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3.281 \(\int \left (\sqrt{9-4 \sqrt{2}} x-\sqrt{2} \sqrt{1+4 x+2 x^2+x^4}\right ) \, dx\)

Optimal. Leaf size=48 \[ \frac{1}{2} \sqrt{9-4 \sqrt{2}} x^2-\sqrt{2} \text{Int}\left (\sqrt{x^4+2 x^2+4 x+1},x\right ) \]

[Out]

(Sqrt[9 - 4*Sqrt[2]]*x^2)/2 - Sqrt[2]*(-Sqrt[1 + 4*x + 2*x^2 + x^4]/3 + ((1 + x)
*Sqrt[1 + 4*x + 2*x^2 + x^4])/3 + ((4*I)*(-13 + 3*Sqrt[33])^(1/3)*Sqrt[1 + 4*x +
 2*x^2 + x^4])/(4*2^(2/3)*(-I + Sqrt[3]) - (2*I)*(-13 + 3*Sqrt[33])^(1/3) + 2^(1
/3)*(I + Sqrt[3])*(-13 + 3*Sqrt[33])^(2/3) + (6*I)*(-13 + 3*Sqrt[33])^(1/3)*x) -
 (8*2^(2/3)*Sqrt[3/(-13 + 3*Sqrt[33] + 4*(-26 + 6*Sqrt[33])^(1/3))]*Sqrt[(I*(-19
899 + 3445*Sqrt[33] + (-26 + 6*Sqrt[33])^(2/3)*(-2574 + 466*Sqrt[33]) + (-26 + 6
*Sqrt[33])^(1/3)*(-19899 + 3445*Sqrt[33]) + (59697 - 10335*Sqrt[33])*x))/((-39 -
 (13*I)*Sqrt[3] + (9*I)*Sqrt[11] + 9*Sqrt[33] + (4*I)*(3*I + Sqrt[3])*(-26 + 6*S
qrt[33])^(1/3))*(26 - 6*Sqrt[33] + (-13 + (13*I)*Sqrt[3] - (9*I)*Sqrt[11] + 3*Sq
rt[33])*(-26 + 6*Sqrt[33])^(1/3) + (-4 - (4*I)*Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3)
 + 6*(-13 + 3*Sqrt[33])*x))]*Sqrt[1 + 4*x + 2*x^2 + x^4]*EllipticE[ArcSin[Sqrt[2
6 - 6*Sqrt[33] + (-13 - (13*I)*Sqrt[3] + (9*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*S
qrt[33])^(1/3) + (4*I)*(I + Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3) + 6*(-13 + 3*Sqrt[
33])*x]/(Sqrt[(39 + (13*I)*Sqrt[3] - (9*I)*Sqrt[11] - 9*Sqrt[33] + 4*(3 - I*Sqrt
[3])*(-26 + 6*Sqrt[33])^(1/3))/(39 - (13*I)*Sqrt[3] + (9*I)*Sqrt[11] - 9*Sqrt[33
] + 4*(3 + I*Sqrt[3])*(-26 + 6*Sqrt[33])^(1/3))]*Sqrt[26 - 6*Sqrt[33] + (-13 + (
13*I)*Sqrt[3] - (9*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3) + (-4 - (4
*I)*Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3) + 6*(-13 + 3*Sqrt[33])*x])], (4*(21 + (7*I
)*Sqrt[3] - (3*I)*Sqrt[11] - 3*Sqrt[33]) + (3 - I*Sqrt[3] - (3*I)*Sqrt[11] + 3*S
qrt[33])*(-26 + 6*Sqrt[33])^(1/3))/(4*(21 - (7*I)*Sqrt[3] + (3*I)*Sqrt[11] - 3*S
qrt[33]) + (3 + I*Sqrt[3] + (3*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3
))])/((4*2^(2/3) - (-13 + 3*Sqrt[33])^(1/3) - 2^(1/3)*(-13 + 3*Sqrt[33])^(2/3) +
 3*(-13 + 3*Sqrt[33])^(1/3)*x)*Sqrt[(I*(1 + x))/((104 - 24*Sqrt[33] + (-13 - (13
*I)*Sqrt[3] + (9*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3) + (4*I)*(I +
 Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3))*(26 - 6*Sqrt[33] + (-13 + (13*I)*Sqrt[3] - (
9*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3) + (-4 - (4*I)*Sqrt[3])*(-26
 + 6*Sqrt[33])^(2/3) + 6*(-13 + 3*Sqrt[33])*x))]*Sqrt[26 - 6*Sqrt[33] + (-13 + (
13*I)*Sqrt[3] - (9*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3) + (-4 - (4
*I)*Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3) + 6*(-13 + 3*Sqrt[33])*x]*Sqrt[26 - 6*Sqrt
[33] + (-13 - (13*I)*Sqrt[3] + (9*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(
1/3) + (4*I)*(I + Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3) + 6*(-13 + 3*Sqrt[33])*x]) +
 ((2^(1/3)*(13 - (13*I)*Sqrt[3] + (9*I)*Sqrt[11] - 3*Sqrt[33]) + 4*2^(2/3)*(1 +
I*Sqrt[3])*(-13 + 3*Sqrt[33])^(1/3) + 20*(-13 + 3*Sqrt[33])^(2/3))*(4*2^(2/3)*(I
 + Sqrt[3]) + (8*I)*(-13 + 3*Sqrt[33])^(1/3) + 2^(1/3)*(-I + Sqrt[3])*(-13 + 3*S
qrt[33])^(2/3))*Sqrt[(52 - 12*Sqrt[33] - 2^(1/3)*(-13 + 3*Sqrt[33])^(4/3) + 4*(-
26 + 6*Sqrt[33])^(2/3))/(-13 + 3*Sqrt[33] + 4*(-26 + 6*Sqrt[33])^(1/3))]*Sqrt[((
-8*I)*(-13 + 3*Sqrt[33]) + (-43*I - 13*Sqrt[3] + 9*Sqrt[11] + (5*I)*Sqrt[33])*(-
26 + 6*Sqrt[33])^(1/3) + (2*I + 4*Sqrt[3] - (2*I)*Sqrt[33])*(-26 + 6*Sqrt[33])^(
2/3) + ((8*I)*(-13 + 3*Sqrt[33]) + (13*I - 13*Sqrt[3] + 9*Sqrt[11] - (3*I)*Sqrt[
33])*(-26 + 6*Sqrt[33])^(1/3) + 4*(I + Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3))*x)/(1
+ x)]*Sqrt[1 + 4*x + 2*x^2 + x^4]*EllipticF[ArcSin[(Sqrt[52 - 12*Sqrt[33] - 2^(1
/3)*(-13 + 3*Sqrt[33])^(4/3) + 4*(-26 + 6*Sqrt[33])^(2/3)]*Sqrt[26 - 6*Sqrt[33]
+ (-13 - (13*I)*Sqrt[3] + (9*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3)
+ (4*I)*(I + Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3) + 6*(-13 + 3*Sqrt[33])*x])/(2^(1/
6)*Sqrt[3]*(-13 + 3*Sqrt[33])^(2/3)*Sqrt[39 + (13*I)*Sqrt[3] - (9*I)*Sqrt[11] -
9*Sqrt[33] + 4*(3 - I*Sqrt[3])*(-26 + 6*Sqrt[33])^(1/3)]*Sqrt[1 + x])], (4*(21*I
 - 7*Sqrt[3] + 3*Sqrt[11] - (3*I)*Sqrt[33]) + (3*I + Sqrt[3] + 3*Sqrt[11] + (3*I
)*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3))/(-56*Sqrt[3] + 24*Sqrt[11] + 2*(Sqrt[3] +
3*Sqrt[11])*(-26 + 6*Sqrt[33])^(1/3))])/(3*2^(2/3)*3^(3/4)*(-13 + 3*Sqrt[33])^(1
/3)*Sqrt[39 + (13*I)*Sqrt[3] - (9*I)*Sqrt[11] - 9*Sqrt[33] + 4*(3 - I*Sqrt[3])*(
-26 + 6*Sqrt[33])^(1/3)]*Sqrt[1 + x]*(4*2^(2/3)*(-I + Sqrt[3]) - (2*I)*(-13 + 3*
Sqrt[33])^(1/3) + 2^(1/3)*(I + Sqrt[3])*(-13 + 3*Sqrt[33])^(2/3) + (6*I)*(-13 +
3*Sqrt[33])^(1/3)*x)*Sqrt[26 - 6*Sqrt[33] + (-13 - (13*I)*Sqrt[3] + (9*I)*Sqrt[1
1] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3) + (4*I)*(I + Sqrt[3])*(-26 + 6*Sqrt[33
])^(2/3) + 6*(-13 + 3*Sqrt[33])*x]*Sqrt[(8*(-13 + 3*Sqrt[33]) - (5 - (3*I)*Sqrt[
3] + (3*I)*Sqrt[11] + Sqrt[33])*(-26 + 6*Sqrt[33])^(2/3) + (-26 + 6*Sqrt[33])^(1
/3)*(-41 + (15*I)*Sqrt[3] - (3*I)*Sqrt[11] + 7*Sqrt[33]) + (104 - 24*Sqrt[33] +
(-13 - (13*I)*Sqrt[3] + (9*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3) +
(4*I)*(I + Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3))*x)/((-39 - (13*I)*Sqrt[3] + (9*I)*
Sqrt[11] + 9*Sqrt[33] + (4*I)*(3*I + Sqrt[3])*(-26 + 6*Sqrt[33])^(1/3))*(1 + x))
]) + ((4*2^(2/3) + 2*(-13 + 3*Sqrt[33])^(1/3) - 2^(1/3)*(-13 + 3*Sqrt[33])^(2/3)
)*(4*2^(2/3)*(I + Sqrt[3]) - (4*I)*(-13 + 3*Sqrt[33])^(1/3) + 2^(1/3)*(-I + Sqrt
[3])*(-13 + 3*Sqrt[33])^(2/3))*(4*2^(2/3)*(-I + Sqrt[3]) + (4*I)*(-13 + 3*Sqrt[3
3])^(1/3) + 2^(1/3)*(I + Sqrt[3])*(-13 + 3*Sqrt[33])^(2/3))*Sqrt[(-39 + (13*I)*S
qrt[3] - (9*I)*Sqrt[11] + 9*Sqrt[33] - (4*I)*(-3*I + Sqrt[3])*(-26 + 6*Sqrt[33])
^(1/3))/(104 - 24*Sqrt[33] + (-13 + (13*I)*Sqrt[3] - (9*I)*Sqrt[11] + 3*Sqrt[33]
)*(-26 + 6*Sqrt[33])^(1/3) + (-4 - (4*I)*Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3))]*Sqr
t[1 + x]*Sqrt[(104 - 24*Sqrt[33] + 2*(1 + (14*I)*Sqrt[3] - (6*I)*Sqrt[11] + Sqrt
[33])*(-26 + 6*Sqrt[33])^(1/3) + (-7 - I*Sqrt[3] - (3*I)*Sqrt[11] + Sqrt[33])*(-
26 + 6*Sqrt[33])^(2/3) + 2*(-52 + 12*Sqrt[33] + 2^(1/3)*(-13 + 3*Sqrt[33])^(4/3)
 - 4*(-26 + 6*Sqrt[33])^(2/3))*x)/((-39 + (13*I)*Sqrt[3] - (9*I)*Sqrt[11] + 9*Sq
rt[33] - (4*I)*(-3*I + Sqrt[3])*(-26 + 6*Sqrt[33])^(1/3))*(1 + x))]*Sqrt[(104 -
24*Sqrt[33] + 2*(1 - (14*I)*Sqrt[3] + (6*I)*Sqrt[11] + Sqrt[33])*(-26 + 6*Sqrt[3
3])^(1/3) + (-7 + I*Sqrt[3] + (3*I)*Sqrt[11] + Sqrt[33])*(-26 + 6*Sqrt[33])^(2/3
) + 2*(-52 + 12*Sqrt[33] + 2^(1/3)*(-13 + 3*Sqrt[33])^(4/3) - 4*(-26 + 6*Sqrt[33
])^(2/3))*x)/((-39 - (13*I)*Sqrt[3] + (9*I)*Sqrt[11] + 9*Sqrt[33] + (4*I)*(3*I +
 Sqrt[3])*(-26 + 6*Sqrt[33])^(1/3))*(1 + x))]*Sqrt[1 + 4*x + 2*x^2 + x^4]*Ellipt
icPi[(2^(1/3)*(4*2^(1/3)*(-3*I + Sqrt[3]) + (3*I + Sqrt[3])*(-13 + 3*Sqrt[33])^(
2/3)))/(4*2^(2/3)*(-I + Sqrt[3]) - (8*I)*(-13 + 3*Sqrt[33])^(1/3) + 2^(1/3)*(I +
 Sqrt[3])*(-13 + 3*Sqrt[33])^(2/3)), ArcSin[Sqrt[13 - 3*Sqrt[33] - 2^(1/3)*(-13
+ 3*Sqrt[33])^(4/3) + 4*(-26 + 6*Sqrt[33])^(2/3) + (-39 + 9*Sqrt[33])*x]/(2^(1/6
)*Sqrt[3]*(-13 + 3*Sqrt[33])^(2/3)*Sqrt[(-39 + (13*I)*Sqrt[3] - (9*I)*Sqrt[11] +
 9*Sqrt[33] - (4*I)*(-3*I + Sqrt[3])*(-26 + 6*Sqrt[33])^(1/3))/(104 - 24*Sqrt[33
] + (-13 + (13*I)*Sqrt[3] - (9*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3
) + (-4 - (4*I)*Sqrt[3])*(-26 + 6*Sqrt[33])^(2/3))]*Sqrt[1 + x])], (4*(21 - (7*I
)*Sqrt[3] + (3*I)*Sqrt[11] - 3*Sqrt[33]) + (3 + I*Sqrt[3] + (3*I)*Sqrt[11] + 3*S
qrt[33])*(-26 + 6*Sqrt[33])^(1/3))/(4*(21 + (7*I)*Sqrt[3] - (3*I)*Sqrt[11] - 3*S
qrt[33]) + (3 - I*Sqrt[3] - (3*I)*Sqrt[11] + 3*Sqrt[33])*(-26 + 6*Sqrt[33])^(1/3
))])/(2^(1/6)*Sqrt[3]*(4*2^(2/3)*(I + Sqrt[3]) + (2*I)*(-13 + 3*Sqrt[33])^(1/3)
+ 2^(1/3)*(-I + Sqrt[3])*(-13 + 3*Sqrt[33])^(2/3) - (6*I)*(-13 + 3*Sqrt[33])^(1/
3)*x)*(4*2^(2/3)*(-I + Sqrt[3]) - (2*I)*(-13 + 3*Sqrt[33])^(1/3) + 2^(1/3)*(I +
Sqrt[3])*(-13 + 3*Sqrt[33])^(2/3) + (6*I)*(-13 + 3*Sqrt[33])^(1/3)*x)*Sqrt[13 -
3*Sqrt[33] - 2^(1/3)*(-13 + 3*Sqrt[33])^(4/3) + 4*(-26 + 6*Sqrt[33])^(2/3) + (-3
9 + 9*Sqrt[33])*x]))

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Rubi [A]  time = 0.0487657, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\sqrt{9-4 \sqrt{2}} x-\sqrt{2} \sqrt{1+4 x+2 x^2+x^4},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[Sqrt[9 - 4*Sqrt[2]]*x - Sqrt[2]*Sqrt[1 + 4*x + 2*x^2 + x^4],x]

[Out]

(Sqrt[9 - 4*Sqrt[2]]*x^2)/2 - Sqrt[2]*Defer[Int][Sqrt[1 + 4*x + 2*x^2 + x^4], x]

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Rubi in Sympy [A]  time = 0., size = 0, normalized size = 0. \[ - \left (- 2 \sqrt{2} + 1\right ) \int x\, dx - \sqrt{2} \int \sqrt{x^{4} + 2 x^{2} + 4 x + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(-2**(1/2)*(x**4+2*x**2+4*x+1)**(1/2)+x*(-1+2*2**(1/2)),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 6.07292, size = 3168, normalized size = 66. \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[9 - 4*Sqrt[2]]*x - Sqrt[2]*Sqrt[1 + 4*x + 2*x^2 + x^4],x]

[Out]

(Sqrt[9 - 4*Sqrt[2]]*x^2)/2 - (Sqrt[2]*x*Sqrt[1 + 4*x + 2*x^2 + x^4])/3 - (2*Sqr
t[2]*((6*(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])^2*(-(EllipticF[ArcSin[Sqrt[
-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3
& , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1
^2 + #1^3 & , 3, 0])))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#
1 - #1^2 + #1^3 & , 2, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((1 + Ro
ot[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Ro
ot[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0]) +
EllipticPi[(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])/(-Root[1 + 3*#1 - #1^2 +
#1^3 & , 1, 0] + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[-(((1 + x)*
(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))
/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 &
 , 3, 0])))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 +
#1^3 & , 2, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((1 + Root[1 + 3*#1
 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1
 - #1^2 + #1^3 & , 3, 0]))]*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0]))*Sqrt[(x
 - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])/((x - Root[1 + 3*#1 - #1^2 + #1^3 & ,
1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0]))]*(-1 - Root[1 + 3*#1 - #1^2
+ #1^3 & , 3, 0])*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])/((x - Root[1
+ 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*Sq
rt[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1
^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 -
 #1^2 + #1^3 & , 3, 0])))])/(Sqrt[1 + 4*x + 2*x^2 + x^4]*(Root[1 + 3*#1 - #1^2 +
 #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])) + (2*EllipticF[ArcSin[
Sqrt[((1 + x)*(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root[1 + 3*#1 - #1^2 + #
1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1
- #1^2 + #1^3 & , 3, 0]))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 +
3*#1 - #1^2 + #1^3 & , 2, 0])*(-1 - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((-1
 - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0]
 - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*(x - Root[1 + 3*#1 - #1^2 + #1^3 & ,
 1, 0])^2*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])/((x - Root[1 + 3*#1 -
 #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0]))]*(-1 - Root
[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 3,
0])/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^
3 & , 3, 0]))]*Sqrt[((1 + x)*(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root[1 +
3*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 +
 Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))])/(Sqrt[1 + 4*x + 2*x^2 + x^4]*(-Root[1
 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])) + ((1
+ x)*(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(x - Root[1 + 3*#1 - #1^2 + #1^
3 & , 3, 0]) + (x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])^2*(1 + Root[1 + 3*#1
- #1^2 + #1^3 & , 1, 0])*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])/((x -
Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0
]))]*Sqrt[(x - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])/((x - Root[1 + 3*#1 - #1^2
 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*Sqrt[-(((1 + x)
*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])
)/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3
& , 3, 0])))]*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])*((EllipticE[ArcSin[Sqr
t[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^
3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 -
#1^2 + #1^3 & , 3, 0])))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3
*#1 - #1^2 + #1^3 & , 2, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((1 +
Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] -
Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2,
 0]))/(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0]) - (EllipticPi[(1 + Root[1 + 3*
#1 - #1^2 + #1^3 & , 3, 0])/(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root[1 + 3
*#1 - #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[-(((1 + x)*(Root[1 + 3*#1 - #1^2 + #1^
3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((x - Root[1 + 3*#1 - #1^2
 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0])))]], ((Root[1 + 3
*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(1 + Root[1
 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 2, 0])*(
Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))]
*(1 - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 2,
 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/(-Root[1 + 3*#1 - #1^2 + #1^3 & ,
1, 0] + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]) + (EllipticF[ArcSin[Sqrt[-(((1 +
x)*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0
]))/((x - Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^
3 & , 3, 0])))]], ((Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3*#1 - #1^2
 + #1^3 & , 2, 0])*(1 + Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((1 + Root[1 + 3
*#1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 + 3
*#1 - #1^2 + #1^3 & , 3, 0]))]*(Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] + Root[1 +
 3*#1 - #1^2 + #1^3 & , 1, 0]*(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] - Root[1 +
 3*#1 - #1^2 + #1^3 & , 3, 0]) - Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))/((1 + R
oot[1 + 3*#1 - #1^2 + #1^3 & , 1, 0])*(-Root[1 + 3*#1 - #1^2 + #1^3 & , 1, 0] +
Root[1 + 3*#1 - #1^2 + #1^3 & , 3, 0]))))/Sqrt[1 + 4*x + 2*x^2 + x^4]))/3

_______________________________________________________________________________________

Maple [A]  time = 0.504, size = 4640, normalized size = 96.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(-2^(1/2)*(x^4+2*x^2+4*x+1)^(1/2)+x*(-1+2*2^(1/2)),x)

[Out]

1/2*x^2*(-1+2*2^(1/2))-2^(1/2)*(1/3*x*(x^4+2*x^2+4*x+1)^(1/2)+4/3*(-4/3-1/6*(26+
6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^
(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^
(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x
)/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(
26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/
3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1
/2))^(1/3)-1/3)^2*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*(x
-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26
+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(2
6+6*33^(1/2))^(1/3)+4/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(
1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)
*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*(x-1/6*(26+6*33^(1/
2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3
)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/
3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+
1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)/(1/2*(26+6*33^(1
/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3
/(26+6*33^(1/2))^(1/3)))/(-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4
/3)/((1+x)*(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3)*(x-1/6*(2
6+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(
1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*3
3^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))
^(1/3))))^(1/2)*EllipticF(((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/
2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(
26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^
(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*
33^(1/2))^(1/3)-1/3))^(1/2),((-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)
-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(-4/3-1/6
*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1
/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6
*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(
1/3)))/(-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(
26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2))+4*(-4/3-1/6*(26+6*33^(1
/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8
/3/(26+6*33^(1/2))^(1/3)))*((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1
/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*
(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33
^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6
*33^(1/2))^(1/3)-1/3))^(1/2)*(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1
/3)-1/3)^2*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*(x-1/6*(2
6+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(
1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^
(1/2))^(1/3)+4/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(
1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*((-1/3
*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*(x-1/6*(26+6*33^(1/2))^(1/
3)+4/3/(26+6*33^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(
26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-
1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26
+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)/(1/2*(26+6*33^(1/2))^(1
/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*
33^(1/2))^(1/3)))/(-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)/((1
+x)*(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3)*(x-1/6*(26+6*33^
(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(
1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2)
)^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))
))^(1/2)*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+1/3)*EllipticF((
(1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^
(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(2
6+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(
1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)
,((-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*
33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/
(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/
2))^(1/3)))/(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1
/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-1/2*(26+6*33^(1/2)
)^(1/3)+4/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(2
6+6*33^(1/2))^(1/3))))^(1/2))+(-4/3+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2)
)^(1/3))*EllipticPi(((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^
(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*3
3^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))
^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/
2))^(1/3)-1/3))^(1/2),(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1
/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/2*(26+6*
33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3
)-8/3/(26+6*33^(1/2))^(1/3))),((-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/
3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(-4/3-1
/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^
(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26
+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))
^(1/3)))/(-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3
*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2)))+2/3*((1+x)*(x-1/6*(2
6+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(
1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*3
3^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))
^(1/3)))+(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)
*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*((1/2*(26+6*33^(1/2))^(
1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6
*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/
3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(
26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))^(1/2)*(x+1/3*(26+6*33^(1/2)
)^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3)^2*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*
33^(1/2))^(1/3)+4/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-
1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6
*33^(1/2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2
))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(
1/2))^(1/3)-1/3))^(1/2)*((-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4
/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1
/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(1/6*(26+6*33^(1/2))^(1/3)-
4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+
6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3))
^(1/2)*((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(
26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))+(1/6*(26+6*33^(1/2))^(1/3)-4/3/(
26+6*33^(1/2))^(1/3)+1/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^
(1/2))^(1/3)))*(-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+1/3)+(-1/3*
(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+1/3)^2)/(1/2*(26+6*33^(1/2))^(1/
3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*3
3^(1/2))^(1/3)))/(-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)*Elli
pticF(((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(2
6+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(1/2))^(1/3)
-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26
+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)-1/3)
)^(1/2),((-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3
*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(-4/3-1/6*(26+6*33^(1/2))^(1/
3)+4/3/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6
*33^(1/2))^(1/3)))/(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/2
*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-1/2*(26+6*3
3^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)
-8/3/(26+6*33^(1/2))^(1/3))))^(1/2))+(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*3
3^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/
3)))*EllipticE(((1/2*(26+6*33^(1/2))^(1/3)-4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1/2)
*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(1+x)/(1/6*(26+6*33^(1/
2))^(1/3)-4/3/(26+6*33^(1/2))^(1/3)+4/3-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3
)-8/3/(26+6*33^(1/2))^(1/3)))/(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(
1/3)-1/3))^(1/2),((-1/2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)-1/2*I*3^(1
/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(-4/3-1/6*(26+6*33^(
1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-
8/3/(26+6*33^(1/2))^(1/3)))/(-4/3-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^
(1/3)-1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))/(-1/
2*(26+6*33^(1/2))^(1/3)+4/(26+6*33^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/
2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2))/(-1/3*(26+6*33^(1/2))^(1/3)+8/3/(2
6+6*33^(1/2))^(1/3)+4/3)))/((1+x)*(x+1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2
))^(1/3)-1/3)*(x-1/6*(26+6*33^(1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3-1/2*I*3
^(1/2)*(-1/3*(26+6*33^(1/2))^(1/3)-8/3/(26+6*33^(1/2))^(1/3)))*(x-1/6*(26+6*33^(
1/2))^(1/3)+4/3/(26+6*33^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(-1/3*(26+6*33^(1/2))^(1
/3)-8/3/(26+6*33^(1/2))^(1/3))))^(1/2))

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Maxima [A]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{2} \,{\left (\left (2 \, \sqrt{2}\right ) - 1\right )} x^{2} - \sqrt{2} \int \sqrt{x^{3} - x^{2} + 3 \, x + 1} \sqrt{x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(2*sqrt(2) - 1) - sqrt(2)*sqrt(x^4 + 2*x^2 + 4*x + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (2 \, \sqrt{2} x - \sqrt{2} \sqrt{x^{4} + 2 \, x^{2} + 4 \, x + 1} - x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(2*sqrt(2) - 1) - sqrt(2)*sqrt(x^4 + 2*x^2 + 4*x + 1),x, algorithm="fricas")

[Out]

integral(2*sqrt(2)*x - sqrt(2)*sqrt(x^4 + 2*x^2 + 4*x + 1) - x, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \int \left (x \left (-1 + 2 \sqrt{2}\right ) - \sqrt{2} \sqrt{x^{4} + 2 x^{2} + 4 x + 1}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-2**(1/2)*(x**4+2*x**2+4*x+1)**(1/2)+x*(-1+2*2**(1/2)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0., size = 0, normalized size = 0. \[ \int x{\left (2 \, \sqrt{2} - 1\right )} - \sqrt{2} \sqrt{x^{4} + 2 \, x^{2} + 4 \, x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(2*sqrt(2) - 1) - sqrt(2)*sqrt(x^4 + 2*x^2 + 4*x + 1),x, algorithm="giac")

[Out]

integrate(x*(2*sqrt(2) - 1) - sqrt(2)*sqrt(x^4 + 2*x^2 + 4*x + 1), x)

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