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

Optimal. Leaf size=49 \[ 2 \tan ^{-1}\left (\frac {x}{\sqrt {x^3-x^2+2}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^3-x^2+2}}\right ) \]

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Rubi [F]  time = 51.17, 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 (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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}}\\ \end {align*}

rest of steps removed due to Latex formating problem.

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Mathematica [C]  time = 6.40, size = 5117, normalized size = 104.43 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.35, size = 49, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {x}{\sqrt {2-x^2+x^3}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {2-x^2+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-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]]

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fricas [B]  time = 0.65, size = 88, normalized size = 1.80 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.64, size = 128, normalized size = 2.61

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}-4 x \sqrt {x^{3}-x^{2}+2}+2 \RootOf \left (\textit {\_Z}^{2}+2\right )}{x^{3}+x^{2}+2}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{3}-x^{2}+2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{3}+2}\right )\)	\(128\)
default	\(\frac {\left (4-2 i\right ) \sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}\, \sqrt {\left (-\frac {2}{5}+\frac {i}{5}\right ) \left (x -1-i\right )}\, \sqrt {\left (-\frac {2}{5}-\frac {i}{5}\right ) \left (x -1+i\right )}\, \EllipticF \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )}{\sqrt {x^{3}-x^{2}+2}}+\left (-4+2 i\right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +1\right ) \sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}\, \sqrt {\left (-\frac {2}{5}+\frac {i}{5}\right ) \left (x -1-i\right )}\, \sqrt {\left (-\frac {2}{5}-\frac {i}{5}\right ) \left (x -1+i\right )}\, \EllipticPi \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, i \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-i \underline {\hspace {1.25 ex}}\alpha +2 \underline {\hspace {1.25 ex}}\alpha +i-2, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )}{\sqrt {x^{3}-x^{2}+2}}\right )+\left (4-2 i\right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+\textit {\_Z}^{2}+2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}\, \sqrt {\left (-\frac {2}{5}+\frac {i}{5}\right ) \left (x -1-i\right )}\, \sqrt {\left (-\frac {2}{5}-\frac {i}{5}\right ) \left (x -1+i\right )}\, \EllipticPi \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, \frac {1}{2} i \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2}, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )}{\sqrt {x^{3}-x^{2}+2}}\right )\)	\(278\)
elliptic	\(\frac {4 \sqrt {\frac {2}{5}+\frac {1}{5} i+\frac {2}{5} x +\frac {1}{5} i x}\, \sqrt {-\frac {2}{5} x +\frac {1}{5} i x +\frac {3}{5}+\frac {1}{5} i}\, \sqrt {-\frac {2}{5} x -\frac {1}{5} i x +\frac {3}{5}-\frac {1}{5} i}\, \EllipticF \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )}{\sqrt {x^{3}-x^{2}+2}}-\frac {2 i \sqrt {\frac {2}{5}+\frac {1}{5} i+\frac {2}{5} x +\frac {1}{5} i x}\, \sqrt {-\frac {2}{5} x +\frac {1}{5} i x +\frac {3}{5}+\frac {1}{5} i}\, \sqrt {-\frac {2}{5} x -\frac {1}{5} i x +\frac {3}{5}-\frac {1}{5} i}\, \EllipticF \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )}{\sqrt {x^{3}-x^{2}+2}}+\frac {4 \sqrt {\frac {2}{5}+\frac {1}{5} i+\frac {2}{5} x +\frac {1}{5} i x}\, \sqrt {-\frac {2}{5} x +\frac {1}{5} i x +\frac {3}{5}+\frac {1}{5} i}\, \sqrt {-\frac {2}{5} x -\frac {1}{5} i x +\frac {3}{5}-\frac {1}{5} i}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha ^{3} \EllipticPi \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, i \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-i \underline {\hspace {1.25 ex}}\alpha +2 \underline {\hspace {1.25 ex}}\alpha +i-2, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )\right )}{\sqrt {x^{3}-x^{2}+2}}-\frac {2 i \sqrt {\frac {2}{5}+\frac {1}{5} i+\frac {2}{5} x +\frac {1}{5} i x}\, \sqrt {-\frac {2}{5} x +\frac {1}{5} i x +\frac {3}{5}+\frac {1}{5} i}\, \sqrt {-\frac {2}{5} x -\frac {1}{5} i x +\frac {3}{5}-\frac {1}{5} i}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha ^{3} \EllipticPi \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, i \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-i \underline {\hspace {1.25 ex}}\alpha +2 \underline {\hspace {1.25 ex}}\alpha +i-2, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )\right )}{\sqrt {x^{3}-x^{2}+2}}-\frac {4 \sqrt {\frac {2}{5}+\frac {1}{5} i+\frac {2}{5} x +\frac {1}{5} i x}\, \sqrt {-\frac {2}{5} x +\frac {1}{5} i x +\frac {3}{5}+\frac {1}{5} i}\, \sqrt {-\frac {2}{5} x -\frac {1}{5} i x +\frac {3}{5}-\frac {1}{5} i}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha ^{2} \EllipticPi \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, i \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-i \underline {\hspace {1.25 ex}}\alpha +2 \underline {\hspace {1.25 ex}}\alpha +i-2, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )\right )}{\sqrt {x^{3}-x^{2}+2}}+\frac {2 i \sqrt {\frac {2}{5}+\frac {1}{5} i+\frac {2}{5} x +\frac {1}{5} i x}\, \sqrt {-\frac {2}{5} x +\frac {1}{5} i x +\frac {3}{5}+\frac {1}{5} i}\, \sqrt {-\frac {2}{5} x -\frac {1}{5} i x +\frac {3}{5}-\frac {1}{5} i}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha ^{2} \EllipticPi \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, i \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-i \underline {\hspace {1.25 ex}}\alpha +2 \underline {\hspace {1.25 ex}}\alpha +i-2, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )\right )}{\sqrt {x^{3}-x^{2}+2}}+\frac {4 \sqrt {\frac {2}{5}+\frac {1}{5} i+\frac {2}{5} x +\frac {1}{5} i x}\, \sqrt {-\frac {2}{5} x +\frac {1}{5} i x +\frac {3}{5}+\frac {1}{5} i}\, \sqrt {-\frac {2}{5} x -\frac {1}{5} i x +\frac {3}{5}-\frac {1}{5} i}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \EllipticPi \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, i \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-i \underline {\hspace {1.25 ex}}\alpha +2 \underline {\hspace {1.25 ex}}\alpha +i-2, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )\right )}{\sqrt {x^{3}-x^{2}+2}}-\frac {2 i \sqrt {\frac {2}{5}+\frac {1}{5} i+\frac {2}{5} x +\frac {1}{5} i x}\, \sqrt {-\frac {2}{5} x +\frac {1}{5} i x +\frac {3}{5}+\frac {1}{5} i}\, \sqrt {-\frac {2}{5} x -\frac {1}{5} i x +\frac {3}{5}-\frac {1}{5} i}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \EllipticPi \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, i \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-i \underline {\hspace {1.25 ex}}\alpha +2 \underline {\hspace {1.25 ex}}\alpha +i-2, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )\right )}{\sqrt {x^{3}-x^{2}+2}}-\frac {4 \sqrt {\frac {2}{5}+\frac {1}{5} i+\frac {2}{5} x +\frac {1}{5} i x}\, \sqrt {-\frac {2}{5} x +\frac {1}{5} i x +\frac {3}{5}+\frac {1}{5} i}\, \sqrt {-\frac {2}{5} x -\frac {1}{5} i x +\frac {3}{5}-\frac {1}{5} i}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+\textit {\_Z}^{2}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha ^{3} \EllipticPi \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, \frac {1}{2} i \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2}, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )\right )}{\sqrt {x^{3}-x^{2}+2}}+\frac {2 i \sqrt {\frac {2}{5}+\frac {1}{5} i+\frac {2}{5} x +\frac {1}{5} i x}\, \sqrt {-\frac {2}{5} x +\frac {1}{5} i x +\frac {3}{5}+\frac {1}{5} i}\, \sqrt {-\frac {2}{5} x -\frac {1}{5} i x +\frac {3}{5}-\frac {1}{5} i}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+\textit {\_Z}^{2}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha ^{3} \EllipticPi \left (\sqrt {\left (\frac {2}{5}+\frac {i}{5}\right ) \left (1+x \right )}, \frac {1}{2} i \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2}, \frac {2 \sqrt {5}}{5}-\frac {i \sqrt {5}}{5}\right )\right )}{\sqrt {x^{3}-x^{2}+2}}\)	\(1019\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(_Z^2+2)*ln((RootOf(_Z^2+2)*x^3-3*RootOf(_Z^2+2)*x^2-4*x*(x^3-x^2+2)^(1/2)+2*RootOf(_Z^2+2))/(x^3+x^2+2)
)+RootOf(_Z^2+1)*ln(-(RootOf(_Z^2+1)*x^3-2*RootOf(_Z^2+1)*x^2+2*x*(x^3-x^2+2)^(1/2)+2*RootOf(_Z^2+1))/(x^3+2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.60, size = 235, normalized size = 4.80 \begin {gather*} \left (\sum _{_{\mathrm {X264}}\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 {X264}}+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 {X264}}}^5+6\,{_{\mathrm {X264}}}^3-2\,{_{\mathrm {X264}}}^2+12\right )\,\left (\frac {4}{25}+\frac {2}{25}{}\mathrm {i}\right )}{_{\mathrm {X264}}\,\left (_{\mathrm {X264}}+1\right )\,\sqrt {x^3-x^2+2}\,\left (6\,{_{\mathrm {X264}}}^4+5\,{_{\mathrm {X264}}}^3+12\,_{\mathrm {X264}}+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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)/(_X264 + 1), asin((5^(1/2)*(x*(2 - 1i) + (2 - 1i))^(1/2))/5), 3/5 + 4i/5)*(6*_X264^3 - 2*_X264
^2 + 2*_X264^5 + 12)*(4/25 + 2i/25))/(_X264*(_X264 + 1)*(x^3 - x^2 + 2)^(1/2)*(12*_X264 + 5*_X264^3 + 6*_X264^
4 + 4)), _X264 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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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