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

Optimal. Leaf size=156 \[ -\frac {7}{3} \log \left (\sqrt [3]{2 x^3+x^2-2}-x\right )+\frac {7}{6} \log \left (x^2+\sqrt [3]{2 x^3+x^2-2} x+\left (2 x^3+x^2-2\right )^{2/3}\right )-\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2 x^3+x^2-2}+x}\right )}{\sqrt {3}}+\frac {\sqrt [3]{2 x^3+x^2-2} \left (-38 x^6-27 x^5+3 x^4+54 x^3-12 x^2+12\right )}{4 x^4 \left (x^3+x^2-2\right )} \]

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Rubi [F]  time = 9.80, 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 (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

((48/5 + (24*I)/5)*(-2 + x^2 + 2*x^3)^(1/3)*Defer[Subst][Defer[Int][((-1/3*(1 + (107 + 6*Sqrt[318])^(2/3))/(10
7 + 6*Sqrt[318])^(1/3) + 2*x)^(1/3)*((-1 + (107 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3))/9 + (2*(1 +
(107 + 6*Sqrt[318])^(2/3))*x)/(3*(107 + 6*Sqrt[318])^(1/3)) + 4*x^2)^(1/3))/((-5/3 + 2*I) - 2*x)^2, x], x, 1/
6 + x])/((1 - (1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 6*x)^(1/3)*(-1 + (107 + 6*Sqrt[318])
^(-2/3) + (107 + 6*Sqrt[318])^(2/3) + ((1 + (107 + 6*Sqrt[318])^(2/3))*(1 + 6*x))/(107 + 6*Sqrt[318])^(1/3) +
(1 + 6*x)^2)^(1/3)) - (((24*I)/5)*(-2 + x^2 + 2*x^3)^(1/3)*Defer[Subst][Defer[Int][((-1/3*(1 + (107 + 6*Sqrt[3
18])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 2*x)^(1/3)*((-1 + (107 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/
3))/9 + (2*(1 + (107 + 6*Sqrt[318])^(2/3))*x)/(3*(107 + 6*Sqrt[318])^(1/3)) + 4*x^2)^(1/3))/((-5/3 + 2*I) - 2*
x), x], x, 1/6 + x])/((1 - (1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 6*x)^(1/3)*(-1 + (107 +
6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3) + ((1 + (107 + 6*Sqrt[318])^(2/3))*(1 + 6*x))/(107 + 6*Sqrt[3
18])^(1/3) + (1 + 6*x)^2)^(1/3)) + (6*(-2 + x^2 + 2*x^3)^(1/3)*Defer[Subst][Defer[Int][((-1/3*(1 + (107 + 6*Sq
rt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 2*x)^(1/3)*((-1 + (107 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])
^(2/3))/9 + (2*(1 + (107 + 6*Sqrt[318])^(2/3))*x)/(3*(107 + 6*Sqrt[318])^(1/3)) + 4*x^2)^(1/3))/(-7/6 + x)^2,
x], x, 1/6 + x])/(5*(1 - (1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 6*x)^(1/3)*(-1 + (107 + 6
*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3) + ((1 + (107 + 6*Sqrt[318])^(2/3))*(1 + 6*x))/(107 + 6*Sqrt[318
])^(1/3) + (1 + 6*x)^2)^(1/3)) - (51*(-2 + x^2 + 2*x^3)^(1/3)*Defer[Subst][Defer[Int][((-1/3*(1 + (107 + 6*Sqr
t[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 2*x)^(1/3)*((-1 + (107 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^
(2/3))/9 + (2*(1 + (107 + 6*Sqrt[318])^(2/3))*x)/(3*(107 + 6*Sqrt[318])^(1/3)) + 4*x^2)^(1/3))/(-7/6 + x), x],
x, 1/6 + x])/(5*(1 - (1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 6*x)^(1/3)*(-1 + (107 + 6*Sq
rt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3) + ((1 + (107 + 6*Sqrt[318])^(2/3))*(1 + 6*x))/(107 + 6*Sqrt[318])^
(1/3) + (1 + 6*x)^2)^(1/3)) + (18*(-2 + x^2 + 2*x^3)^(1/3)*Defer[Subst][Defer[Int][((-1/3*(1 + (107 + 6*Sqrt[3
18])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 2*x)^(1/3)*((-1 + (107 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/
3))/9 + (2*(1 + (107 + 6*Sqrt[318])^(2/3))*x)/(3*(107 + 6*Sqrt[318])^(1/3)) + 4*x^2)^(1/3))/(-1/6 + x)^5, x],
x, 1/6 + x])/((1 - (1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 6*x)^(1/3)*(-1 + (107 + 6*Sqrt[
318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3) + ((1 + (107 + 6*Sqrt[318])^(2/3))*(1 + 6*x))/(107 + 6*Sqrt[318])^(1/
3) + (1 + 6*x)^2)^(1/3)) - (3*(-2 + x^2 + 2*x^3)^(1/3)*Defer[Subst][Defer[Int][((-1/3*(1 + (107 + 6*Sqrt[318])
^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 2*x)^(1/3)*((-1 + (107 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3))/
9 + (2*(1 + (107 + 6*Sqrt[318])^(2/3))*x)/(3*(107 + 6*Sqrt[318])^(1/3)) + 4*x^2)^(1/3))/(-1/6 + x)^3, x], x, 1
/6 + x])/((1 - (1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 6*x)^(1/3)*(-1 + (107 + 6*Sqrt[318]
)^(-2/3) + (107 + 6*Sqrt[318])^(2/3) + ((1 + (107 + 6*Sqrt[318])^(2/3))*(1 + 6*x))/(107 + 6*Sqrt[318])^(1/3) +
(1 + 6*x)^2)^(1/3)) + (27*(-2 + x^2 + 2*x^3)^(1/3)*Defer[Subst][Defer[Int][((-1/3*(1 + (107 + 6*Sqrt[318])^(2
/3))/(107 + 6*Sqrt[318])^(1/3) + 2*x)^(1/3)*((-1 + (107 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3))/9 +
(2*(1 + (107 + 6*Sqrt[318])^(2/3))*x)/(3*(107 + 6*Sqrt[318])^(1/3)) + 4*x^2)^(1/3))/(-1/6 + x)^2, x], x, 1/6
+ x])/((1 - (1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 6*x)^(1/3)*(-1 + (107 + 6*Sqrt[318])^(
-2/3) + (107 + 6*Sqrt[318])^(2/3) + ((1 + (107 + 6*Sqrt[318])^(2/3))*(1 + 6*x))/(107 + 6*Sqrt[318])^(1/3) + (1
+ 6*x)^2)^(1/3)) + ((51/5 + (39*I)/5)*(-2 + x^2 + 2*x^3)^(1/3)*Defer[Subst][Defer[Int][((-1/3*(1 + (107 + 6*S
qrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 2*x)^(1/3)*((-1 + (107 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318]
)^(2/3))/9 + (2*(1 + (107 + 6*Sqrt[318])^(2/3))*x)/(3*(107 + 6*Sqrt[318])^(1/3)) + 4*x^2)^(1/3))/((5/3 - 2*I)
+ 2*x), x], x, 1/6 + x])/((1 - (1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 6*x)^(1/3)*(-1 + (1
07 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3) + ((1 + (107 + 6*Sqrt[318])^(2/3))*(1 + 6*x))/(107 + 6*Sq
rt[318])^(1/3) + (1 + 6*x)^2)^(1/3)) + ((48/5 - (24*I)/5)*(-2 + x^2 + 2*x^3)^(1/3)*Defer[Subst][Defer[Int][((-
1/3*(1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 2*x)^(1/3)*((-1 + (107 + 6*Sqrt[318])^(-2/3) +
(107 + 6*Sqrt[318])^(2/3))/9 + (2*(1 + (107 + 6*Sqrt[318])^(2/3))*x)/(3*(107 + 6*Sqrt[318])^(1/3)) + 4*x^2)^(
1/3))/((5/3 + 2*I) + 2*x)^2, x], x, 1/6 + x])/((1 - (1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3)
+ 6*x)^(1/3)*(-1 + (107 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3) + ((1 + (107 + 6*Sqrt[318])^(2/3))*(
1 + 6*x))/(107 + 6*Sqrt[318])^(1/3) + (1 + 6*x)^2)^(1/3)) + ((51/5 - (63*I)/5)*(-2 + x^2 + 2*x^3)^(1/3)*Defer[
Subst][Defer[Int][((-1/3*(1 + (107 + 6*Sqrt[318])^(2/3))/(107 + 6*Sqrt[318])^(1/3) + 2*x)^(1/3)*((-1 + (107 +
6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3))/9 + (2*(1 + (107 + 6*Sqrt[318])^(2/3))*x)/(3*(107 + 6*Sqrt[31
8])^(1/3)) + 4*x^2)^(1/3))/((5/3 + 2*I) + 2*x), x], x, 1/6 + x])/((1 - (1 + (107 + 6*Sqrt[318])^(2/3))/(107 +
6*Sqrt[318])^(1/3) + 6*x)^(1/3)*(-1 + (107 + 6*Sqrt[318])^(-2/3) + (107 + 6*Sqrt[318])^(2/3) + ((1 + (107 + 6*
Sqrt[318])^(2/3))*(1 + 6*x))/(107 + 6*Sqrt[318])^(1/3) + (1 + 6*x)^2)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx &=\int \left (\frac {2 \sqrt [3]{-2+x^2+2 x^3}}{5 (-1+x)^2}-\frac {17 \sqrt [3]{-2+x^2+2 x^3}}{5 (-1+x)}+\frac {6 \sqrt [3]{-2+x^2+2 x^3}}{x^5}-\frac {\sqrt [3]{-2+x^2+2 x^3}}{x^3}+\frac {9 \sqrt [3]{-2+x^2+2 x^3}}{x^2}-\frac {8 (3+x) \sqrt [3]{-2+x^2+2 x^3}}{5 \left (2+2 x+x^2\right )^2}+\frac {(4+17 x) \sqrt [3]{-2+x^2+2 x^3}}{5 \left (2+2 x+x^2\right )}\right ) \, dx\\ &=\frac {1}{5} \int \frac {(4+17 x) \sqrt [3]{-2+x^2+2 x^3}}{2+2 x+x^2} \, dx+\frac {2}{5} \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{(-1+x)^2} \, dx-\frac {8}{5} \int \frac {(3+x) \sqrt [3]{-2+x^2+2 x^3}}{\left (2+2 x+x^2\right )^2} \, dx-\frac {17}{5} \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{-1+x} \, dx+6 \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{x^5} \, dx+9 \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{x^2} \, dx-\int \frac {\sqrt [3]{-2+x^2+2 x^3}}{x^3} \, dx\\ &=\frac {1}{5} \int \left (\frac {(17+13 i) \sqrt [3]{-2+x^2+2 x^3}}{(2-2 i)+2 x}+\frac {(17-13 i) \sqrt [3]{-2+x^2+2 x^3}}{(2+2 i)+2 x}\right ) \, dx+\frac {2}{5} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (-\frac {7}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )-\frac {8}{5} \int \left (\frac {3 \sqrt [3]{-2+x^2+2 x^3}}{\left (2+2 x+x^2\right )^2}+\frac {x \sqrt [3]{-2+x^2+2 x^3}}{\left (2+2 x+x^2\right )^2}\right ) \, dx-\frac {17}{5} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{-\frac {7}{6}+x} \, dx,x,\frac {1}{6}+x\right )+6 \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (-\frac {1}{6}+x\right )^5} \, dx,x,\frac {1}{6}+x\right )+9 \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (-\frac {1}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )-\operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (-\frac {1}{6}+x\right )^3} \, dx,x,\frac {1}{6}+x\right )\\ &=-\left (\frac {8}{5} \int \frac {x \sqrt [3]{-2+x^2+2 x^3}}{\left (2+2 x+x^2\right )^2} \, dx\right )+\left (\frac {17}{5}-\frac {13 i}{5}\right ) \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{(2+2 i)+2 x} \, dx+\left (\frac {17}{5}+\frac {13 i}{5}\right ) \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{(2-2 i)+2 x} \, dx-\frac {24}{5} \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{\left (2+2 x+x^2\right )^2} \, dx+\frac {\left (6 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {7}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (3 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^3} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (51 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{-\frac {7}{6}+x} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (18 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^5} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (27 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}\\ &=-\left (\frac {8}{5} \int \left (\frac {(1-i) \sqrt [3]{-2+x^2+2 x^3}}{((-2+2 i)-2 x)^2}-\frac {i \sqrt [3]{-2+x^2+2 x^3}}{2 ((-2+2 i)-2 x)}+\frac {(1+i) \sqrt [3]{-2+x^2+2 x^3}}{((2+2 i)+2 x)^2}-\frac {i \sqrt [3]{-2+x^2+2 x^3}}{2 ((2+2 i)+2 x)}\right ) \, dx\right )+\left (\frac {17}{5}-\frac {13 i}{5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (\frac {5}{3}+2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )+\left (\frac {17}{5}+\frac {13 i}{5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (\frac {5}{3}-2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )-\frac {24}{5} \int \left (-\frac {\sqrt [3]{-2+x^2+2 x^3}}{((-2+2 i)-2 x)^2}+\frac {i \sqrt [3]{-2+x^2+2 x^3}}{2 ((-2+2 i)-2 x)}-\frac {\sqrt [3]{-2+x^2+2 x^3}}{((2+2 i)+2 x)^2}+\frac {i \sqrt [3]{-2+x^2+2 x^3}}{2 ((2+2 i)+2 x)}\right ) \, dx+\frac {\left (6 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {7}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (3 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^3} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (51 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{-\frac {7}{6}+x} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (18 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^5} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (27 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}\\ &=\frac {4}{5} i \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{(-2+2 i)-2 x} \, dx+\frac {4}{5} i \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{(2+2 i)+2 x} \, dx-\frac {12}{5} i \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{(-2+2 i)-2 x} \, dx-\frac {12}{5} i \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{(2+2 i)+2 x} \, dx-\left (\frac {8}{5}-\frac {8 i}{5}\right ) \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{((-2+2 i)-2 x)^2} \, dx-\left (\frac {8}{5}+\frac {8 i}{5}\right ) \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{((2+2 i)+2 x)^2} \, dx+\frac {24}{5} \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{((-2+2 i)-2 x)^2} \, dx+\frac {24}{5} \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{((2+2 i)+2 x)^2} \, dx+\frac {\left (6 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {7}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (3 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^3} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (51 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{-\frac {7}{6}+x} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (\left (\frac {51}{5}-\frac {39 i}{5}\right ) \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\frac {5}{3}+2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (\left (\frac {51}{5}+\frac {39 i}{5}\right ) \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\frac {5}{3}-2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (18 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^5} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (27 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}\\ &=\frac {4}{5} i \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (-\frac {5}{3}+2 i\right )-2 x} \, dx,x,\frac {1}{6}+x\right )+\frac {4}{5} i \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (\frac {5}{3}+2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )-\frac {12}{5} i \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (-\frac {5}{3}+2 i\right )-2 x} \, dx,x,\frac {1}{6}+x\right )-\frac {12}{5} i \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (\frac {5}{3}+2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )-\left (\frac {8}{5}-\frac {8 i}{5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (\left (-\frac {5}{3}+2 i\right )-2 x\right )^2} \, dx,x,\frac {1}{6}+x\right )-\left (\frac {8}{5}+\frac {8 i}{5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (\left (\frac {5}{3}+2 i\right )+2 x\right )^2} \, dx,x,\frac {1}{6}+x\right )+\frac {24}{5} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (\left (-\frac {5}{3}+2 i\right )-2 x\right )^2} \, dx,x,\frac {1}{6}+x\right )+\frac {24}{5} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (\left (\frac {5}{3}+2 i\right )+2 x\right )^2} \, dx,x,\frac {1}{6}+x\right )+\frac {\left (6 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {7}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (3 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^3} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (51 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{-\frac {7}{6}+x} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (\left (\frac {51}{5}-\frac {39 i}{5}\right ) \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\frac {5}{3}+2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (\left (\frac {51}{5}+\frac {39 i}{5}\right ) \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\frac {5}{3}-2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (18 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^5} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (27 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}\\ &=\frac {\left (12 i \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {5}{3}+2 i\right )-2 x} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (12 i \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\frac {5}{3}+2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (36 i \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {5}{3}+2 i\right )-2 x} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (36 i \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\frac {5}{3}+2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (6 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {7}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (3 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^3} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (\left (\frac {24}{5}-\frac {24 i}{5}\right ) \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\left (-\frac {5}{3}+2 i\right )-2 x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (\left (\frac {24}{5}+\frac {24 i}{5}\right ) \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\left (\frac {5}{3}+2 i\right )+2 x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}-\frac {\left (51 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{-\frac {7}{6}+x} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (\left (\frac {51}{5}-\frac {39 i}{5}\right ) \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\frac {5}{3}+2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (\left (\frac {51}{5}+\frac {39 i}{5}\right ) \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\frac {5}{3}-2 i\right )+2 x} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (72 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\left (-\frac {5}{3}+2 i\right )-2 x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (72 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (\left (\frac {5}{3}+2 i\right )+2 x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{5 \sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (18 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^5} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}+\frac {\left (27 \sqrt [3]{-2+x^2+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{3 \sqrt [3]{107+6 \sqrt {318}}}+2 x} \sqrt [3]{\frac {1}{9} \left (-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}\right )+\frac {2 \left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) x}{3 \sqrt [3]{107+6 \sqrt {318}}}+4 x^2}}{\left (-\frac {1}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )}{\sqrt [3]{1-\frac {1+\left (107+6 \sqrt {318}\right )^{2/3}}{\sqrt [3]{107+6 \sqrt {318}}}+6 x} \sqrt [3]{-1+\frac {1}{\left (107+6 \sqrt {318}\right )^{2/3}}+\left (107+6 \sqrt {318}\right )^{2/3}+36 \left (\frac {1}{6}+x\right )^2+\frac {\left (1+\left (107+6 \sqrt {318}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{107+6 \sqrt {318}}}}}\\ \end {align*}

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Mathematica [F]  time = 0.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.01, size = 156, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{-2+x^2+2 x^3} \left (12-12 x^2+54 x^3+3 x^4-27 x^5-38 x^6\right )}{4 x^4 \left (-2+x^2+x^3\right )}-\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-2+x^2+2 x^3}}\right )}{\sqrt {3}}-\frac {7}{3} \log \left (-x+\sqrt [3]{-2+x^2+2 x^3}\right )+\frac {7}{6} \log \left (x^2+x \sqrt [3]{-2+x^2+2 x^3}+\left (-2+x^2+2 x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2 + x^2 + 2*x^3)^(1/3)*(12 - 12*x^2 + 54*x^3 + 3*x^4 - 27*x^5 - 38*x^6))/(4*x^4*(-2 + x^2 + x^3)) - (7*ArcT
an[(Sqrt[3]*x)/(x + 2*(-2 + x^2 + 2*x^3)^(1/3))])/Sqrt[3] - (7*Log[-x + (-2 + x^2 + 2*x^3)^(1/3)])/3 + (7*Log[
x^2 + x*(-2 + x^2 + 2*x^3)^(1/3) + (-2 + x^2 + 2*x^3)^(2/3)])/6

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fricas [A]  time = 2.68, size = 212, normalized size = 1.36 \begin {gather*} \frac {28 \, \sqrt {3} {\left (x^{7} + x^{6} - 2 \, x^{4}\right )} \arctan \left (\frac {1078 \, \sqrt {3} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} x^{2} + 196 \, \sqrt {3} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (669 \, x^{3} + 32 \, x^{2} - 64\right )}}{1315 \, x^{3} - 8 \, x^{2} + 16}\right ) - 14 \, {\left (x^{7} + x^{6} - 2 \, x^{4}\right )} \log \left (\frac {x^{3} + 3 \, {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} x^{2} + x^{2} - 3 \, {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {2}{3}} x - 2}{x^{3} + x^{2} - 2}\right ) - 3 \, {\left (38 \, x^{6} + 27 \, x^{5} - 3 \, x^{4} - 54 \, x^{3} + 12 \, x^{2} - 12\right )} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}}}{12 \, {\left (x^{7} + x^{6} - 2 \, x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(28*sqrt(3)*(x^7 + x^6 - 2*x^4)*arctan((1078*sqrt(3)*(2*x^3 + x^2 - 2)^(1/3)*x^2 + 196*sqrt(3)*(2*x^3 + x
^2 - 2)^(2/3)*x + sqrt(3)*(669*x^3 + 32*x^2 - 64))/(1315*x^3 - 8*x^2 + 16)) - 14*(x^7 + x^6 - 2*x^4)*log((x^3
+ 3*(2*x^3 + x^2 - 2)^(1/3)*x^2 + x^2 - 3*(2*x^3 + x^2 - 2)^(2/3)*x - 2)/(x^3 + x^2 - 2)) - 3*(38*x^6 + 27*x^5
- 3*x^4 - 54*x^3 + 12*x^2 - 12)*(2*x^3 + x^2 - 2)^(1/3))/(x^7 + x^6 - 2*x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 4.74, size = 707, normalized size = 4.53

method	result	size
trager	\(-\frac {\left (38 x^{6}+27 x^{5}-3 x^{4}-54 x^{3}+12 x^{2}-12\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}}{4 \left (x^{3}+x^{2}-2\right ) x^{4}}-\frac {7 \ln \left (-\frac {-26052480000 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x -6566790240 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}[phone] \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{2}[phone] \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{3}+5160045 \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x +8520768 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} x^{2}-754819680 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}-20074174 x^{3}-208419840000 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2}-5827986 x^{2}[phone] \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )[phone]}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )}{3}+\frac {7 \ln \left (\frac {-149196441600 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x [phone] \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}[phone] \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{2}-9270501600 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{3}-13680813 \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x +8520768 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} x^{2}-4257840960 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}+3392250 x^{3}-1193571532800 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2}+1469975 x^{2}[phone] \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )-2939950}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )}{3}-1120 \ln \left (\frac {-149196441600 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x [phone] \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}[phone] \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{2}-9270501600 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{3}-13680813 \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x +8520768 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} x^{2}-4257840960 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}+3392250 x^{3}-1193571532800 \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2}+1469975 x^{2}[phone] \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )-2939950}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right ) \RootOf \left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )\)	\(707\)
risch	\(-\frac {76 x^{9}+92 x^{8}+21 x^{7}-187 x^{6}-84 x^{5}+18 x^{4}+84 x^{3}-36 x^{2}+24}{4 x^{4} \left (x^{3}+x^{2}-2\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}}}+\frac {\left (\frac {7 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{5}+6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{6}+6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x^{4}+7 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{5}-8 x^{6}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {2}{3}} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-12 \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x^{4}-8 x^{5}-14 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}-6 \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {2}{3}} x^{2}-6 \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x^{3}-2 x^{4}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x -8 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+16 x^{3}+12 \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x +8 x^{2}+8 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-8}{\left (x^{2}+2 x +2\right ) \left (-1+x \right ) \left (2 x^{3}+x^{2}-2\right )}\right )}{6}-\frac {7 \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{5}-14 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{6}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x^{4}-11 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{5}+12 x^{6}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {2}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x^{3}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+10 x^{5}+22 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+2 x^{4}+6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x +8 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-20 x^{3}-8 x^{2}-8 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+8}{\left (x^{2}+2 x +2\right ) \left (-1+x \right ) \left (2 x^{3}+x^{2}-2\right )}\right ) \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )}{6}+\frac {7 \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{5}-14 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{6}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x^{4}-11 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{5}+12 x^{6}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {2}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x^{3}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+10 x^{5}+22 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+2 x^{4}+6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (4 x^{6}+4 x^{5}+x^{4}-8 x^{3}-4 x^{2}+4\right )^{\frac {1}{3}} x +8 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-20 x^{3}-8 x^{2}-8 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+8}{\left (x^{2}+2 x +2\right ) \left (-1+x \right ) \left (2 x^{3}+x^{2}-2\right )}\right )}{3}\right ) \left (\left (2 x^{3}+x^{2}-2\right )^{2}\right )^{\frac {1}{3}}}{\left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}}}\)	\(1287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-6)*(x^2-2)*(x^3-x^2+2)*(2*x^3+x^2-2)^(1/3)/x^5/(x^3+x^2-2)^2,x,method=_RETURNVERBOSE)

[Out]

-1/4*(38*x^6+27*x^5-3*x^4-54*x^3+12*x^2-12)/(x^3+x^2-2)/x^4*(2*x^3+x^2-2)^(1/3)-7/3*ln(-(-26052480000*RootOf(2
30400*_Z^2-480*_Z+1)^2*x^3+4089968640*RootOf(230400*_Z^2-480*_Z+1)*(2*x^3+x^2-2)^(2/3)*x-6566790240*(2*x^3+x^2
-2)^(1/3)*RootOf(230400*_Z^2-480*_Z+1)*x^2+104209920000*RootOf(230400*_Z^2-480*_Z+1)^2*x^2+1993381920*RootOf(2
30400*_Z^2-480*_Z+1)*x^3+5160045*(2*x^3+x^2-2)^(2/3)*x+8520768*(2*x^3+x^2-2)^(1/3)*x^2-754819680*RootOf(230400
*_Z^2-480*_Z+1)*x^2-20074174*x^3-208419840000*RootOf(230400*_Z^2-480*_Z+1)^2-5827986*x^2+1509639360*RootOf(230
400*_Z^2-480*_Z+1)[phone])/(-1+x)/(x^2+2*x+2))+7/3*ln((-149196441600*RootOf(230400*_Z^2-480*_Z+1)^2*x^3+4089
968640*RootOf(230400*_Z^2-480*_Z+1)*(2*x^3+x^2-2)^(2/3)*x+2476821600*(2*x^3+x^2-2)^(1/3)*RootOf(230400*_Z^2-48
0*_Z+1)*x^2+596785766400*RootOf(230400*_Z^2-480*_Z+1)^2*x^2-9270501600*RootOf(230400*_Z^2-480*_Z+1)*x^3-136808
13*(2*x^3+x^2-2)^(2/3)*x+8520768*(2*x^3+x^2-2)^(1/3)*x^2-4257840960*RootOf(230400*_Z^2-480*_Z+1)*x^2+3392250*x
^3-1193571532800*RootOf(230400*_Z^2-480*_Z+1)^2+1469975*x^2+8515681920*RootOf(230400*_Z^2-480*_Z+1)-2939950)/(
-1+x)/(x^2+2*x+2))-1120*ln((-149196441600*RootOf(230400*_Z^2-480*_Z+1)^2*x^3+4089968640*RootOf(230400*_Z^2-480
*_Z+1)*(2*x^3+x^2-2)^(2/3)*x+2476821600*(2*x^3+x^2-2)^(1/3)*RootOf(230400*_Z^2-480*_Z+1)*x^2+596785766400*Root
Of(230400*_Z^2-480*_Z+1)^2*x^2-9270501600*RootOf(230400*_Z^2-480*_Z+1)*x^3-13680813*(2*x^3+x^2-2)^(2/3)*x+8520
768*(2*x^3+x^2-2)^(1/3)*x^2-4257840960*RootOf(230400*_Z^2-480*_Z+1)*x^2+3392250*x^3-1193571532800*RootOf(23040
0*_Z^2-480*_Z+1)^2+1469975*x^2+8515681920*RootOf(230400*_Z^2-480*_Z+1)-2939950)/(-1+x)/(x^2+2*x+2))*RootOf(230
400*_Z^2-480*_Z+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2-2\right )\,\left (x^2-6\right )\,\left (x^3-x^2+2\right )\,{\left (2\,x^3+x^2-2\right )}^{1/3}}{x^5\,{\left (x^3+x^2-2\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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