∫ (2+x3)√ _______ −1−x2+x3 __________________________________

3.12.52 \(\int \frac {(2+x^3) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx\)

Optimal. Leaf size=93 \[ \frac {1}{5} \left (-5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {x^3-x^2-1}}\right )+\frac {1}{5} \left (5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {2}{3+\sqrt {5}}} x}{\sqrt {x^3-x^2-1}}\right ) \]

________________________________________________________________________________________

Rubi [F] time = 0.67, 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 (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[((2 + x^3)*Sqrt[-1 - x^2 + x^3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6),x]

[Out]

2*Defer[Int][Sqrt[-1 - x^2 + x^3]/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6), x] + Defer[Int][(x^3*Sqrt[-1 - x^2 + x^

3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6), x]

Rubi steps

\begin {align*} \int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx &=\int \left (\frac {2 \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6}+\frac {x^3 \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6}\right ) \, dx\\ &=2 \int \frac {\sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx+\int \frac {x^3 \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] time = 5.18, size = 25746, normalized size = 276.84 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((2 + x^3)*Sqrt[-1 - x^2 + x^3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6),x]

[Out]

Result too large to show

________________________________________________________________________________________

IntegrateAlgebraic [A] time = 0.54, size = 81, normalized size = 0.87 \begin {gather*} \frac {1}{5} \left (5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\left (\sqrt {5}-1\right ) x}{2 \sqrt {x^3-x^2-1}}\right )+\frac {1}{5} \left (-5-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\left (1+\sqrt {5}\right ) x}{2 \sqrt {x^3-x^2-1}}\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

IntegrateAlgebraic[((2 + x^3)*Sqrt[-1 - x^2 + x^3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6),x]

[Out]

((5 - Sqrt[5])*ArcTan[((-1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^3])])/5 + ((-5 - Sqrt[5])*ArcTan[((1 + Sqrt[5])*

x)/(2*Sqrt[-1 - x^2 + x^3])])/5

________________________________________________________________________________________

fricas [A] time = 0.52, size = 79, normalized size = 0.85 \begin {gather*} -\frac {1}{5} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 6} \arctan \left (\frac {x \sqrt {2 \, \sqrt {5} + 6}}{2 \, \sqrt {x^{3} - x^{2} - 1}}\right ) + \frac {1}{5} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} + 6} \arctan \left (\frac {x \sqrt {-2 \, \sqrt {5} + 6}}{2 \, \sqrt {x^{3} - x^{2} - 1}}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x, algorithm="fricas")

[Out]

-1/5*sqrt(5)*sqrt(2*sqrt(5) + 6)*arctan(1/2*x*sqrt(2*sqrt(5) + 6)/sqrt(x^3 - x^2 - 1)) + 1/5*sqrt(5)*sqrt(-2*s

qrt(5) + 6)*arctan(1/2*x*sqrt(-2*sqrt(5) + 6)/sqrt(x^3 - x^2 - 1))

________________________________________________________________________________________

giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{3} - x^{2} - 1} {\left (x^{3} + 2\right )}}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x, algorithm="giac")

[Out]

integrate(sqrt(x^3 - x^2 - 1)*(x^3 + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1), x)

________________________________________________________________________________________

maple [C] time = 5.74, size = 2877, normalized size = 30.94 \begin {gather*} \text {output too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x)

[Out]

2/3*I*3^(1/2)*(1/6*(116+12*93^(1/2))^(1/3)-2/3/(116+12*93^(1/2))^(1/3))*(-I*(x+1/12*(116+12*93^(1/2))^(1/3)+1/

3/(116+12*93^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(1/6*(116+12*93^(1/2))^(1/3)-2/3/(116+12*93^(1/2))^(1/3)))*3^(1/2)

/(1/6*(116+12*93^(1/2))^(1/3)-2/3/(116+12*93^(1/2))^(1/3)))^(1/2)*((x-1/6*(116+12*93^(1/2))^(1/3)-2/3/(116+12*

93^(1/2))^(1/3)-1/3)/(-1/4*(116+12*93^(1/2))^(1/3)-1/(116+12*93^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/6*(116+12*93^(1/

2))^(1/3)-2/3/(116+12*93^(1/2))^(1/3))))^(1/2)*(I*(x+1/12*(116+12*93^(1/2))^(1/3)+1/3/(116+12*93^(1/2))^(1/3)-

1/3-1/2*I*3^(1/2)*(1/6*(116+12*93^(1/2))^(1/3)-2/3/(116+12*93^(1/2))^(1/3)))*3^(1/2)/(1/6*(116+12*93^(1/2))^(1

/3)-2/3/(116+12*93^(1/2))^(1/3)))^(1/2)/(x^3-x^2-1)^(1/2)*EllipticF(1/3*3^(1/2)*(-I*(x+1/12*(116+12*93^(1/2))^

(1/3)+1/3/(116+12*93^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(1/6*(116+12*93^(1/2))^(1/3)-2/3/(116+12*93^(1/2))^(1/3)))

*3^(1/2)/(1/6*(116+12*93^(1/2))^(1/3)-2/3/(116+12*93^(1/2))^(1/3)))^(1/2),(-I*3^(1/2)*(1/6*(116+12*93^(1/2))^(

1/3)-2/3/(116+12*93^(1/2))^(1/3))/(-1/4*(116+12*93^(1/2))^(1/3)-1/(116+12*93^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/6*(

116+12*93^(1/2))^(1/3)-2/3/(116+12*93^(1/2))^(1/3))))^(1/2))-1/360*I*2^(1/2)*4^(1/3)*sum(_alpha*(3*_alpha^4+3*

_alpha^3-3*_alpha^2-3*_alpha-4)*((29+3*93^(1/2))^(1/3)-4^(1/3)/(29+3*93^(1/2))^(1/3))*(-1/8*I*(-4+12*x+4^(1/3)

*((29+3*93^(1/2))^(1/3)+4^(1/3)/(29+3*93^(1/2))^(1/3)+I*3^(1/2)*((29+3*93^(1/2))^(1/3)-4^(1/3)/(29+3*93^(1/2))

^(1/3))))/((29+3*93^(1/2))^(1/3)-4^(1/3)/(29+3*93^(1/2))^(1/3)))^(1/2)*((-2+6*x+4^(1/3)*(-(29+3*93^(1/2))^(1/3

)-4^(1/3)/(29+3*93^(1/2))^(1/3)))/(-3*(29+3*93^(1/2))^(1/3)-3*4^(1/3)/(29+3*93^(1/2))^(1/3)-I*3^(1/2)*((29+3*9

3^(1/2))^(1/3)-4^(1/3)/(29+3*93^(1/2))^(1/3))))^(1/2)*(1/8*I*(-4+12*x+4^(1/3)*((29+3*93^(1/2))^(1/3)+4^(1/3)/(

29+3*93^(1/2))^(1/3)-I*3^(1/2)*((29+3*93^(1/2))^(1/3)-4^(1/3)/(29+3*93^(1/2))^(1/3))))/((29+3*93^(1/2))^(1/3)-

4^(1/3)/(29+3*93^(1/2))^(1/3)))^(1/2)/(x^3-x^2-1)^(1/2)*(32*_alpha^5+80*_alpha^4-16*_alpha^3-144*_alpha^2-48*_

alpha+80+4^(1/3)*(4*(3*3^(1/2)*31^(1/2)+29)^(1/3)+4*I*3^(1/2)*(3*3^(1/2)*31^(1/2)+29)^(1/3)+18*4^(1/3)*_alpha*

(3*3^(1/2)*31^(1/2)+29)^(2/3)-76*4^(1/3)*_alpha^3*(3*3^(1/2)*31^(1/2)+29)^(2/3)-87*4^(1/3)*_alpha^2*(3*3^(1/2)

*31^(1/2)+29)^(2/3)+29*4^(1/3)*_alpha^4*(3*3^(1/2)*31^(1/2)+29)^(2/3)+38*4^(1/3)*_alpha^5*(3*3^(1/2)*31^(1/2)+

29)^(2/3)+4*_alpha^4*(3*3^(1/2)*31^(1/2)+29)^(1/3)-14*_alpha^5*(3*3^(1/2)*31^(1/2)+29)^(1/3)+29*4^(1/3)*(3*3^(

1/2)*31^(1/2)+29)^(2/3)-36*_alpha*(3*3^(1/2)*31^(1/2)+29)^(1/3)+8*4^(1/3)*31^(1/2)*3^(1/2)*_alpha^3*(3*3^(1/2)

*31^(1/2)+29)^(2/3)-2*4^(1/3)*31^(1/2)*3^(1/2)*_alpha*(3*3^(1/2)*31^(1/2)+29)^(2/3)+9*4^(1/3)*31^(1/2)*3^(1/2)

*_alpha^2*(3*3^(1/2)*31^(1/2)+29)^(2/3)-4*4^(1/3)*31^(1/2)*3^(1/2)*_alpha^5*(3*3^(1/2)*31^(1/2)+29)^(2/3)-3*4^

(1/3)*31^(1/2)*3^(1/2)*_alpha^4*(3*3^(1/2)*31^(1/2)+29)^(2/3)+9*I*4^(1/3)*31^(1/2)*_alpha^4*(3*3^(1/2)*31^(1/2

)+29)^(2/3)-29*I*3^(1/2)*4^(1/3)*_alpha^4*(3*3^(1/2)*31^(1/2)+29)^(2/3)-24*I*4^(1/3)*31^(1/2)*_alpha^3*(3*3^(1

/2)*31^(1/2)+29)^(2/3)+76*I*3^(1/2)*4^(1/3)*_alpha^3*(3*3^(1/2)*31^(1/2)+29)^(2/3)-27*I*4^(1/3)*31^(1/2)*_alph

a^2*(3*3^(1/2)*31^(1/2)+29)^(2/3)+87*I*3^(1/2)*4^(1/3)*_alpha^2*(3*3^(1/2)*31^(1/2)+29)^(2/3)-18*I*3^(1/2)*4^(

1/3)*_alpha*(3*3^(1/2)*31^(1/2)+29)^(2/3)+6*I*4^(1/3)*31^(1/2)*_alpha*(3*3^(1/2)*31^(1/2)+29)^(2/3)+12*I*4^(1/

3)*31^(1/2)*_alpha^5*(3*3^(1/2)*31^(1/2)+29)^(2/3)-38*I*3^(1/2)*4^(1/3)*_alpha^5*(3*3^(1/2)*31^(1/2)+29)^(2/3)

-12*_alpha^2*(3*3^(1/2)*31^(1/2)+29)^(1/3)+28*_alpha^3*(3*3^(1/2)*31^(1/2)+29)^(1/3)-3*4^(1/3)*31^(1/2)*3^(1/2

)*(3*3^(1/2)*31^(1/2)+29)^(2/3)-4*3^(1/2)*31^(1/2)*_alpha^3*(3*3^(1/2)*31^(1/2)+29)^(1/3)+4*3^(1/2)*31^(1/2)*_

alpha*(3*3^(1/2)*31^(1/2)+29)^(1/3)-29*I*3^(1/2)*4^(1/3)*(3*3^(1/2)*31^(1/2)+29)^(2/3)+9*I*4^(1/3)*31^(1/2)*(3

*3^(1/2)*31^(1/2)+29)^(2/3)+2*3^(1/2)*31^(1/2)*_alpha^5*(3*3^(1/2)*31^(1/2)+29)^(1/3)+4*I*3^(1/2)*_alpha^4*(3*

3^(1/2)*31^(1/2)+29)^(1/3)-12*I*31^(1/2)*_alpha^3*(3*3^(1/2)*31^(1/2)+29)^(1/3)+28*I*3^(1/2)*_alpha^3*(3*3^(1/

2)*31^(1/2)+29)^(1/3)-12*I*3^(1/2)*_alpha^2*(3*3^(1/2)*31^(1/2)+29)^(1/3)+12*I*31^(1/2)*_alpha*(3*3^(1/2)*31^(

1/2)+29)^(1/3)-36*I*3^(1/2)*_alpha*(3*3^(1/2)*31^(1/2)+29)^(1/3)+6*I*31^(1/2)*_alpha^5*(3*3^(1/2)*31^(1/2)+29)

^(1/3)-14*I*3^(1/2)*_alpha^5*(3*3^(1/2)*31^(1/2)+29)^(1/3)))*EllipticPi(1/3*3^(1/2)*(-I*(x+1/12*(116+12*93^(1/

2))^(1/3)+1/3/(116+12*93^(1/2))^(1/3)-1/3+1/2*I*3^(1/2)*(1/6*(116+12*93^(1/2))^(1/3)-2/3/(116+12*93^(1/2))^(1/

3)))*3^(1/2)/(1/6*(116+12*93^(1/2))^(1/3)-2/3/(116+12*93^(1/2))^(1/3)))^(1/2),-1/3+3/4*_alpha*(116+12*3^(1/2)*

31^(1/2))^(1/3)-3/8*_alpha*(116+12*3^(1/2)*31^(1/2))^(2/3)-29/8*_alpha^2*(116+12*3^(1/2)*31^(1/2))^(1/3)-19/6*

_alpha^3*(116+12*3^(1/2)*31^(1/2))^(1/3)+7/24*_alpha^3*(116+12*3^(1/2)*31^(1/2))^(2/3)-1/8*_alpha^2*(116+12*3^

(1/2)*31^(1/2))^(2/3)+1/24*_alpha^4*(116+12*3^(1/2)*31^(1/2))^(2/3)+29/24*_alpha^4*(116+12*3^(1/2)*31^(1/2))^(

1/3)+19/12*_alpha^5*(116+12*3^(1/2)*31^(1/2))^(1/3)-7/48*_alpha^5*(116+12*3^(1/2)*31^(1/2))^(2/3)-1/8*3^(1/2)*

31^(1/2)*(116+12*3^(1/2)*31^(1/2))^(1/3)+1/3*I*_alpha*31^(1/2)+7/6*_alpha^5-1/3*_alpha^4-7/3*_alpha^3+_alpha^2

+3*_alpha-1/8*I*(116+12*3^(1/2)*31^(1/2))^(1/3)*31^(1/2)+1/8*I*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(1/3)+2*I*3^(

1/2)*(116+12*3^(1/2)*31^(1/2))^(2/3)-5/8*I*(116+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2)+29/24*(116+12*3^(1/2)*31^(

1/2))^(1/3)+1/24*(116+12*3^(1/2)*31^(1/2))^(2/3)+1/3*_alpha^3*(116+12*3^(1/2)*31^(1/2))^(1/3)*31^(1/2)*3^(1/2)

-1/24*_alpha^3*(116+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2)*3^(1/2)+1/24*_alpha*(116+12*3^(1/2)*31^(1/2))^(2/3)*31

^(1/2)*3^(1/2)+3/8*_alpha^2*(116+12*3^(1/2)*31^(1/2))^(1/3)*31^(1/2)*3^(1/2)-1/12*_alpha*(116+12*3^(1/2)*31^(1

/2))^(1/3)*31^(1/2)*3^(1/2)-1/3*I*_alpha^3*31^(1/2)+1/6*I*_alpha^5*31^(1/2)-1/6*_alpha^5*(116+12*3^(1/2)*31^(1

/2))^(1/3)*31^(1/2)*3^(1/2)+1/48*_alpha^5*(116+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2)*3^(1/2)-1/8*_alpha^4*(116+1

2*3^(1/2)*31^(1/2))^(1/3)*31^(1/2)*3^(1/2)+1/8*I*_alpha^4*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(1/3)-5/8*I*_alpha

^4*(116+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2)+2*I*_alpha^4*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(2/3)+1/3*I*_alpha^

3*(116+12*3^(1/2)*31^(1/2))^(1/3)*31^(1/2)-I*_alpha^3*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(1/3)+1/6*I*_alpha^3*(

116+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2)-1/2*I*_alpha^3*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(2/3)-17/24*I*_alpha^

2*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(1/3)+9/8*I*_alpha^2*(116+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2)-43/12*I*_alp

ha^2*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(2/3)-13/12*I*_alpha*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(2/3)+1/3*I*_alp

ha*(116+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2)-1/12*I*_alpha*(116+12*3^(1/2)*31^(1/2))^(1/3)*31^(1/2)+5/12*I*_alp

ha*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(1/3)-1/6*I*_alpha^5*(116+12*3^(1/2)*31^(1/2))^(1/3)*31^(1/2)+5/12*I*_alp

ha^5*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(1/3)-13/48*I*_alpha^5*(116+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2)+41/48*I

*_alpha^5*3^(1/2)*(116+12*3^(1/2)*31^(1/2))^(2/3)+3/8*I*(116+12*3^(1/2)*31^(1/2))^(1/3)*_alpha^2*31^(1/2)-1/8*

I*(116+12*3^(1/2)*31^(1/2))^(1/3)*_alpha^4*31^(1/2),(-I*3^(1/2)*(1/6*(116+12*93^(1/2))^(1/3)-2/3/(116+12*93^(1

/2))^(1/3))/(-1/4*(116+12*93^(1/2))^(1/3)-1/(116+12*93^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/6*(116+12*93^(1/2))^(1/3)

-2/3/(116+12*93^(1/2))^(1/3))))^(1/2)),_alpha=RootOf(_Z^6+_Z^5-_Z^4-2*_Z^3-_Z^2+1))

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{3} - x^{2} - 1} {\left (x^{3} + 2\right )}}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^3 - x^2 - 1)*(x^3 + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1), x)

________________________________________________________________________________________

mupad [B] time = 3.20, size = 2803, normalized size = 30.14

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^3 + 2)*(x^3 - x^2 - 1)^(1/2))/(x^2 + 2*x^3 + x^4 - x^5 - x^6 - 1),x)

[Out]

symsum(-(2*((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)

^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 -

1/3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3

)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)*

((1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - x + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)/(1/(6*(

(31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/

2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)*ellipticPi((1

/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((3

1^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)/(root(z^6 + z^5

- z^4 - 2*z^3 - z^2 + 1, z, k) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1

/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 +

29/54)^(1/3)/2 - 1/3), asin(((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(

1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108

+ 29/54)^(1/3)/2 - 1/3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))

/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)

^(1/3))/2))^(1/2)), (3^(1/2)*(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(

1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 2

9/54)^(1/3))/2)*1i)/(3*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1

/3))))*(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/

3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)*(-(3^(

1/2)*(x + (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))

*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)*1

i)/(3*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))))^(1/2)*(roo

t(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^2 - 3*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^3 - root(z^6 +

z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^4 + 2*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^5 + 3))/((x^3 - x^2 - x

*(((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2

- 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3)*((3^(1/2

)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((

31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3) - (1/(9*((31^(1/2)

*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*1

08^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/

108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29

/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/5

4)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))

+ ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^

(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1

/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1

/2))/108 + 29/54)^(1/3)/2 + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1

/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 +

29/54)^(1/3)/2 - 1/3))^(1/2)*(root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k) - (3^(1/2)*(1/(9*((31^(1/2)*108^(

1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108

+ 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)*(6*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1,

z, k)^2 + 4*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^3 - 5*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)

^4 - 6*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^5 + 2*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k))), k,

1, 6) + (2*((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54

)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2

- 1/3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/

3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)

*((1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - x + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)/(1/(6*

((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1

/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)*ellipticF(as

in(((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*

1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)/(1

/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((3

1^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)), (3^(1/

2)*(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))

- ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)*1i)/(3*(1/

(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))))*(1/(6*((31^(1/2)*10

8^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2)

)/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)*(-(3^(1/2)*(x + (3^(1/2)*(1/(9*(

(31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*

108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)*1i)/(3*(1/(9*((31^(1/2)*108

^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))))^(1/2))/(x^3 - x^2 - x*(((3^(1/2)*(1/

(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1

/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3)*((3^(1/2)*(1/(9*((31^(1

/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1

/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3) - (1/(9*((31^(1/2)*108^(1/2))/108

+ 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 +

29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1

/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + (

(31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((3

1^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*10

8^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2)

)/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/

108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/5

4)^(1/3)/2 + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/5

4)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2

- 1/3))^(1/2)

________________________________________________________________________________________

sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{3} + 2\right ) \sqrt {x^{3} - x^{2} - 1}}{x^{6} + x^{5} - x^{4} - 2 x^{3} - x^{2} + 1}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+2)*(x**3-x**2-1)**(1/2)/(x**6+x**5-x**4-2*x**3-x**2+1),x)

[Out]

Integral((x**3 + 2)*sqrt(x**3 - x**2 - 1)/(x**6 + x**5 - x**4 - 2*x**3 - x**2 + 1), x)

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]