\(\int \frac {(-4+x^3) \sqrt {2-x^2+x^3}}{(2+x^3) (2+x^2+x^3)} \, dx\) [620]

Optimal result

Integrand size = 37, antiderivative size = 49 \[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt {2-x^2+x^3}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {2-x^2+x^3}}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx \]

[In]

Int[((-4 + x^3)*Sqrt[2 - x^2 + x^3])/((2 + x^3)*(2 + x^2 + x^3)),x]

[Out]

-1/3*((-2)^(2/3)*Sqrt[2 - x^2 + x^3]) - (2^(2/3)*Sqrt[2 - x^2 + x^3])/3 + ((-1)^(1/3)*2^(2/3)*Sqrt[2 - x^2 + x
^3])/3 + ((1 - 3*(-2)^(1/3))*(-2*(26 + 15*Sqrt[3]))^(1/6)*Sqrt[2 - x^2 + x^3]*EllipticE[ArcSin[((26 - 15*Sqrt[
3])^(1/6)*Sqrt[(-I)*(2 + (26 - 15*Sqrt[3])^(1/3)*(1 - I*Sqrt[3]) + (1 + I*Sqrt[3])/(26 - 15*Sqrt[3])^(1/3) - 6
*x)])/(3^(1/4)*Sqrt[2*(1 - (26 - 15*Sqrt[3])^(2/3))])], (-2*Sqrt[3]*(1 - (26 - 15*Sqrt[3])^(2/3)))/(3*I - Sqrt
[3] + (26 - 15*Sqrt[3])^(2/3)*(3*I + Sqrt[3]))])/(Sqrt[-((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(1/
3) - 3*x)/(3 + I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt[3])))]*Sqrt[-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]) + (I
*((-1)^(1/3) - 3*2^(1/3))*(2*(26 + 15*Sqrt[3]))^(1/6)*Sqrt[2 - x^2 + x^3]*EllipticE[ArcSin[((26 - 15*Sqrt[3])^
(1/6)*Sqrt[(-I)*(2 + (26 - 15*Sqrt[3])^(1/3)*(1 - I*Sqrt[3]) + (1 + I*Sqrt[3])/(26 - 15*Sqrt[3])^(1/3) - 6*x)]
)/(3^(1/4)*Sqrt[2*(1 - (26 - 15*Sqrt[3])^(2/3))])], (-2*Sqrt[3]*(1 - (26 - 15*Sqrt[3])^(2/3)))/(3*I - Sqrt[3]
+ (26 - 15*Sqrt[3])^(2/3)*(3*I + Sqrt[3]))])/(Sqrt[-((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(1/3) -
3*x)/(3 + I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt[3])))]*Sqrt[-1 + (26 - 15*Sqrt[3])^(-2/3) + (26 - 1
5*Sqrt[3])^(2/3) + ((1 + (26 - 15*Sqrt[3])^(2/3))*(1 - 3*x))/(26 - 15*Sqrt[3])^(1/3) + (-1 + 3*x)^2]) - (I*(1
+ 3*2^(1/3))*(2*(26 + 15*Sqrt[3]))^(1/6)*Sqrt[2 - x^2 + x^3]*EllipticE[ArcSin[((26 - 15*Sqrt[3])^(1/6)*Sqrt[(-
I)*(2 + (26 - 15*Sqrt[3])^(1/3)*(1 - I*Sqrt[3]) + (1 + I*Sqrt[3])/(26 - 15*Sqrt[3])^(1/3) - 6*x)])/(3^(1/4)*Sq
rt[2*(1 - (26 - 15*Sqrt[3])^(2/3))])], (-2*Sqrt[3]*(1 - (26 - 15*Sqrt[3])^(2/3)))/(3*I - Sqrt[3] + (26 - 15*Sq
rt[3])^(2/3)*(3*I + Sqrt[3]))])/(Sqrt[-((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(1/3) - 3*x)/(3 + I*
Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt[3])))]*Sqrt[-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*(1 + 6*(-2)^(1/3
) - 9*(-2)^(2/3))*(-2*(26 - 15*Sqrt[3]))^(1/6)*Sqrt[-((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(1/3)
- 3*x)/(3 + I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt[3])))]*Sqrt[2 - x^2 + x^3]*EllipticF[ArcSin[((26 -
15*Sqrt[3])^(1/6)*Sqrt[(-I)*(2 + (26 - 15*Sqrt[3])^(1/3)*(1 - I*Sqrt[3]) + (1 + I*Sqrt[3])/(26 - 15*Sqrt[3])^
(1/3) - 6*x)])/(3^(1/4)*Sqrt[2*(1 - (26 - 15*Sqrt[3])^(2/3))])], (-2*Sqrt[3]*(1 - (26 - 15*Sqrt[3])^(2/3)))/(3
*I - Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3*I + Sqrt[3]))])/((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(
1/3) - 3*x)*Sqrt[-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*I)*(1 - 6*2^(1/3) - 9*2^(2/3))*(2*(26 - 15*Sqrt[3]))^(1/
6)*Sqrt[-((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(1/3) - 3*x)/(3 + I*Sqrt[3] + (26 - 15*Sqrt[3])^(2
/3)*(3 - I*Sqrt[3])))]*Sqrt[2 - x^2 + x^3]*EllipticF[ArcSin[((26 - 15*Sqrt[3])^(1/6)*Sqrt[(-I)*(2 + (26 - 15*S
qrt[3])^(1/3)*(1 - I*Sqrt[3]) + (1 + I*Sqrt[3])/(26 - 15*Sqrt[3])^(1/3) - 6*x)])/(3^(1/4)*Sqrt[2*(1 - (26 - 15
*Sqrt[3])^(2/3))])], (-2*Sqrt[3]*(1 - (26 - 15*Sqrt[3])^(2/3)))/(3*I - Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3*I
+ Sqrt[3]))])/((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(1/3) - 3*x)*Sqrt[-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*(-1)^(5/6)*(1 - 6*(-1)^(2/3)*2^(1/3) + 9*(-1)^(1/3)*2^(2/3))*(2*(26 - 15*Sqrt[3]))^(1/6)*Sqrt[-((1 -
(26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(1/3) - 3*x)/(3 + I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt
[3])))]*Sqrt[2 - x^2 + x^3]*EllipticF[ArcSin[((26 - 15*Sqrt[3])^(1/6)*Sqrt[(-I)*(2 + (26 - 15*Sqrt[3])^(1/3)*(
1 - I*Sqrt[3]) + (1 + I*Sqrt[3])/(26 - 15*Sqrt[3])^(1/3) - 6*x)])/(3^(1/4)*Sqrt[2*(1 - (26 - 15*Sqrt[3])^(2/3)
)])], (-2*Sqrt[3]*(1 - (26 - 15*Sqrt[3])^(2/3)))/(3*I - Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3*I + Sqrt[3]))])/(
(1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(1/3) - 3*x)*Sqrt[-1 + (26 - 15*Sqrt[3])^(-2/3) + (26 - 15*S
qrt[3])^(2/3) + ((1 + (26 - 15*Sqrt[3])^(2/3))*(1 - 3*x))/(26 - 15*Sqrt[3])^(1/3) + (-1 + 3*x)^2]) + (2*(1 - 3
*(-2)^(1/3))*(-2*(26 + 15*Sqrt[3]))^(1/6)*(1 + (26 - 15*Sqrt[3])^(2/3))*Sqrt[-((1 - (26 - 15*Sqrt[3])^(-1/3) -
(26 - 15*Sqrt[3])^(1/3) - 3*x)/(3 + I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt[3])))]*Sqrt[2 - x^2 + x^3
]*EllipticF[ArcSin[((26 - 15*Sqrt[3])^(1/6)*Sqrt[(-I)*(2 + (26 - 15*Sqrt[3])^(1/3)*(1 - I*Sqrt[3]) + (1 + I*Sq
rt[3])/(26 - 15*Sqrt[3])^(1/3) - 6*x)])/(3^(1/4)*Sqrt[2*(1 - (26 - 15*Sqrt[3])^(2/3))])], (-2*Sqrt[3]*(1 - (26
- 15*Sqrt[3])^(2/3)))/(3*I - Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3*I + Sqrt[3]))])/((1 - (26 - 15*Sqrt[3])^(-1
/3) - (26 - 15*Sqrt[3])^(1/3) - 3*x)*Sqrt[-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*I)*((-1)^(1/3) - 3*2^(1/3))*(2*
(26 + 15*Sqrt[3]))^(1/6)*(1 + (26 - 15*Sqrt[3])^(2/3))*Sqrt[-((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3]
)^(1/3) - 3*x)/(3 + I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt[3])))]*Sqrt[2 - x^2 + x^3]*EllipticF[ArcSi
n[((26 - 15*Sqrt[3])^(1/6)*Sqrt[(-I)*(2 + (26 - 15*Sqrt[3])^(1/3)*(1 - I*Sqrt[3]) + (1 + I*Sqrt[3])/(26 - 15*S
qrt[3])^(1/3) - 6*x)])/(3^(1/4)*Sqrt[2*(1 - (26 - 15*Sqrt[3])^(2/3))])], (-2*Sqrt[3]*(1 - (26 - 15*Sqrt[3])^(2
/3)))/(3*I - Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3*I + Sqrt[3]))])/((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sq
rt[3])^(1/3) - 3*x)*Sqrt[-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*I)*(1 + 3*2^(1/3))*(2*(26 + 15*Sqrt[3]))^(1/6)*(
1 + (26 - 15*Sqrt[3])^(2/3))*Sqrt[-((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(1/3) - 3*x)/(3 + I*Sqrt
[3] + (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt[3])))]*Sqrt[2 - x^2 + x^3]*EllipticF[ArcSin[((26 - 15*Sqrt[3])^(1/6)
*Sqrt[(-I)*(2 + (26 - 15*Sqrt[3])^(1/3)*(1 - I*Sqrt[3]) + (1 + I*Sqrt[3])/(26 - 15*Sqrt[3])^(1/3) - 6*x)])/(3^
(1/4)*Sqrt[2*(1 - (26 - 15*Sqrt[3])^(2/3))])], (-2*Sqrt[3]*(1 - (26 - 15*Sqrt[3])^(2/3)))/(3*I - Sqrt[3] + (26
- 15*Sqrt[3])^(2/3)*(3*I + Sqrt[3]))])/((1 - (26 - 15*Sqrt[3])^(-1/3) - (26 - 15*Sqrt[3])^(1/3) - 3*x)*Sqrt[-
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*Sq
rt[3])^(1/3) + (-1 + 3*x)^2]) + (27*2^(5/6)*Sqrt[(26 - 15*Sqrt[3])*(3 - I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3
+ I*Sqrt[3]))]*Sqrt[1 - (2*(1 + (26 - 15*Sqrt[3])^(2/3) - (26 - 15*Sqrt[3])^(1/3)*(1 - 3*x)))/(3 + I*Sqrt[3]
+ (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt[3]))]*Sqrt[1 - (2*(1 + (26 - 15*Sqrt[3])^(2/3) - (26 - 15*Sqrt[3])^(1/3)
*(1 - 3*x)))/(3 - I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 + I*Sqrt[3]))]*Sqrt[-2 - (26 - 15*Sqrt[3])^(1/3)*(1 +
I*Sqrt[3]) + (I*(I + Sqrt[3]))/(26 - 15*Sqrt[3])^(1/3) + 6*x]*Sqrt[-2 - (1 + I*Sqrt[3])/(26 - 15*Sqrt[3])^(1/
3) + I*(26 - 15*Sqrt[3])^(1/3)*(I + Sqrt[3]) + 6*x]*Sqrt[2 - x^2 + x^3]*EllipticPi[(3 - I*Sqrt[3] + (26 - 15*S
qrt[3])^(2/3)*(3 + I*Sqrt[3]))/(2*(1 - 3*(52 - 30*Sqrt[3])^(1/3) - (26 - 15*Sqrt[3])^(1/3) + (26 - 15*Sqrt[3])
^(2/3))), ArcSin[(Sqrt[2]*(26 - 15*Sqrt[3])^(1/6)*Sqrt[-1 + (26 - 15*Sqrt[3])^(-1/3) + (26 - 15*Sqrt[3])^(1/3)
+ 3*x])/Sqrt[3 - I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 + I*Sqrt[3])]], (3*I + Sqrt[3] + (26 - 15*Sqrt[3])^(2
/3)*(3*I - Sqrt[3]))/(3*I - Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3*I + Sqrt[3]))])/((1 - 3*(52 - 30*Sqrt[3])^(1/
3) - (26 - 15*Sqrt[3])^(1/3) + (26 - 15*Sqrt[3])^(2/3))*Sqrt[-1 + I*Sqrt[3] - (26 - 15*Sqrt[3])^(2/3)*(1 + I*S
qrt[3]) - 2*(26 - 15*Sqrt[3])^(1/3)*(1 - 3*x)]*Sqrt[-1 - I*Sqrt[3] + I*(26 - 15*Sqrt[3])^(2/3)*(I + Sqrt[3]) -
2*(26 - 15*Sqrt[3])^(1/3)*(1 - 3*x)]*(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)*Sqrt[-1 + (26 - 15*Sqrt[3])^(-1/3) + (26 -
15*Sqrt[3])^(1/3) + 3*x]) + (27*2^(5/6)*(26 - 15*Sqrt[3])^(1/6)*Sqrt[3 - I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(
3 + I*Sqrt[3])]*Sqrt[1 - (2*(1 + (26 - 15*Sqrt[3])^(2/3) - (26 - 15*Sqrt[3])^(1/3)*(1 - 3*x)))/(3 + I*Sqrt[3]
+ (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt[3]))]*Sqrt[1 - (2*(1 + (26 - 15*Sqrt[3])^(2/3) - (26 - 15*Sqrt[3])^(1/3)
*(1 - 3*x)))/(3 - I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 + I*Sqrt[3]))]*Sqrt[-2 - (26 - 15*Sqrt[3])^(1/3)*(1 +
I*Sqrt[3]) + (I*(I + Sqrt[3]))/(26 - 15*Sqrt[3])^(1/3) + 6*x]*Sqrt[-2 - (1 + I*Sqrt[3])/(26 - 15*Sqrt[3])^(1/
3) + I*(26 - 15*Sqrt[3])^(1/3)*(I + Sqrt[3]) + 6*x]*Sqrt[2 - x^2 + x^3]*EllipticPi[-1/2*((-26 - 15*Sqrt[3])^(1
/3)*(3 - I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 + I*Sqrt[3])))/((-1)^(1/3) - 3*2^(1/3) - (-26 - 15*Sqrt[3])^(1
/3) - (-26 + 15*Sqrt[3])^(1/3)), ArcSin[(Sqrt[2]*(26 - 15*Sqrt[3])^(1/6)*Sqrt[-1 + (26 - 15*Sqrt[3])^(-1/3) +
(26 - 15*Sqrt[3])^(1/3) + 3*x])/Sqrt[3 - I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 + I*Sqrt[3])]], (3*I + Sqrt[3]
+ (26 - 15*Sqrt[3])^(2/3)*(3*I - Sqrt[3]))/(3*I - Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3*I + Sqrt[3]))])/(((-1)
^(1/3) - 3*2^(1/3) - (-26 - 15*Sqrt[3])^(1/3) - (-26 + 15*Sqrt[3])^(1/3))*Sqrt[-1 + I*Sqrt[3] - (26 - 15*Sqrt[
3])^(2/3)*(1 + I*Sqrt[3]) - 2*(26 - 15*Sqrt[3])^(1/3)*(1 - 3*x)]*Sqrt[-1 - I*Sqrt[3] + I*(26 - 15*Sqrt[3])^(2/
3)*(I + Sqrt[3]) - 2*(26 - 15*Sqrt[3])^(1/3)*(1 - 3*x)]*(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)*Sqrt[-1 + (26 - 15*Sqrt[3
])^(-1/3) + (26 - 15*Sqrt[3])^(1/3) + 3*x]) + (27*2^(5/6)*Sqrt[(26 - 15*Sqrt[3])*(3 - I*Sqrt[3] + (26 - 15*Sqr
t[3])^(2/3)*(3 + I*Sqrt[3]))]*Sqrt[1 - (2*(1 + (26 - 15*Sqrt[3])^(2/3) - (26 - 15*Sqrt[3])^(1/3)*(1 - 3*x)))/(
3 + I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 - I*Sqrt[3]))]*Sqrt[1 - (2*(1 + (26 - 15*Sqrt[3])^(2/3) - (26 - 15*
Sqrt[3])^(1/3)*(1 - 3*x)))/(3 - I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 + I*Sqrt[3]))]*Sqrt[-2 - (26 - 15*Sqrt[
3])^(1/3)*(1 + I*Sqrt[3]) + (I*(I + Sqrt[3]))/(26 - 15*Sqrt[3])^(1/3) + 6*x]*Sqrt[-2 - (1 + I*Sqrt[3])/(26 - 1
5*Sqrt[3])^(1/3) + I*(26 - 15*Sqrt[3])^(1/3)*(I + Sqrt[3]) + 6*x]*Sqrt[2 - x^2 + x^3]*EllipticPi[((-1)^(1/6)*(
3*I + Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3*I - Sqrt[3])))/(2*((-1)^(2/3) - 3*(52 - 30*Sqrt[3])^(1/3) - (-1)^(2
/3)*(26 - 15*Sqrt[3])^(1/3) + (-26 + 15*Sqrt[3])^(2/3))), ArcSin[(Sqrt[2]*(26 - 15*Sqrt[3])^(1/6)*Sqrt[-1 + (2
6 - 15*Sqrt[3])^(-1/3) + (26 - 15*Sqrt[3])^(1/3) + 3*x])/Sqrt[3 - I*Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3 + I*S
qrt[3])]], (3*I + Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*(3*I - Sqrt[3]))/(3*I - Sqrt[3] + (26 - 15*Sqrt[3])^(2/3)*
(3*I + Sqrt[3]))])/(((-1)^(2/3) - 3*(52 - 30*Sqrt[3])^(1/3) - (-1)^(2/3)*(26 - 15*Sqrt[3])^(1/3) + (-26 + 15*S
qrt[3])^(2/3))*Sqrt[-1 + I*Sqrt[3] - (26 - 15*Sqrt[3])^(2/3)*(1 + I*Sqrt[3]) - 2*(26 - 15*Sqrt[3])^(1/3)*(1 -
3*x)]*Sqrt[-1 - I*Sqrt[3] + I*(26 - 15*Sqrt[3])^(2/3)*(I + Sqrt[3]) - 2*(26 - 15*Sqrt[3])^(1/3)*(1 - 3*x)]*(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)*Sqrt[-1 + (26 - 15*Sqrt[3])^(-1/3) + (26 - 15*Sqrt[3])^(1/3) + 3*x]) - 2*Defer[Int][
Sqrt[2 - x^2 + x^3]/(2 + x^2 + x^3), x] - 3*Defer[Int][(x*Sqrt[2 - x^2 + x^3])/(2 + x^2 + x^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 x \sqrt {2-x^2+x^3}}{2+x^3}+\frac {(-2-3 x) \sqrt {2-x^2+x^3}}{2+x^2+x^3}\right ) \, dx \\ & = 3 \int \frac {x \sqrt {2-x^2+x^3}}{2+x^3} \, dx+\int \frac {(-2-3 x) \sqrt {2-x^2+x^3}}{2+x^2+x^3} \, dx \\ & = 3 \int \left (-\frac {\sqrt {2-x^2+x^3}}{3 \sqrt [3]{2} \left (\sqrt [3]{2}+x\right )}-\frac {(-1)^{2/3} \sqrt {2-x^2+x^3}}{3 \sqrt [3]{2} \left (\sqrt [3]{2}-\sqrt [3]{-1} x\right )}+\frac {\sqrt [3]{-\frac {1}{2}} \sqrt {2-x^2+x^3}}{3 \left (\sqrt [3]{2}+(-1)^{2/3} x\right )}\right ) \, dx+\int \left (-\frac {2 \sqrt {2-x^2+x^3}}{2+x^2+x^3}-\frac {3 x \sqrt {2-x^2+x^3}}{2+x^2+x^3}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt {2-x^2+x^3}}{2+x^2+x^3} \, dx\right )-3 \int \frac {x \sqrt {2-x^2+x^3}}{2+x^2+x^3} \, dx+\sqrt [3]{-\frac {1}{2}} \int \frac {\sqrt {2-x^2+x^3}}{\sqrt [3]{2}+(-1)^{2/3} x} \, dx-\frac {\int \frac {\sqrt {2-x^2+x^3}}{\sqrt [3]{2}+x} \, dx}{\sqrt [3]{2}}-\frac {(-1)^{2/3} \int \frac {\sqrt {2-x^2+x^3}}{\sqrt [3]{2}-\sqrt [3]{-1} x} \, dx}{\sqrt [3]{2}} \\ & = -\left (2 \int \frac {\sqrt {2-x^2+x^3}}{2+x^2+x^3} \, dx\right )-3 \int \frac {x \sqrt {2-x^2+x^3}}{2+x^2+x^3} \, dx+\sqrt [3]{-\frac {1}{2}} \text {Subst}\left (\int \frac {\sqrt {\frac {52}{27}-\frac {x}{3}+x^3}}{\frac {1}{3} \left ((-1)^{2/3}+3 \sqrt [3]{2}\right )+(-1)^{2/3} x} \, dx,x,-\frac {1}{3}+x\right )-\frac {\text {Subst}\left (\int \frac {\sqrt {\frac {52}{27}-\frac {x}{3}+x^3}}{\frac {1}{3} \left (1+3 \sqrt [3]{2}\right )+x} \, dx,x,-\frac {1}{3}+x\right )}{\sqrt [3]{2}}-\frac {(-1)^{2/3} \text {Subst}\left (\int \frac {\sqrt {\frac {52}{27}-\frac {x}{3}+x^3}}{\frac {1}{3} \left (-\sqrt [3]{-1}+3 \sqrt [3]{2}\right )-\sqrt [3]{-1} x} \, dx,x,-\frac {1}{3}+x\right )}{\sqrt [3]{2}} \\ & = -\left (2 \int \frac {\sqrt {2-x^2+x^3}}{2+x^2+x^3} \, dx\right )-3 \int \frac {x \sqrt {2-x^2+x^3}}{2+x^2+x^3} \, dx+\frac {\left (3 \sqrt [3]{-\frac {1}{2}} \sqrt {2-x^2+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x} \sqrt {\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}}{\frac {1}{3} \left ((-1)^{2/3}+3 \sqrt [3]{2}\right )+(-1)^{2/3} x} \, dx,x,-\frac {1}{3}+x\right )}{\sqrt {\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x} \sqrt {-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}}}}}-\frac {\left (3 \sqrt {2-x^2+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x} \sqrt {\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}}{\frac {1}{3} \left (1+3 \sqrt [3]{2}\right )+x} \, dx,x,-\frac {1}{3}+x\right )}{\sqrt [3]{2} \sqrt {\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x} \sqrt {-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}}}}}-\frac {\left (3 (-1)^{2/3} \sqrt {2-x^2+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x} \sqrt {\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}}{\frac {1}{3} \left (-\sqrt [3]{-1}+3 \sqrt [3]{2}\right )-\sqrt [3]{-1} x} \, dx,x,-\frac {1}{3}+x\right )}{\sqrt [3]{2} \sqrt {\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x} \sqrt {-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}}}}} \\ & = -\frac {1}{3} (-2)^{2/3} \sqrt {2-x^2+x^3}-\frac {1}{3} 2^{2/3} \sqrt {2-x^2+x^3}+\frac {1}{3} \sqrt [3]{-1} 2^{2/3} \sqrt {2-x^2+x^3}-2 \int \frac {\sqrt {2-x^2+x^3}}{2+x^2+x^3} \, dx-3 \int \frac {x \sqrt {2-x^2+x^3}}{2+x^2+x^3} \, dx-\frac {\left (\sqrt [3]{-\frac {1}{2}} \sqrt {2-x^2+x^3}\right ) \text {Subst}\left (\int \frac {\frac {1}{9} \left (53 \sqrt [3]{-1}-3 \sqrt [3]{2}\right )-\frac {2}{3} \sqrt [3]{-1} x-\left (\sqrt [3]{-1}-3 \sqrt [3]{2}\right ) x^2}{\sqrt {\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x} \left (\frac {1}{3} \left (-\sqrt [3]{-1}+3 \sqrt [3]{2}\right )-\sqrt [3]{-1} x\right ) \sqrt {\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}} \, dx,x,-\frac {1}{3}+x\right )}{\sqrt {\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x} \sqrt {-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}}}}}+\frac {\sqrt {2-x^2+x^3} \text {Subst}\left (\int \frac {\frac {1}{9} \left (-53-3 \sqrt [3]{2}\right )+\frac {2 x}{3}+\left (1+3 \sqrt [3]{2}\right ) x^2}{\left (\frac {1}{3} \left (1+3 \sqrt [3]{2}\right )+x\right ) \sqrt {\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x} \sqrt {\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}} \, dx,x,-\frac {1}{3}+x\right )}{\sqrt [3]{2} \sqrt {\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x} \sqrt {-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}}}}}+\frac {\left ((-1)^{2/3} \sqrt {2-x^2+x^3}\right ) \text {Subst}\left (\int \frac {\frac {1}{9} \left (-53 (-1)^{2/3}-3 \sqrt [3]{2}\right )+\frac {2}{3} (-1)^{2/3} x+\left ((-1)^{2/3}+3 \sqrt [3]{2}\right ) x^2}{\sqrt {\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x} \left (\frac {1}{3} \left ((-1)^{2/3}+3 \sqrt [3]{2}\right )+(-1)^{2/3} x\right ) \sqrt {\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}} \, dx,x,-\frac {1}{3}+x\right )}{\sqrt [3]{2} \sqrt {\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x} \sqrt {-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}}}}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt {2-x^2+x^3}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {2-x^2+x^3}}\right ) \]

[In]

Integrate[((-4 + x^3)*Sqrt[2 - x^2 + x^3])/((2 + x^3)*(2 + x^2 + x^3)),x]

[Out]

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

Maple [A] (verified)

Time = 3.78 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.96

method	result	size
default	\(2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{3}-x^{2}+2}}{2 x}\right )-2 \arctan \left (\frac {\sqrt {x^{3}-x^{2}+2}}{x}\right )\)	\(47\)
pseudoelliptic	\(2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{3}-x^{2}+2}}{2 x}\right )-2 \arctan \left (\frac {\sqrt {x^{3}-x^{2}+2}}{x}\right )\)	\(47\)
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 x \sqrt {x^{3}-x^{2}+2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{x^{3}+x^{2}+2}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{3}-x^{2}+2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}+2}\right )\)	\(130\)
elliptic	\(\text {Expression too large to display}\)	\(1019\)

[In]

int((x^3-4)*(x^3-x^2+2)^(1/2)/(x^3+2)/(x^3+x^2+2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (41) = 82\).

Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.80 \[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x^{3} - x^{2} + 2} {\left (x^{3} - 3 \, x^{2} + 2\right )}}{4 \, {\left (x^{4} - x^{3} + 2 \, x\right )}}\right ) - \arctan \left (\frac {\sqrt {x^{3} - x^{2} + 2} {\left (x^{3} - 2 \, x^{2} + 2\right )}}{2 \, {\left (x^{4} - x^{3} + 2 \, x\right )}}\right ) \]

[In]

integrate((x^3-4)*(x^3-x^2+2)^(1/2)/(x^3+2)/(x^3+x^2+2),x, algorithm="fricas")

[Out]

sqrt(2)*arctan(1/4*sqrt(2)*sqrt(x^3 - x^2 + 2)*(x^3 - 3*x^2 + 2)/(x^4 - x^3 + 2*x)) - arctan(1/2*sqrt(x^3 - x^
2 + 2)*(x^3 - 2*x^2 + 2)/(x^4 - x^3 + 2*x))

Sympy [F]

\[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\int \frac {\sqrt {\left (x + 1\right ) \left (x^{2} - 2 x + 2\right )} \left (x^{3} - 4\right )}{\left (x^{3} + 2\right ) \left (x^{3} + x^{2} + 2\right )}\, dx \]

[In]

integrate((x**3-4)*(x**3-x**2+2)**(1/2)/(x**3+2)/(x**3+x**2+2),x)

[Out]

Integral(sqrt((x + 1)*(x**2 - 2*x + 2))*(x**3 - 4)/((x**3 + 2)*(x**3 + x**2 + 2)), x)

Maxima [F]

\[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} + 2} {\left (x^{3} - 4\right )}}{{\left (x^{3} + x^{2} + 2\right )} {\left (x^{3} + 2\right )}} \,d x } \]

[In]

integrate((x^3-4)*(x^3-x^2+2)^(1/2)/(x^3+2)/(x^3+x^2+2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^3 - x^2 + 2)*(x^3 - 4)/((x^3 + x^2 + 2)*(x^3 + 2)), x)

Giac [F]

\[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} + 2} {\left (x^{3} - 4\right )}}{{\left (x^{3} + x^{2} + 2\right )} {\left (x^{3} + 2\right )}} \,d x } \]

[In]

integrate((x^3-4)*(x^3-x^2+2)^(1/2)/(x^3+2)/(x^3+x^2+2),x, algorithm="giac")

[Out]

integrate(sqrt(x^3 - x^2 + 2)*(x^3 - 4)/((x^3 + x^2 + 2)*(x^3 + 2)), x)

Mupad [B] (verification not implemented)

Time = 6.02 (sec) , antiderivative size = 235, normalized size of antiderivative = 4.80 \[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\left (\sum _{_{\mathrm {X494}}\in \left \{-2^{1/3},2^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right ),-2^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right \}\cup \mathrm {root}\left (z^3+z^2+2,z\right )}\frac {\sqrt {5}\,\sqrt {x\,\left (2-\mathrm {i}\right )+2-\mathrm {i}}\,\sqrt {3+x\,\left (-2+1{}\mathrm {i}\right )+1{}\mathrm {i}}\,\sqrt {3+x\,\left (-2-\mathrm {i}\right )-\mathrm {i}}\,\Pi \left (\frac {2+1{}\mathrm {i}}{_{\mathrm {X494}}+1};\mathrm {asin}\left (\frac {\sqrt {5}\,\sqrt {x\,\left (2-\mathrm {i}\right )+2-\mathrm {i}}}{5}\right )\middle |\frac {3}{5}+\frac {4}{5}{}\mathrm {i}\right )\,\left (2\,{_{\mathrm {X494}}}^5+6\,{_{\mathrm {X494}}}^3-2\,{_{\mathrm {X494}}}^2+12\right )\,\left (\frac {4}{25}+\frac {2}{25}{}\mathrm {i}\right )}{_{\mathrm {X494}}\,\left (_{\mathrm {X494}}+1\right )\,\sqrt {x^3-x^2+2}\,\left (6\,{_{\mathrm {X494}}}^4+5\,{_{\mathrm {X494}}}^3+12\,_{\mathrm {X494}}+4\right )}\right )+\frac {\sqrt {x\,\left (\frac {2}{5}-\frac {1}{5}{}\mathrm {i}\right )+\frac {2}{5}-\frac {1}{5}{}\mathrm {i}}\,\sqrt {\frac {3}{5}+x\,\left (-\frac {2}{5}+\frac {1}{5}{}\mathrm {i}\right )+\frac {1}{5}{}\mathrm {i}}\,\sqrt {\frac {3}{5}+x\,\left (-\frac {2}{5}-\frac {1}{5}{}\mathrm {i}\right )-\frac {1}{5}{}\mathrm {i}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {x\,\left (\frac {2}{5}-\frac {1}{5}{}\mathrm {i}\right )+\frac {2}{5}-\frac {1}{5}{}\mathrm {i}}\right )\middle |\frac {3}{5}+\frac {4}{5}{}\mathrm {i}\right )\,\left (4+2{}\mathrm {i}\right )}{\sqrt {x^3-x^2+2}} \]

[In]

int(((x^3 - 4)*(x^3 - x^2 + 2)^(1/2))/((x^3 + 2)*(x^2 + x^3 + 2)),x)

[Out]

symsum((5^(1/2)*(x*(2 - 1i) + (2 - 1i))^(1/2)*((3 + 1i) - x*(2 - 1i))^(1/2)*((3 - 1i) - x*(2 + 1i))^(1/2)*elli
pticPi((2 + 1i)/(_X494 + 1), asin((5^(1/2)*(x*(2 - 1i) + (2 - 1i))^(1/2))/5), 3/5 + 4i/5)*(6*_X494^3 - 2*_X494
^2 + 2*_X494^5 + 12)*(4/25 + 2i/25))/(_X494*(_X494 + 1)*(x^3 - x^2 + 2)^(1/2)*(12*_X494 + 5*_X494^3 + 6*_X494^
4 + 4)), _X494 in {-2^(1/3), 2^(1/3)*((3^(1/2)*1i)/2 + 1/2), -2^(1/3)*((3^(1/2)*1i)/2 - 1/2)} union root(z^3 +
z^2 + 2, z)) + ((x*(2/5 - 1i/5) + (2/5 - 1i/5))^(1/2)*((3/5 + 1i/5) - x*(2/5 - 1i/5))^(1/2)*((3/5 - 1i/5) - x
*(2/5 + 1i/5))^(1/2)*ellipticF(asin((x*(2/5 - 1i/5) + (2/5 - 1i/5))^(1/2)), 3/5 + 4i/5)*(4 + 2i))/(x^3 - x^2 +
2)^(1/2)