\(\int \frac {(-6+x^2) (2-x^2+x^3)^{2/3}}{x^3 (-2+x^2+x^3)} \, dx\) [2086]

Optimal result

Integrand size = 33, antiderivative size = 151 \[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx=-\frac {3 \left (2-x^2+x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{2-x^2+x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{2-x^2+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{2-x^2+x^3}+\sqrt [3]{2} \left (2-x^2+x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

((9*I)*(2 - x^2 + x^3)^(2/3)*Defer[Subst][Defer[Int][(((1 + (26 - 15*Sqrt[3])^(2/3))/(3*(26 - 15*Sqrt[3])^(1/3
)) + x)^(2/3)*((-1 + (26 - 15*Sqrt[3])^(-2/3) + (26 - 15*Sqrt[3])^(2/3))/9 - ((1 + (26 - 15*Sqrt[3])^(2/3))*x)
/(3*(26 - 15*Sqrt[3])^(1/3)) + x^2)^(2/3))/((-8/3 + 2*I) - 2*x), x], x, -1/3 + x])/((-1 + (26 - 15*Sqrt[3])^(-
1/3) + (26 - 15*Sqrt[3])^(1/3) + 3*x)^(2/3)*(-1 + (26 - 15*Sqrt[3])^(-2/3) + (26 - 15*Sqrt[3])^(2/3) + ((1 + (
26 - 15*Sqrt[3])^(2/3))*(1 - 3*x))/(26 - 15*Sqrt[3])^(1/3) + (-1 + 3*x)^2)^(2/3)) + (9*(2 - x^2 + x^3)^(2/3)*D
efer[Subst][Defer[Int][(((1 + (26 - 15*Sqrt[3])^(2/3))/(3*(26 - 15*Sqrt[3])^(1/3)) + x)^(2/3)*((-1 + (26 - 15*
Sqrt[3])^(-2/3) + (26 - 15*Sqrt[3])^(2/3))/9 - ((1 + (26 - 15*Sqrt[3])^(2/3))*x)/(3*(26 - 15*Sqrt[3])^(1/3)) +
x^2)^(2/3))/(2/3 - x), x], x, -1/3 + x])/((-1 + (26 - 15*Sqrt[3])^(-1/3) + (26 - 15*Sqrt[3])^(1/3) + 3*x)^(2/
3)*(-1 + (26 - 15*Sqrt[3])^(-2/3) + (26 - 15*Sqrt[3])^(2/3) + ((1 + (26 - 15*Sqrt[3])^(2/3))*(1 - 3*x))/(26 -
15*Sqrt[3])^(1/3) + (-1 + 3*x)^2)^(2/3)) + (27*(2 - x^2 + x^3)^(2/3)*Defer[Subst][Defer[Int][(((1 + (26 - 15*S
qrt[3])^(2/3))/(3*(26 - 15*Sqrt[3])^(1/3)) + x)^(2/3)*((-1 + (26 - 15*Sqrt[3])^(-2/3) + (26 - 15*Sqrt[3])^(2/3
))/9 - ((1 + (26 - 15*Sqrt[3])^(2/3))*x)/(3*(26 - 15*Sqrt[3])^(1/3)) + x^2)^(2/3))/(1/3 + x)^3, x], x, -1/3 +
x])/((-1 + (26 - 15*Sqrt[3])^(-1/3) + (26 - 15*Sqrt[3])^(1/3) + 3*x)^(2/3)*(-1 + (26 - 15*Sqrt[3])^(-2/3) + (2
6 - 15*Sqrt[3])^(2/3) + ((1 + (26 - 15*Sqrt[3])^(2/3))*(1 - 3*x))/(26 - 15*Sqrt[3])^(1/3) + (-1 + 3*x)^2)^(2/3
)) + (9*(2 - x^2 + x^3)^(2/3)*Defer[Subst][Defer[Int][(((1 + (26 - 15*Sqrt[3])^(2/3))/(3*(26 - 15*Sqrt[3])^(1/
3)) + x)^(2/3)*((-1 + (26 - 15*Sqrt[3])^(-2/3) + (26 - 15*Sqrt[3])^(2/3))/9 - ((1 + (26 - 15*Sqrt[3])^(2/3))*x
)/(3*(26 - 15*Sqrt[3])^(1/3)) + x^2)^(2/3))/(1/3 + x), x], x, -1/3 + x])/((-1 + (26 - 15*Sqrt[3])^(-1/3) + (26
- 15*Sqrt[3])^(1/3) + 3*x)^(2/3)*(-1 + (26 - 15*Sqrt[3])^(-2/3) + (26 - 15*Sqrt[3])^(2/3) + ((1 + (26 - 15*Sq
rt[3])^(2/3))*(1 - 3*x))/(26 - 15*Sqrt[3])^(1/3) + (-1 + 3*x)^2)^(2/3)) + ((9*I)*(2 - x^2 + x^3)^(2/3)*Defer[S
ubst][Defer[Int][(((1 + (26 - 15*Sqrt[3])^(2/3))/(3*(26 - 15*Sqrt[3])^(1/3)) + x)^(2/3)*((-1 + (26 - 15*Sqrt[3
])^(-2/3) + (26 - 15*Sqrt[3])^(2/3))/9 - ((1 + (26 - 15*Sqrt[3])^(2/3))*x)/(3*(26 - 15*Sqrt[3])^(1/3)) + x^2)^
(2/3))/((8/3 + 2*I) + 2*x), x], x, -1/3 + x])/((-1 + (26 - 15*Sqrt[3])^(-1/3) + (26 - 15*Sqrt[3])^(1/3) + 3*x)
^(2/3)*(-1 + (26 - 15*Sqrt[3])^(-2/3) + (26 - 15*Sqrt[3])^(2/3) + ((1 + (26 - 15*Sqrt[3])^(2/3))*(1 - 3*x))/(2
6 - 15*Sqrt[3])^(1/3) + (-1 + 3*x)^2)^(2/3))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (2-x^2+x^3\right )^{2/3}}{1-x}+\frac {3 \left (2-x^2+x^3\right )^{2/3}}{x^3}+\frac {\left (2-x^2+x^3\right )^{2/3}}{x}+\frac {\left (2-x^2+x^3\right )^{2/3}}{2+2 x+x^2}\right ) \, dx \\ & = 3 \int \frac {\left (2-x^2+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (2-x^2+x^3\right )^{2/3}}{1-x} \, dx+\int \frac {\left (2-x^2+x^3\right )^{2/3}}{x} \, dx+\int \frac {\left (2-x^2+x^3\right )^{2/3}}{2+2 x+x^2} \, dx \\ & = 3 \text {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{2/3}}{\left (\frac {1}{3}+x\right )^3} \, dx,x,-\frac {1}{3}+x\right )+\int \left (\frac {i \left (2-x^2+x^3\right )^{2/3}}{(-2+2 i)-2 x}+\frac {i \left (2-x^2+x^3\right )^{2/3}}{(2+2 i)+2 x}\right ) \, dx+\text {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{2/3}}{\frac {2}{3}-x} \, dx,x,-\frac {1}{3}+x\right )+\text {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{2/3}}{\frac {1}{3}+x} \, dx,x,-\frac {1}{3}+x\right ) \\ & = i \int \frac {\left (2-x^2+x^3\right )^{2/3}}{(-2+2 i)-2 x} \, dx+i \int \frac {\left (2-x^2+x^3\right )^{2/3}}{(2+2 i)+2 x} \, dx+\frac {\left (3 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\frac {2}{3}-x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (3 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\frac {1}{3}+x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (9 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\left (\frac {1}{3}+x\right )^3} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}} \\ & = i \text {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{2/3}}{\left (-\frac {8}{3}+2 i\right )-2 x} \, dx,x,-\frac {1}{3}+x\right )+i \text {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{2/3}}{\left (\frac {8}{3}+2 i\right )+2 x} \, dx,x,-\frac {1}{3}+x\right )+\frac {\left (3 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\frac {2}{3}-x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (3 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\frac {1}{3}+x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (9 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\left (\frac {1}{3}+x\right )^3} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx=-\frac {3 \left (2-x^2+x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{2-x^2+x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{2-x^2+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{2-x^2+x^3}+\sqrt [3]{2} \left (2-x^2+x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 16.01 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}} x +\left (x^{3}-x^{2}+2\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-2 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-x^{2}+2\right )^{\frac {1}{3}}}{x}\right ) x^{2}-3 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(140\)
risch	\(\text {Expression too large to display}\)	\(735\)
trager	\(\text {Expression too large to display}\)	\(1526\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (122) = 244\).

Time = 8.45 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.78 \[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx=-\frac {2 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{6} - x^{5} - 8 \, x^{4} + 4 \, x^{3} - 4 \, x\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{7} + x^{6} + 32 \, x^{5} - 4 \, x^{4} + 4 \, x^{2}\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{8} + 33 \, x^{7} + 221 \, x^{6} - 132 \, x^{5} + 6 \, x^{4} + 132 \, x^{3} - 12 \, x^{2} + 8\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{8} + 3 \, x^{7} + 211 \, x^{6} - 12 \, x^{5} - 6 \, x^{4} + 12 \, x^{3} + 12 \, x^{2} - 8\right )}}\right ) - 2 \, \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - x^{2} + 2\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + x^{2} - 2\right )}}{x^{3} + x^{2} - 2}\right ) + \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x^{3} + 2 \, x\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{5} + x^{4} + 32 \, x^{3} - 4 \, x^{2} + 4\right )} - 24 \, {\left (2 \, x^{5} - x^{4} + 2 \, x^{2}\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{5} + x^{4} - 4 \, x^{3} - 4 \, x^{2} + 4}\right ) + 9 \, {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]

[In]

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

[Out]

-1/6*(2*sqrt(3)*(-4)^(1/3)*x^2*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(5*x^7 + 4*x^6 - x^5 - 8*x^4 + 4*x^3 - 4*x)*(x
^3 - x^2 + 2)^(2/3) + 6*sqrt(3)*(-4)^(1/3)*(19*x^8 - 16*x^7 + x^6 + 32*x^5 - 4*x^4 + 4*x^2)*(x^3 - x^2 + 2)^(1
/3) - sqrt(3)*(71*x^9 - 111*x^8 + 33*x^7 + 221*x^6 - 132*x^5 + 6*x^4 + 132*x^3 - 12*x^2 + 8))/(109*x^9 - 105*x
^8 + 3*x^7 + 211*x^6 - 12*x^5 - 6*x^4 + 12*x^3 + 12*x^2 - 8)) - 2*(-4)^(1/3)*x^2*log(-(3*(-4)^(2/3)*(x^3 - x^2
+ 2)^(1/3)*x^2 - 6*(x^3 - x^2 + 2)^(2/3)*x + (-4)^(1/3)*(x^3 + x^2 - 2))/(x^3 + x^2 - 2)) + (-4)^(1/3)*x^2*lo
g(-(6*(-4)^(1/3)*(5*x^4 - x^3 + 2*x)*(x^3 - x^2 + 2)^(2/3) - (-4)^(2/3)*(19*x^6 - 16*x^5 + x^4 + 32*x^3 - 4*x^
2 + 4) - 24*(2*x^5 - x^4 + 2*x^2)*(x^3 - x^2 + 2)^(1/3))/(x^6 + 2*x^5 + x^4 - 4*x^3 - 4*x^2 + 4)) + 9*(x^3 - x
^2 + 2)^(2/3))/x^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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