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

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

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Rubi [F]  time = 0.96, 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 [C]  time = 147.28, size = 5248, normalized size = 26.37 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[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]

Result too large to show

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IntegrateAlgebraic [A]  time = 1.57, size = 313, normalized size = 1.57 \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 [1-2 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-7 \log (x)+7 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1-6 \text {$\#$1}+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]-\frac {2}{5} \text {RootSum}\left [1-2 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-19 \log (x)+19 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}+\log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1-6 \text {$\#$1}+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-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[1 - 2*#1 - 6*#1^2 + 2*#1^3 +
#1^4 & , (-7*Log[x] + 7*Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1] - Log[x]*#1^2 + Log[1 + x^2 + Sqrt[1 + 3*x
^2 + x^4] - x*#1]*#1^2)/(-1 - 6*#1 + 3*#1^2 + 2*#1^3) & ] - (2*RootSum[1 - 2*#1 - 6*#1^2 + 2*#1^3 + #1^4 & , (
-19*Log[x] + 19*Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1] - Log[x]*#1 + Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4]
- x*#1]*#1 - Log[x]*#1^2 + Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1]*#1^2)/(-1 - 6*#1 + 3*#1^2 + 2*#1^3) & ]
)/5

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fricas [B]  time = 0.70, 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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} {\left (x^{2} - 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]

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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maple [C]  time = 3.14, size = 335, normalized size = 1.68

method	result	size
risch	\(-\frac {\left (x^{2}+3 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{5 \left (x^{4}+x^{3}+x^{2}+x +1\right )}+\frac {3 \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ), \frac {3}{2}+\frac {\sqrt {5}}{2}\right )}{5 \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ) \sqrt {x^{4}+3 x^{2}+1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+3\right ) \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}+9 \underline {\hspace {1.25 ex}}\alpha ^{2}+11 x^{2}+6 \underline {\hspace {1.25 ex}}\alpha +12\right )}{22 \sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }}-\frac {\sqrt {2}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {3 x^{2}+2-\sqrt {5}\, x^{2}}\, \sqrt {3 x^{2}+2+\sqrt {5}\, x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}}{2}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\sqrt {5}-3}\, \sqrt {x^{4}+3 x^{2}+1}}\right )\right )}{10}\)	\(335\)
default	\(\frac {3 \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ), \frac {3}{2}+\frac {\sqrt {5}}{2}\right )}{5 \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ) \sqrt {x^{4}+3 x^{2}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\left (-6 \underline {\hspace {1.25 ex}}\alpha ^{3}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha -4\right ) \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+3\right ) \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}+9 \underline {\hspace {1.25 ex}}\alpha ^{2}+11 x^{2}+6 \underline {\hspace {1.25 ex}}\alpha +12\right )}{22 \sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }}-\frac {\sqrt {2}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {3 x^{2}+2-\sqrt {5}\, x^{2}}\, \sqrt {3 x^{2}+2+\sqrt {5}\, x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}}{2}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\sqrt {5}-3}\, \sqrt {x^{4}+3 x^{2}+1}}\right )\right )}{10}+\frac {\left (-\frac {1}{5}-\frac {1}{5} x^{2}-\frac {3}{5} x \right ) \sqrt {x^{4}+3 x^{2}+1}}{x^{4}+x^{3}+x^{2}+x +1}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\left (-6 \underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-4 \underline {\hspace {1.25 ex}}\alpha -3\right ) \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+3\right ) \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}+9 \underline {\hspace {1.25 ex}}\alpha ^{2}+11 x^{2}+6 \underline {\hspace {1.25 ex}}\alpha +12\right )}{22 \sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }}-\frac {\sqrt {2}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {3 x^{2}+2-\sqrt {5}\, x^{2}}\, \sqrt {3 x^{2}+2+\sqrt {5}\, x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}}{2}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\sqrt {5}-3}\, \sqrt {x^{4}+3 x^{2}+1}}\right )\right )}{10}\)	\(571\)
trager	\(-\frac {\left (x^{2}+3 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{5 \left (x^{4}+x^{3}+x^{2}+x +1\right )}-\frac {\RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {25 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{5} x -20 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x^{2}-70 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x -20 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3}+28 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x^{2}+49 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x +16 \sqrt {x^{4}+3 x^{2}+1}+28 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )}{4 x^{2}+5 x \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{2}-3 x +4}\right )}{5}+\frac {\ln \left (-\frac {25 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{5} x +20 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x^{2}-70 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x +20 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3}-44 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x^{2}+33 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x +16 \sqrt {x^{4}+3 x^{2}+1}-44 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )}{5 x \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{2}-4 x^{2}-7 x -4}\right ) \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3}}{2}-\frac {9 \ln \left (-\frac {25 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{5} x +20 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x^{2}-70 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x +20 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3}-44 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x^{2}+33 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x +16 \sqrt {x^{4}+3 x^{2}+1}-44 \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )}{5 x \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{2}-4 x^{2}-7 x -4}\right ) \RootOf \left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )}{10}\)	\(595\)
elliptic	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-2 \textit {\_Z}^{7}+3 \textit {\_Z}^{6}-14 \textit {\_Z}^{5}+50 \textit {\_Z}^{4}-124 \textit {\_Z}^{3}+198 \textit {\_Z}^{2}-172 \textit {\_Z} +61\right )}{\sum }\frac {\left (\textit {\_R}^{6}-8 \textit {\_R}^{5}+17 \textit {\_R}^{4}+32 \textit {\_R}^{3}-191 \textit {\_R}^{2}+286 \textit {\_R} -142\right ) \ln \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-7 \textit {\_R}^{6}+9 \textit {\_R}^{5}-35 \textit {\_R}^{4}+100 \textit {\_R}^{3}-186 \textit {\_R}^{2}+198 \textit {\_R} -86}\right )}{2}+\frac {\frac {\left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{7}}{5}-\frac {18 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{6}}{5}+\frac {46 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{5}}{5}-4 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{4}-12 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{3}+\frac {66 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{2}}{5}+\frac {12 \sqrt {x^{4}+3 x^{2}+1}}{5}-\frac {12 x^{2}}{5}-\frac {29}{5}}{\left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{8}-2 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{7}+3 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{6}-14 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{5}+50 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{4}-124 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{3}+198 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{2}-172 \sqrt {x^{4}+3 x^{2}+1}+172 x^{2}+61}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-2 \textit {\_Z}^{7}+3 \textit {\_Z}^{6}-14 \textit {\_Z}^{5}+50 \textit {\_Z}^{4}-124 \textit {\_Z}^{3}+198 \textit {\_Z}^{2}-172 \textit {\_Z} +61\right )}{\sum }\frac {\left (\textit {\_R}^{6}-34 \textit {\_R}^{5}+100 \textit {\_R}^{4}+80 \textit {\_R}^{3}-790 \textit {\_R}^{2}+1256 \textit {\_R} -634\right ) \ln \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-7 \textit {\_R}^{6}+9 \textit {\_R}^{5}-35 \textit {\_R}^{4}+100 \textit {\_R}^{3}-186 \textit {\_R}^{2}+198 \textit {\_R} -86}\right )}{10}+\frac {\left (\frac {-\frac {\left (x^{4}+3 x^{2}+1\right )^{\frac {3}{2}} \sqrt {2}}{10 x^{3}}+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{4 x}}{\frac {\left (x^{4}+3 x^{2}+1\right )^{2}}{4 x^{4}}-\frac {5 \left (x^{4}+3 x^{2}+1\right )}{4 x^{2}}+\frac {5}{4}}-\frac {\left (5+3 \sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x \sqrt {5+\sqrt {5}}}\right )}{25 \sqrt {5+\sqrt {5}}}-\frac {\left (3 \sqrt {5}-5\right ) \sqrt {5}\, \arctanh \left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x \sqrt {5-\sqrt {5}}}\right )}{25 \sqrt {5-\sqrt {5}}}\right ) \sqrt {2}}{2}\)	\(777\)

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]

-1/5*(x^2+3*x+1)/(x^4+x^3+x^2+x+1)*(x^4+3*x^2+1)^(1/2)+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))-1/1
0*sum((_alpha^2-_alpha+1)*(-1/(-_alpha^3+2*_alpha^2-_alpha)^(1/2)*arctanh(1/22*(2*_alpha^2+3)*(2*_alpha^3+9*_a
lpha^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)*Elli
pticPi((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*} \int \frac {\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} {\left (x^{2} - 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]

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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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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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**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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