\(\int \frac {(-6+x^2) (-2+x^2) (2-x^2+x^3) \sqrt [3]{-2+x^2+2 x^3}}{x^5 (-2+x^2+x^3)^2} \, dx\) [2137]

Optimal result

Integrand size = 48, antiderivative size = 156 \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\frac {\sqrt [3]{-2+x^2+2 x^3} \left (12-12 x^2+54 x^3+3 x^4-27 x^5-38 x^6\right )}{4 x^4 \left (-2+x^2+x^3\right )}-\frac {7 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-2+x^2+2 x^3}}\right )}{\sqrt {3}}-\frac {7}{3} \log \left (-x+\sqrt [3]{-2+x^2+2 x^3}\right )+\frac {7}{6} \log \left (x^2+x \sqrt [3]{-2+x^2+2 x^3}+\left (-2+x^2+2 x^3\right )^{2/3}\right ) \]

[Out]

1/4*(2*x^3+x^2-2)^(1/3)*(-38*x^6-27*x^5+3*x^4+54*x^3-12*x^2+12)/x^4/(x^3+x^2-2)-7/3*arctan(3^(1/2)*x/(x+2*(2*x
^3+x^2-2)^(1/3)))*3^(1/2)-7/3*ln(-x+(2*x^3+x^2-2)^(1/3))+7/6*ln(x^2+x*(2*x^3+x^2-2)^(1/3)+(2*x^3+x^2-2)^(2/3))

Rubi [F]

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

[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*} \text {integral}& = \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} \text {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} \text {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 \text {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 \text {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 )-\text {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 {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\frac {\sqrt [3]{-2+x^2+2 x^3} \left (12-12 x^2+54 x^3+3 x^4-27 x^5-38 x^6\right )}{4 x^4 \left (-2+x^2+x^3\right )}-\frac {7 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-2+x^2+2 x^3}}\right )}{\sqrt {3}}-\frac {7}{3} \log \left (-x+\sqrt [3]{-2+x^2+2 x^3}\right )+\frac {7}{6} \log \left (x^2+x \sqrt [3]{-2+x^2+2 x^3}+\left (-2+x^2+2 x^3\right )^{2/3}\right ) \]

[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]

((-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

Maple [A] (verified)

Time = 3.16 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.28

method	result	size
pseudoelliptic	\(\frac {\left (114 x^{6}+81 x^{5}-9 x^{4}-162 x^{3}+36 x^{2}-36\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}-14 x^{4} \left (x^{3}+x^{2}-2\right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}\right )}{3 x}\right )+\ln \left (\frac {x^{2}+x \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}+\left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-x +\left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}}{x}\right )\right )}{12 \left (x^{2}+x \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}+\left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}}\right ) x^{4} \left (x -\left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}\right )}\)	\(199\)
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 {20044800 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{3}+468480 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x -1586400 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}-80179200 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{2}+1292640 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{3}-3305 \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x +2329 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} x^{2}+383520 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}+2490 x^{3}[phone] \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2}+1079 x^{2}-767040 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )-2158}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )}{3}+1120 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \ln \left (-\frac {149196441600 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{3}[phone] \operatorname {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}} \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}-596785766400 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{2}-9892153440 \operatorname {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 -13680813 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} x^{2}-1771233600 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}[phone] x^{3}[phone] \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2}+4810311 x^{2}[phone] \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )-9620622}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )\)	\(495\)
risch	\(\text {Expression too large to display}\)	\(1291\)

[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]

1/12*((114*x^6+81*x^5-9*x^4-162*x^3+36*x^2-36)*(2*x^3+x^2-2)^(1/3)-14*x^4*(x^3+x^2-2)*(2*3^(1/2)*arctan(1/3*3^
(1/2)/x*(x+2*(2*x^3+x^2-2)^(1/3)))+ln((x^2+x*(2*x^3+x^2-2)^(1/3)+(2*x^3+x^2-2)^(2/3))/x^2)-2*ln((-x+(2*x^3+x^2
-2)^(1/3))/x)))/(x^2+x*(2*x^3+x^2-2)^(1/3)+(2*x^3+x^2-2)^(2/3))/x^4/(x-(2*x^3+x^2-2)^(1/3))

Fricas [A] (verification not implemented)

none

Time = 1.01 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\frac {28 \, \sqrt {3} {\left (x^{7} + x^{6} - 2 \, x^{4}\right )} \arctan \left (\frac {1078 \, \sqrt {3} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} x^{2} + 196 \, \sqrt {3} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (669 \, x^{3} + 32 \, x^{2} - 64\right )}}{1315 \, x^{3} - 8 \, x^{2} + 16}\right ) - 14 \, {\left (x^{7} + x^{6} - 2 \, x^{4}\right )} \log \left (\frac {x^{3} + 3 \, {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} x^{2} + x^{2} - 3 \, {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {2}{3}} x - 2}{x^{3} + x^{2} - 2}\right ) - 3 \, {\left (38 \, x^{6} + 27 \, x^{5} - 3 \, x^{4} - 54 \, x^{3} + 12 \, x^{2} - 12\right )} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}}}{12 \, {\left (x^{7} + x^{6} - 2 \, x^{4}\right )}} \]

[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)

Sympy [F]

\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int \frac {\left (x + 1\right ) \left (x^{2} - 6\right ) \left (x^{2} - 2\right ) \left (x^{2} - 2 x + 2\right ) \sqrt [3]{2 x^{3} + x^{2} - 2}}{x^{5} \left (x - 1\right )^{2} \left (x^{2} + 2 x + 2\right )^{2}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int { \frac {{\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} + 2\right )} {\left (x^{2} - 2\right )} {\left (x^{2} - 6\right )}}{{\left (x^{3} + x^{2} - 2\right )}^{2} x^{5}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int { \frac {{\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} + 2\right )} {\left (x^{2} - 2\right )} {\left (x^{2} - 6\right )}}{{\left (x^{3} + x^{2} - 2\right )}^{2} x^{5}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int \frac {\left (x^2-2\right )\,\left (x^2-6\right )\,\left (x^3-x^2+2\right )\,{\left (2\,x^3+x^2-2\right )}^{1/3}}{x^5\,{\left (x^3+x^2-2\right )}^2} \,d x \]

[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)