3.25.54 \(\int \frac {(-1+x^2) (1+x+x^2) \sqrt {1+3 x^2+x^4}}{(1+x+x^2+x^3+x^4)^2} \, dx\) [2454]

Optimal. Leaf size=199 \[ \frac {\left (-1-3 x-x^2\right ) \sqrt {1+3 x^2+x^4}}{5 \left (1+x+x^2+x^3+x^4\right )}+\frac {1}{5} \text {RootSum}\left [1-2 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\& ,\frac {3 \log (x)-3 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}-2 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-3 \log (x) \text {$\#$1}^2+3 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1-6 \text {$\#$1}+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 0.64, 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 (-1+x^2\right ) \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.61, size = 367, normalized size = 1.84 \begin {gather*} \frac {\left (-1-3 x-x^2\right ) \sqrt {1+3 x^2+x^4}}{5 \left (1+x+x^2+x^3+x^4\right )}+\text {RootSum}\left [-4-4 \text {$\#$1}+6 \text {$\#$1}^2+6 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-8 \log (x)+8 \log \left (1-x+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right )-2 \log (x) \text {$\#$1}+2 \log \left (1-x+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1-x+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2+6 \text {$\#$1}+9 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]-\frac {2}{5} \text {RootSum}\left [-4-4 \text {$\#$1}+6 \text {$\#$1}^2+6 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-21 \log (x)+21 \log \left (1-x+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right )-3 \log (x) \text {$\#$1}+3 \log \left (1-x+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1-x+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2+6 \text {$\#$1}+9 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 - 3*x - x^2)*Sqrt[1 + 3*x^2 + x^4])/(5*(1 + x + x^2 + x^3 + x^4)) + RootSum[-4 - 4*#1 + 6*#1^2 + 6*#1^3 +
#1^4 & , (-8*Log[x] + 8*Log[1 - x + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1] - 2*Log[x]*#1 + 2*Log[1 - x + x^2 + S
qrt[1 + 3*x^2 + x^4] - x*#1]*#1 - Log[x]*#1^2 + Log[1 - x + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1]*#1^2)/(-2 + 6*
#1 + 9*#1^2 + 2*#1^3) & ] - (2*RootSum[-4 - 4*#1 + 6*#1^2 + 6*#1^3 + #1^4 & , (-21*Log[x] + 21*Log[1 - x + x^2
+ Sqrt[1 + 3*x^2 + x^4] - x*#1] - 3*Log[x]*#1 + 3*Log[1 - x + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1]*#1 - Log[x]
*#1^2 + Log[1 - x + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1]*#1^2)/(-2 + 6*#1 + 9*#1^2 + 2*#1^3) & ])/5

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 2.25, size = 571, normalized size = 2.87 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5/(1/2*I*5^(1/2)-1/2*I)*(1-(1/2*5^(1/2)-3/2)*x^2)^(1/2)*(1-(-3/2-1/2*5^(1/2))*x^2)^(1/2)/(x^4+3*x^2+1)^(1/2)
*EllipticF(x*(1/2*I*5^(1/2)-1/2*I),3/2+1/2*5^(1/2))+1/10*sum((-6*_alpha^3-2*_alpha^2-3*_alpha-4)*(-1/(-_alpha^
3+2*_alpha^2-_alpha)^(1/2)*arctanh(1/22*(2*_alpha^2+3)*(2*_alpha^3+9*_alpha^2+11*x^2+6*_alpha+12)/(-_alpha^3+2
*_alpha^2-_alpha)^(1/2)/(x^4+3*x^2+1)^(1/2))-2^(1/2)*(-_alpha^3-_alpha^2-_alpha-1)/(5^(1/2)-3)^(1/2)*(3*x^2+2-
5^(1/2)*x^2)^(1/2)*(3*x^2+2+5^(1/2)*x^2)^(1/2)/(x^4+3*x^2+1)^(1/2)*EllipticPi((1/2*5^(1/2)-3/2)^(1/2)*x,-3/2*_
alpha^3-1/2*_alpha^3*5^(1/2),(-3/2-1/2*5^(1/2))^(1/2)/(1/2*5^(1/2)-3/2)^(1/2))),_alpha=RootOf(_Z^4+_Z^3+_Z^2+_
Z+1))+(-1/5-1/5*x^2-3/5*x)*(x^4+3*x^2+1)^(1/2)/(x^4+x^3+x^2+x+1)-1/10*sum((-6*_alpha^3-_alpha^2-4*_alpha-3)*(-
1/(-_alpha^3+2*_alpha^2-_alpha)^(1/2)*arctanh(1/22*(2*_alpha^2+3)*(2*_alpha^3+9*_alpha^2+11*x^2+6*_alpha+12)/(
-_alpha^3+2*_alpha^2-_alpha)^(1/2)/(x^4+3*x^2+1)^(1/2))-2^(1/2)*(-_alpha^3-_alpha^2-_alpha-1)/(5^(1/2)-3)^(1/2
)*(3*x^2+2-5^(1/2)*x^2)^(1/2)*(3*x^2+2+5^(1/2)*x^2)^(1/2)/(x^4+3*x^2+1)^(1/2)*EllipticPi((1/2*5^(1/2)-3/2)^(1/
2)*x,-3/2*_alpha^3-1/2*_alpha^3*5^(1/2),(-3/2-1/2*5^(1/2))^(1/2)/(1/2*5^(1/2)-3/2)^(1/2))),_alpha=RootOf(_Z^4+
_Z^3+_Z^2+_Z+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.56, size = 536, normalized size = 2.69 \begin {gather*} -\frac {\sqrt {5} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \sqrt {-8 \, \sqrt {5} + 20} \log \left (-\frac {10 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{2} + \sqrt {5} x + x + 2\right )} + {\left (5 \, x^{4} + 10 \, x^{3} + 20 \, x^{2} + \sqrt {5} {\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + 6 \, x + 1\right )} + 10 \, x + 5\right )} \sqrt {-8 \, \sqrt {5} + 20}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) - \sqrt {5} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \sqrt {-8 \, \sqrt {5} + 20} \log \left (-\frac {10 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{2} + \sqrt {5} x + x + 2\right )} - {\left (5 \, x^{4} + 10 \, x^{3} + 20 \, x^{2} + \sqrt {5} {\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + 6 \, x + 1\right )} + 10 \, x + 5\right )} \sqrt {-8 \, \sqrt {5} + 20}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) - 2 \, \sqrt {5} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \sqrt {2 \, \sqrt {5} + 5} \log \left (-\frac {2 \, {\left (5 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{2} - \sqrt {5} x + x + 2\right )} + {\left (5 \, x^{4} + 10 \, x^{3} + 20 \, x^{2} - \sqrt {5} {\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + 6 \, x + 1\right )} + 10 \, x + 5\right )} \sqrt {2 \, \sqrt {5} + 5}\right )}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) + 2 \, \sqrt {5} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \sqrt {2 \, \sqrt {5} + 5} \log \left (-\frac {2 \, {\left (5 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{2} - \sqrt {5} x + x + 2\right )} - {\left (5 \, x^{4} + 10 \, x^{3} + 20 \, x^{2} - \sqrt {5} {\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + 6 \, x + 1\right )} + 10 \, x + 5\right )} \sqrt {2 \, \sqrt {5} + 5}\right )}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) + 20 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + 3 \, x + 1\right )}}{100 \, {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/100*(sqrt(5)*(x^4 + x^3 + x^2 + x + 1)*sqrt(-8*sqrt(5) + 20)*log(-(10*sqrt(x^4 + 3*x^2 + 1)*(2*x^2 + sqrt(5
)*x + x + 2) + (5*x^4 + 10*x^3 + 20*x^2 + sqrt(5)*(x^4 + 6*x^3 + 6*x^2 + 6*x + 1) + 10*x + 5)*sqrt(-8*sqrt(5)
+ 20))/(x^4 + x^3 + x^2 + x + 1)) - sqrt(5)*(x^4 + x^3 + x^2 + x + 1)*sqrt(-8*sqrt(5) + 20)*log(-(10*sqrt(x^4
+ 3*x^2 + 1)*(2*x^2 + sqrt(5)*x + x + 2) - (5*x^4 + 10*x^3 + 20*x^2 + sqrt(5)*(x^4 + 6*x^3 + 6*x^2 + 6*x + 1)
+ 10*x + 5)*sqrt(-8*sqrt(5) + 20))/(x^4 + x^3 + x^2 + x + 1)) - 2*sqrt(5)*(x^4 + x^3 + x^2 + x + 1)*sqrt(2*sqr
t(5) + 5)*log(-2*(5*sqrt(x^4 + 3*x^2 + 1)*(2*x^2 - sqrt(5)*x + x + 2) + (5*x^4 + 10*x^3 + 20*x^2 - sqrt(5)*(x^
4 + 6*x^3 + 6*x^2 + 6*x + 1) + 10*x + 5)*sqrt(2*sqrt(5) + 5))/(x^4 + x^3 + x^2 + x + 1)) + 2*sqrt(5)*(x^4 + x^
3 + x^2 + x + 1)*sqrt(2*sqrt(5) + 5)*log(-2*(5*sqrt(x^4 + 3*x^2 + 1)*(2*x^2 - sqrt(5)*x + x + 2) - (5*x^4 + 10
*x^3 + 20*x^2 - sqrt(5)*(x^4 + 6*x^3 + 6*x^2 + 6*x + 1) + 10*x + 5)*sqrt(2*sqrt(5) + 5))/(x^4 + x^3 + x^2 + x
+ 1)) + 20*sqrt(x^4 + 3*x^2 + 1)*(x^2 + 3*x + 1))/(x^4 + x^3 + x^2 + x + 1)

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Sympy [F(-1)] Timed out
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**2-1)*(x**2+x+1)*(x**4+3*x**2+1)**(1/2)/(x**4+x**3+x**2+x+1)**2,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {x^4+3\,x^2+1}\,\left (x^2+x+1\right )}{{\left (x^4+x^3+x^2+x+1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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