∫ (∘ ______ 9 − 4√ _ 2 x −√ _ 2 √ _________

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

Optimal. Leaf size=4030 \[ \text {result too large to display} \]

[Out]

1/2*x^2*(-1+2*2^(1/2))-2^(1/2)*(-1/3*(x^4+2*x^2+4*x+1)^(1/2)+1/3*(1+x)*(x^4+2*x^2+4*x+1)^(1/2)+4*I*(-13+3*33^(

1/2))^(1/3)*(x^4+2*x^2+4*x+1)^(1/2)/(4*2^(2/3)*(-I+3^(1/2))-2*I*(-13+3*33^(1/2))^(1/3)+6*I*x*(-13+3*33^(1/2))^

(1/3)+2^(1/3)*(3^(1/2)+I)*(-13+3*33^(1/2))^(2/3))-8*2^(2/3)*EllipticE((26-6*33^(1/2)+6*x*(-13+3*33^(1/2))+(-13

-13*I*3^(1/2)+9*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+4*I*(3^(1/2)+I)*(-26+6*33^(1/2))^(2/3))^(1/2)/((

39+13*I*3^(1/2)-9*I*11^(1/2)-9*33^(1/2)+4*(3-I*3^(1/2))*(-26+6*33^(1/2))^(1/3))/(39-13*I*3^(1/2)+9*I*11^(1/2)-

9*33^(1/2)+4*(3+I*3^(1/2))*(-26+6*33^(1/2))^(1/3)))^(1/2)/(26-6*33^(1/2)+6*x*(-13+3*33^(1/2))+(-13+13*I*3^(1/2

)-9*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+(-4-4*I*3^(1/2))*(-26+6*33^(1/2))^(2/3))^(1/2),((84+28*I*3^(

1/2)-12*I*11^(1/2)-12*33^(1/2)+(3-I*3^(1/2)-3*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3))/(84-28*I*3^(1/2)+

12*I*11^(1/2)-12*33^(1/2)+(3+I*3^(1/2)+3*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)))^(1/2))*(x^4+2*x^2+4*x

+1)^(1/2)*3^(1/2)/(-13+3*33^(1/2)+4*(-26+6*33^(1/2))^(1/3))^(1/2)*(I*(-19899+x*(59697-10335*33^(1/2))+3445*33^

(1/2)+(-26+6*33^(1/2))^(2/3)*(-2574+466*33^(1/2))+(-26+6*33^(1/2))^(1/3)*(-19899+3445*33^(1/2)))/(-39-13*I*3^(

1/2)+9*I*11^(1/2)+9*33^(1/2)+4*I*(3*I+3^(1/2))*(-26+6*33^(1/2))^(1/3))/(26-6*33^(1/2)+6*x*(-13+3*33^(1/2))+(-1

3+13*I*3^(1/2)-9*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+(-4-4*I*3^(1/2))*(-26+6*33^(1/2))^(2/3)))^(1/2)

/(4*2^(2/3)-(-13+3*33^(1/2))^(1/3)+3*x*(-13+3*33^(1/2))^(1/3)-2^(1/3)*(-13+3*33^(1/2))^(2/3))/(26-6*33^(1/2)+6

*x*(-13+3*33^(1/2))+(-13+13*I*3^(1/2)-9*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+(-4-4*I*3^(1/2))*(-26+6*

33^(1/2))^(2/3))^(1/2)/(I*(1+x)/(26-6*33^(1/2)+6*x*(-13+3*33^(1/2))+(-13+13*I*3^(1/2)-9*I*11^(1/2)+3*33^(1/2))

*(-26+6*33^(1/2))^(1/3)+(-4-4*I*3^(1/2))*(-26+6*33^(1/2))^(2/3))/(104-24*33^(1/2)+(-13-13*I*3^(1/2)+9*I*11^(1/

2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+4*I*(3^(1/2)+I)*(-26+6*33^(1/2))^(2/3)))^(1/2)/(26-6*33^(1/2)+6*x*(-13+3

*33^(1/2))+(-13-13*I*3^(1/2)+9*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+4*I*(3^(1/2)+I)*(-26+6*33^(1/2))^

(2/3))^(1/2)+1/6*EllipticPi(1/6*(13-3*33^(1/2)-2^(1/3)*(-13+3*33^(1/2))^(4/3)+4*(-26+6*33^(1/2))^(2/3)+x*(-39+

9*33^(1/2)))^(1/2)*2^(5/6)*3^(1/2)/(-13+3*33^(1/2))^(2/3)/(1+x)^(1/2)/((-39+13*I*3^(1/2)-9*I*11^(1/2)+9*33^(1/

2)-4*I*(-3*I+3^(1/2))*(-26+6*33^(1/2))^(1/3))/(104-24*33^(1/2)+(-13+13*I*3^(1/2)-9*I*11^(1/2)+3*33^(1/2))*(-26

+6*33^(1/2))^(1/3)+(-4-4*I*3^(1/2))*(-26+6*33^(1/2))^(2/3)))^(1/2),2^(1/3)*(4*2^(1/3)*(-3*I+3^(1/2))+(3*I+3^(1

/2))*(-13+3*33^(1/2))^(2/3))/(4*2^(2/3)*(-I+3^(1/2))-8*I*(-13+3*33^(1/2))^(1/3)+2^(1/3)*(3^(1/2)+I)*(-13+3*33^

(1/2))^(2/3)),((84-28*I*3^(1/2)+12*I*11^(1/2)-12*33^(1/2)+(3+I*3^(1/2)+3*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2

))^(1/3))/(84+28*I*3^(1/2)-12*I*11^(1/2)-12*33^(1/2)+(3-I*3^(1/2)-3*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1

/3)))^(1/2))*(4*2^(2/3)+2*(-13+3*33^(1/2))^(1/3)-2^(1/3)*(-13+3*33^(1/2))^(2/3))*(4*2^(2/3)*(3^(1/2)+I)-4*I*(-

13+3*33^(1/2))^(1/3)+2^(1/3)*(-I+3^(1/2))*(-13+3*33^(1/2))^(2/3))*(4*2^(2/3)*(-I+3^(1/2))+4*I*(-13+3*33^(1/2))

^(1/3)+2^(1/3)*(3^(1/2)+I)*(-13+3*33^(1/2))^(2/3))*(1+x)^(1/2)*(x^4+2*x^2+4*x+1)^(1/2)*((-39+13*I*3^(1/2)-9*I*

11^(1/2)+9*33^(1/2)-4*I*(-3*I+3^(1/2))*(-26+6*33^(1/2))^(1/3))/(104-24*33^(1/2)+(-13+13*I*3^(1/2)-9*I*11^(1/2)

+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+(-4-4*I*3^(1/2))*(-26+6*33^(1/2))^(2/3)))^(1/2)*((104-24*33^(1/2)+2*(1+14*

I*3^(1/2)-6*I*11^(1/2)+33^(1/2))*(-26+6*33^(1/2))^(1/3)+(-7-I*3^(1/2)-3*I*11^(1/2)+33^(1/2))*(-26+6*33^(1/2))^

(2/3)+2*x*(-52+12*33^(1/2)+2^(1/3)*(-13+3*33^(1/2))^(4/3)-4*(-26+6*33^(1/2))^(2/3)))/(1+x)/(-39+13*I*3^(1/2)-9

*I*11^(1/2)+9*33^(1/2)-4*I*(-3*I+3^(1/2))*(-26+6*33^(1/2))^(1/3)))^(1/2)*((104-24*33^(1/2)+2*(1-14*I*3^(1/2)+6

*I*11^(1/2)+33^(1/2))*(-26+6*33^(1/2))^(1/3)+(-7+I*3^(1/2)+3*I*11^(1/2)+33^(1/2))*(-26+6*33^(1/2))^(2/3)+2*x*(

-52+12*33^(1/2)+2^(1/3)*(-13+3*33^(1/2))^(4/3)-4*(-26+6*33^(1/2))^(2/3)))/(1+x)/(-39-13*I*3^(1/2)+9*I*11^(1/2)

+9*33^(1/2)+4*I*(3*I+3^(1/2))*(-26+6*33^(1/2))^(1/3)))^(1/2)*2^(5/6)*3^(1/2)/(4*2^(2/3)*(3^(1/2)+I)+2*I*(-13+3

*33^(1/2))^(1/3)-6*I*x*(-13+3*33^(1/2))^(1/3)+2^(1/3)*(-I+3^(1/2))*(-13+3*33^(1/2))^(2/3))/(4*2^(2/3)*(-I+3^(1

/2))-2*I*(-13+3*33^(1/2))^(1/3)+6*I*x*(-13+3*33^(1/2))^(1/3)+2^(1/3)*(3^(1/2)+I)*(-13+3*33^(1/2))^(2/3))/(13-3

*33^(1/2)-2^(1/3)*(-13+3*33^(1/2))^(4/3)+4*(-26+6*33^(1/2))^(2/3)+x*(-39+9*33^(1/2)))^(1/2)+1/18*I*EllipticF(1

/6*I*(-52+12*33^(1/2)+2^(1/3)*(-13+3*33^(1/2))^(4/3)-4*(-26+6*33^(1/2))^(2/3))^(1/2)*(26-6*33^(1/2)+6*x*(-13+3

*33^(1/2))+(-13-13*I*3^(1/2)+9*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+4*I*(3^(1/2)+I)*(-26+6*33^(1/2))^

(2/3))^(1/2)*2^(5/6)*3^(1/2)/(-13+3*33^(1/2))^(2/3)/(1+x)^(1/2)/(39+13*I*3^(1/2)-9*I*11^(1/2)-9*33^(1/2)+4*(3-

I*3^(1/2))*(-26+6*33^(1/2))^(1/3))^(1/2),((84*I-28*3^(1/2)+12*11^(1/2)-12*I*33^(1/2)+(3*I+3^(1/2)+3*11^(1/2)+3

*I*33^(1/2))*(-26+6*33^(1/2))^(1/3))/(-56*3^(1/2)+24*11^(1/2)+2*(-26+6*33^(1/2))^(1/3)*(3^(1/2)+3*11^(1/2))))^

(1/2))*(2^(1/3)*(13-13*I*3^(1/2)+9*I*11^(1/2)-3*33^(1/2))+4*2^(2/3)*(1+I*3^(1/2))*(-13+3*33^(1/2))^(1/3)+20*(-

13+3*33^(1/2))^(2/3))*(4*2^(2/3)*(3^(1/2)+I)+8*I*(-13+3*33^(1/2))^(1/3)+2^(1/3)*(-I+3^(1/2))*(-13+3*33^(1/2))^

(2/3))*(x^4+2*x^2+4*x+1)^(1/2)*(1/(-13+3*33^(1/2)+4*(-26+6*33^(1/2))^(1/3))*(-52+12*33^(1/2)+2^(1/3)*(-13+3*33

^(1/2))^(4/3)-4*(-26+6*33^(1/2))^(2/3)))^(1/2)*((-8*I*(-13+3*33^(1/2))+(-43*I-13*3^(1/2)+9*11^(1/2)+5*I*33^(1/

2))*(-26+6*33^(1/2))^(1/3)+(2*I+4*3^(1/2)-2*I*33^(1/2))*(-26+6*33^(1/2))^(2/3)+x*(8*I*(-13+3*33^(1/2))+(13*I-1

3*3^(1/2)+9*11^(1/2)-3*I*33^(1/2))*(-26+6*33^(1/2))^(1/3)+4*(3^(1/2)+I)*(-26+6*33^(1/2))^(2/3)))/(1+x))^(1/2)*

2^(1/3)*3^(1/4)/(-13+3*33^(1/2))^(1/3)/(4*2^(2/3)*(-I+3^(1/2))-2*I*(-13+3*33^(1/2))^(1/3)+6*I*x*(-13+3*33^(1/2

))^(1/3)+2^(1/3)*(3^(1/2)+I)*(-13+3*33^(1/2))^(2/3))/(1+x)^(1/2)/(39+13*I*3^(1/2)-9*I*11^(1/2)-9*33^(1/2)+4*(3

-I*3^(1/2))*(-26+6*33^(1/2))^(1/3))^(1/2)/(26-6*33^(1/2)+6*x*(-13+3*33^(1/2))+(-13-13*I*3^(1/2)+9*I*11^(1/2)+3

*33^(1/2))*(-26+6*33^(1/2))^(1/3)+4*I*(3^(1/2)+I)*(-26+6*33^(1/2))^(2/3))^(1/2)/((-104+24*33^(1/2)-(5-3*I*3^(1

/2)+3*I*11^(1/2)+33^(1/2))*(-26+6*33^(1/2))^(2/3)+(-26+6*33^(1/2))^(1/3)*(-41+15*I*3^(1/2)-3*I*11^(1/2)+7*33^(

1/2))+x*(104-24*33^(1/2)+(-13-13*I*3^(1/2)+9*I*11^(1/2)+3*33^(1/2))*(-26+6*33^(1/2))^(1/3)+4*I*(3^(1/2)+I)*(-2

6+6*33^(1/2))^(2/3)))/(1+x)/(-39-13*I*3^(1/2)+9*I*11^(1/2)+9*33^(1/2)+4*I*(3*I+3^(1/2))*(-26+6*33^(1/2))^(1/3)

))^(1/2))

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Rubi [F] time = 0.03, 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 = {} \[ \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx \]

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]

Rubi steps

\begin {align*} \int \left (\sqrt {9-4 \sqrt {2}} x-\sqrt {2} \sqrt {1+4 x+2 x^2+x^4}\right ) \, dx &=\frac {1}{2} \sqrt {9-4 \sqrt {2}} x^2-\sqrt {2} \int \sqrt {1+4 x+2 x^2+x^4} \, dx\\ \end {align*}

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Mathematica [C] time = 6.04, size = 3168, normalized size = 0.79 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[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*Sqrt[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 + 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]) + 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]))]*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])) + (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[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 & , 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 + 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]))))/Sqrt[1 + 4*x + 2*x^2 + x^4]))/3

________________________________________________________________________________________

fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (2 \, \sqrt {2} x - \sqrt {2} \sqrt {x^{4} + 2 \, x^{2} + 4 \, x + 1} - x, x\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\left (2 \, \sqrt {2} - 1\right )} - \sqrt {2} \sqrt {x^{4} + 2 \, x^{2} + 4 \, x + 1}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A] time = 1.16, size = 4640, normalized size = 1.15 \[ \text {output too large to display} \]

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)))*(x+1)/(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)))/(-1/3*(26+6*33^(1/2))^(1/3)+8/3/(26+6*33^(1/2))^(1/3)+4/3)/((x+1)*(x+1/3*(2

6+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*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)))*(x+1)/(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)))*(x+1)/(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*3

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

6+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))+

(-4/3+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)))*(x+1)/(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),(1/6*(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/(2

6+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*((x+1)*(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)))+(-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)))*(x+1)/(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*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)*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)))*(x+1)/(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*(2

6+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))+(-4/3-1/6*(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)))*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)))*(x+1)/(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/(26+6*33^(1/2))^(1/3)+4/3)))/((x+1)*(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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} {\left (2 \, \sqrt {2} - 1\right )} - \sqrt {2} \int \sqrt {x^{3} - x^{2} + 3 \, x + 1} \sqrt {x + 1}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,\left (2\,\sqrt {2}-1\right )-\sqrt {2}\,\sqrt {x^4+2\,x^2+4\,x+1} \,d x \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (x \left (-1 + 2 \sqrt {2}\right ) - \sqrt {2} \sqrt {x^{4} + 2 x^{2} + 4 x + 1}\right )\, dx \]

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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