^3+x^8}\right )+\sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{-x-x^3+x^8}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x-x^3+x^8}+\left (-x-x^3+x^8\right )^{2/3}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{-x-x^3+x^8}-\sqrt [3]{2} \left (-x-x^3+x^8\right )^{2/3}\right )}{2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-x - x^3 + x^8)^(1/3))] - 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)
*(-x - x^3 + x^8)^(1/3))] - Log[x + (-x - x^3 + x^8)^(1/3)] + 2^(1/3)*Log[2*x + 2^(2/3)*(-x - x^3 + x^8)^(1/3)
] + Log[x^2 - x*(-x - x^3 + x^8)^(1/3) + (-x - x^3 + x^8)^(2/3)]/2 - Log[-2*x^2 + 2^(2/3)*x*(-x - x^3 + x^8)^(
1/3) - 2^(1/3)*(-x - x^3 + x^8)^(2/3)]/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((5*x^7+2)*(x^8-x^3-x)^(1/3)/(x^7-1)/(x^7+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^{8} - x^{3} - x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} + 2\right )}}{{\left (x^{7} + x^{2} - 1\right )} {\left (x^{7} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (5 x^{7}+2\right ) \left (x^{8}-x^{3}-x \right )^{\frac {1}{3}}}{\left (x^{7}-1\right ) \left (x^{7}+x^{2}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((5*x^7+2)*(x^8-x^3-x)^(1/3)/(x^7-1)/(x^7+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^{8} - x^{3} - x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} + 2\right )}}{{\left (x^{7} + x^{2} - 1\right )} {\left (x^{7} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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