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

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 )\\ &=\text {rest of steps removed due to Latex formating problem} \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 = 0.95, size = 156, normalized size = 1.00 \begin {gather*} -\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 )} \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 = 2.76, 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 = 1.62, size = 1291, normalized size = 8.28

result too large to display

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)

[Out]

-1/4*(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)/x^4/(x^3+x^2-2)/(2*x^3+x^2-2)^(2/3)+(7/6*Ro
otOf(_Z^2-2*_Z+4)*ln((2*RootOf(_Z^2-2*_Z+4)^2*x^6+RootOf(_Z^2-2*_Z+4)^2*x^5+6*RootOf(_Z^2-2*_Z+4)*x^6+6*RootOf
(_Z^2-2*_Z+4)*(4*x^6+4*x^5+x^4-8*x^3-4*x^2+4)^(1/3)*x^4+7*RootOf(_Z^2-2*_Z+4)*x^5-8*x^6+3*RootOf(_Z^2-2*_Z+4)*
(4*x^6+4*x^5+x^4-8*x^3-4*x^2+4)^(2/3)*x^2+3*RootOf(_Z^2-2*_Z+4)*(4*x^6+4*x^5+x^4-8*x^3-4*x^2+4)^(1/3)*x^3-12*(
4*x^6+4*x^5+x^4-8*x^3-4*x^2+4)^(1/3)*x^4-2*RootOf(_Z^2-2*_Z+4)^2*x^3+2*RootOf(_Z^2-2*_Z+4)*x^4-8*x^5-6*(4*x^6+
4*x^5+x^4-8*x^3-4*x^2+4)^(2/3)*x^2-6*(4*x^6+4*x^5+x^4-8*x^3-4*x^2+4)^(1/3)*x^3-14*RootOf(_Z^2-2*_Z+4)*x^3-2*x^
4-6*RootOf(_Z^2-2*_Z+4)*(4*x^6+4*x^5+x^4-8*x^3-4*x^2+4)^(1/3)*x-8*RootOf(_Z^2-2*_Z+4)*x^2+16*x^3+12*(4*x^6+4*x
^5+x^4-8*x^3-4*x^2+4)^(1/3)*x+8*x^2+8*RootOf(_Z^2-2*_Z+4)-8)/(x^2+2*x+2)/(2*x^3+x^2-2)/(-1+x))-7/6*ln(-(-2*Roo
tOf(_Z^2-2*_Z+4)^2*x^6-RootOf(_Z^2-2*_Z+4)^2*x^5+14*RootOf(_Z^2-2*_Z+4)*x^6+6*RootOf(_Z^2-2*_Z+4)*(4*x^6+4*x^5
+x^4-8*x^3-4*x^2+4)^(1/3)*x^4+11*RootOf(_Z^2-2*_Z+4)*x^5-12*x^6+3*RootOf(_Z^2-2*_Z+4)*(4*x^6+4*x^5+x^4-8*x^3-4
*x^2+4)^(2/3)*x^2+3*RootOf(_Z^2-2*_Z+4)*(4*x^6+4*x^5+x^4-8*x^3-4*x^2+4)^(1/3)*x^3+2*RootOf(_Z^2-2*_Z+4)^2*x^3+
2*RootOf(_Z^2-2*_Z+4)*x^4-10*x^5-22*RootOf(_Z^2-2*_Z+4)*x^3-2*x^4-6*RootOf(_Z^2-2*_Z+4)*(4*x^6+4*x^5+x^4-8*x^3
-4*x^2+4)^(1/3)*x-8*RootOf(_Z^2-2*_Z+4)*x^2+20*x^3+8*x^2+8*RootOf(_Z^2-2*_Z+4)-8)/(x^2+2*x+2)/(2*x^3+x^2-2)/(-
1+x))*RootOf(_Z^2-2*_Z+4)+7/3*ln(-(-2*RootOf(_Z^2-2*_Z+4)^2*x^6-RootOf(_Z^2-2*_Z+4)^2*x^5+14*RootOf(_Z^2-2*_Z+
4)*x^6+6*RootOf(_Z^2-2*_Z+4)*(4*x^6+4*x^5+x^4-8*x^3-4*x^2+4)^(1/3)*x^4+11*RootOf(_Z^2-2*_Z+4)*x^5-12*x^6+3*Roo
tOf(_Z^2-2*_Z+4)*(4*x^6+4*x^5+x^4-8*x^3-4*x^2+4)^(2/3)*x^2+3*RootOf(_Z^2-2*_Z+4)*(4*x^6+4*x^5+x^4-8*x^3-4*x^2+
4)^(1/3)*x^3+2*RootOf(_Z^2-2*_Z+4)^2*x^3+2*RootOf(_Z^2-2*_Z+4)*x^4-10*x^5-22*RootOf(_Z^2-2*_Z+4)*x^3-2*x^4-6*R
ootOf(_Z^2-2*_Z+4)*(4*x^6+4*x^5+x^4-8*x^3-4*x^2+4)^(1/3)*x-8*RootOf(_Z^2-2*_Z+4)*x^2+20*x^3+8*x^2+8*RootOf(_Z^
2-2*_Z+4)-8)/(x^2+2*x+2)/(2*x^3+x^2-2)/(-1+x)))/(2*x^3+x^2-2)^(2/3)*((2*x^3+x^2-2)^2)^(1/3)

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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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