3.17.24 \(\int \frac {(-3+2 x+2 x^5) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx\)

Optimal. Leaf size=110 \[ \sqrt {\frac {1}{5} \left (2 \sqrt {5}-2\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {x^6-x^2+x}}{x^2}\right )-\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} \sqrt {x^6-x^2+x}}{x^2}\right ) \]

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Rubi [F]  time = 2.34, 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 (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-3 + 2*x + 2*x^5)*Sqrt[x - x^2 + x^6])/(1 - 2*x + x^2 - x^3 + x^4 + 2*x^5 - 3*x^6 - x^8 + x^10),x]

[Out]

(-6*Sqrt[x - x^2 + x^6]*Defer[Subst][Defer[Int][(x^2*Sqrt[1 - x^2 + x^10])/(1 - 2*x^2 + x^4 - x^6 + x^8 + 2*x^
10 - 3*x^12 - x^16 + x^20), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 - x + x^5]) + (4*Sqrt[x - x^2 + x^6]*Defer[Subst]
[Defer[Int][(x^4*Sqrt[1 - x^2 + x^10])/(1 - 2*x^2 + x^4 - x^6 + x^8 + 2*x^10 - 3*x^12 - x^16 + x^20), x], x, S
qrt[x]])/(Sqrt[x]*Sqrt[1 - x + x^5]) + (4*Sqrt[x - x^2 + x^6]*Defer[Subst][Defer[Int][(x^12*Sqrt[1 - x^2 + x^1
0])/(1 - 2*x^2 + x^4 - x^6 + x^8 + 2*x^10 - 3*x^12 - x^16 + x^20), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 - x + x^5]
)

Rubi steps

\begin {align*} \int \frac {\left (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx &=\frac {\sqrt {x-x^2+x^6} \int \frac {\sqrt {x} \sqrt {1-x+x^5} \left (-3+2 x+2 x^5\right )}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx}{\sqrt {x} \sqrt {1-x+x^5}}\\ &=\frac {\left (2 \sqrt {x-x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-x^2+x^{10}} \left (-3+2 x^2+2 x^{10}\right )}{1-2 x^2+x^4-x^6+x^8+2 x^{10}-3 x^{12}-x^{16}+x^{20}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x+x^5}}\\ &=\frac {\left (2 \sqrt {x-x^2+x^6}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x^2 \sqrt {1-x^2+x^{10}}}{1-2 x^2+x^4-x^6+x^8+2 x^{10}-3 x^{12}-x^{16}+x^{20}}+\frac {2 x^4 \sqrt {1-x^2+x^{10}}}{1-2 x^2+x^4-x^6+x^8+2 x^{10}-3 x^{12}-x^{16}+x^{20}}+\frac {2 x^{12} \sqrt {1-x^2+x^{10}}}{1-2 x^2+x^4-x^6+x^8+2 x^{10}-3 x^{12}-x^{16}+x^{20}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x+x^5}}\\ &=\frac {\left (4 \sqrt {x-x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1-x^2+x^{10}}}{1-2 x^2+x^4-x^6+x^8+2 x^{10}-3 x^{12}-x^{16}+x^{20}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x+x^5}}+\frac {\left (4 \sqrt {x-x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt {1-x^2+x^{10}}}{1-2 x^2+x^4-x^6+x^8+2 x^{10}-3 x^{12}-x^{16}+x^{20}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x+x^5}}-\frac {\left (6 \sqrt {x-x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-x^2+x^{10}}}{1-2 x^2+x^4-x^6+x^8+2 x^{10}-3 x^{12}-x^{16}+x^{20}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+2 x+2 x^5\right ) \sqrt {x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-3 + 2*x + 2*x^5)*Sqrt[x - x^2 + x^6])/(1 - 2*x + x^2 - x^3 + x^4 + 2*x^5 - 3*x^6 - x^8 + x^10),x]

[Out]

Integrate[((-3 + 2*x + 2*x^5)*Sqrt[x - x^2 + x^6])/(1 - 2*x + x^2 - x^3 + x^4 + 2*x^5 - 3*x^6 - x^8 + x^10), x
]

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IntegrateAlgebraic [A]  time = 9.46, size = 110, normalized size = 1.00 \begin {gather*} \sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {x-x^2+x^6}}{x^2}\right )-\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {x-x^2+x^6}}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-3 + 2*x + 2*x^5)*Sqrt[x - x^2 + x^6])/(1 - 2*x + x^2 - x^3 + x^4 + 2*x^5 - 3*x^6 - x^8 +
x^10),x]

[Out]

Sqrt[(-2 + 2*Sqrt[5])/5]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*Sqrt[x - x^2 + x^6])/x^2] - Sqrt[(2 + 2*Sqrt[5])/5]*Arc
Tanh[(Sqrt[-1/2 + Sqrt[5]/2]*Sqrt[x - x^2 + x^6])/x^2]

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fricas [B]  time = 0.96, size = 566, normalized size = 5.15 \begin {gather*} -\frac {1}{5} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \arctan \left (\frac {2 \, {\left (2 \, x^{6} + \sqrt {5} x^{4} + x^{4} - 2 \, x^{2} + 2 \, x\right )} \sqrt {x^{6} - x^{2} + x} \sqrt {2 \, \sqrt {5} - 2} + {\left (3 \, x^{10} + 5 \, x^{8} - 3 \, x^{6} + 6 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} + 3 \, x^{2} + \sqrt {5} {\left (x^{10} + 3 \, x^{8} - x^{6} + 2 \, x^{5} - 3 \, x^{4} + 3 \, x^{3} + x^{2} - 2 \, x + 1\right )} - 6 \, x + 3\right )} \sqrt {2 \, \sqrt {5} - 2} \sqrt {\sqrt {5} - 2}}{4 \, {\left (x^{10} + x^{8} - 3 \, x^{6} + 2 \, x^{5} - x^{4} + x^{3} + x^{2} - 2 \, x + 1\right )}}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{6} + x^{4} - 3 \, x^{2} + \sqrt {5} {\left (x^{6} + x^{4} - x^{2} + x\right )} + 3 \, x\right )} \sqrt {x^{6} - x^{2} + x} + {\left (x^{10} + 5 \, x^{8} - x^{6} + 2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} + x^{2} + \sqrt {5} {\left (x^{10} + x^{8} - x^{6} + 2 \, x^{5} - x^{4} + x^{3} + x^{2} - 2 \, x + 1\right )} - 2 \, x + 1\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{10} - x^{8} - 3 \, x^{6} + 2 \, x^{5} + x^{4} - x^{3} + x^{2} - 2 \, x + 1}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{6} + x^{4} - 3 \, x^{2} + \sqrt {5} {\left (x^{6} + x^{4} - x^{2} + x\right )} + 3 \, x\right )} \sqrt {x^{6} - x^{2} + x} - {\left (x^{10} + 5 \, x^{8} - x^{6} + 2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} + x^{2} + \sqrt {5} {\left (x^{10} + x^{8} - x^{6} + 2 \, x^{5} - x^{4} + x^{3} + x^{2} - 2 \, x + 1\right )} - 2 \, x + 1\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{10} - x^{8} - 3 \, x^{6} + 2 \, x^{5} + x^{4} - x^{3} + x^{2} - 2 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+2*x-3)*(x^6-x^2+x)^(1/2)/(x^10-x^8-3*x^6+2*x^5+x^4-x^3+x^2-2*x+1),x, algorithm="fricas")

[Out]

-1/5*sqrt(5)*sqrt(2*sqrt(5) - 2)*arctan(1/4*(2*(2*x^6 + sqrt(5)*x^4 + x^4 - 2*x^2 + 2*x)*sqrt(x^6 - x^2 + x)*s
qrt(2*sqrt(5) - 2) + (3*x^10 + 5*x^8 - 3*x^6 + 6*x^5 - 5*x^4 + 5*x^3 + 3*x^2 + sqrt(5)*(x^10 + 3*x^8 - x^6 + 2
*x^5 - 3*x^4 + 3*x^3 + x^2 - 2*x + 1) - 6*x + 3)*sqrt(2*sqrt(5) - 2)*sqrt(sqrt(5) - 2))/(x^10 + x^8 - 3*x^6 +
2*x^5 - x^4 + x^3 + x^2 - 2*x + 1)) - 1/20*sqrt(5)*sqrt(2*sqrt(5) + 2)*log(-(4*(3*x^6 + x^4 - 3*x^2 + sqrt(5)*
(x^6 + x^4 - x^2 + x) + 3*x)*sqrt(x^6 - x^2 + x) + (x^10 + 5*x^8 - x^6 + 2*x^5 - 5*x^4 + 5*x^3 + x^2 + sqrt(5)
*(x^10 + x^8 - x^6 + 2*x^5 - x^4 + x^3 + x^2 - 2*x + 1) - 2*x + 1)*sqrt(2*sqrt(5) + 2))/(x^10 - x^8 - 3*x^6 +
2*x^5 + x^4 - x^3 + x^2 - 2*x + 1)) + 1/20*sqrt(5)*sqrt(2*sqrt(5) + 2)*log(-(4*(3*x^6 + x^4 - 3*x^2 + sqrt(5)*
(x^6 + x^4 - x^2 + x) + 3*x)*sqrt(x^6 - x^2 + x) - (x^10 + 5*x^8 - x^6 + 2*x^5 - 5*x^4 + 5*x^3 + x^2 + sqrt(5)
*(x^10 + x^8 - x^6 + 2*x^5 - x^4 + x^3 + x^2 - 2*x + 1) - 2*x + 1)*sqrt(2*sqrt(5) + 2))/(x^10 - x^8 - 3*x^6 +
2*x^5 + x^4 - x^3 + x^2 - 2*x + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - x^{2} + x} {\left (2 \, x^{5} + 2 \, x - 3\right )}}{x^{10} - x^{8} - 3 \, x^{6} + 2 \, x^{5} + x^{4} - x^{3} + x^{2} - 2 \, x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+2*x-3)*(x^6-x^2+x)^(1/2)/(x^10-x^8-3*x^6+2*x^5+x^4-x^3+x^2-2*x+1),x, algorithm="giac")

[Out]

integrate(sqrt(x^6 - x^2 + x)*(2*x^5 + 2*x - 3)/(x^10 - x^8 - 3*x^6 + 2*x^5 + x^4 - x^3 + x^2 - 2*x + 1), x)

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maple [C]  time = 8.79, size = 943, normalized size = 8.57

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right ) \ln \left (-\frac {225 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{4} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right ) x^{5}-225 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{4} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right ) x^{3}+85 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right ) x^{5}-225 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{4} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right ) x -40 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right ) x^{3}+8 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right ) x^{5}+225 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{4} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right )+350 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2} \sqrt {x^{6}-x^{2}+x}\, x -85 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right ) x +85 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right )+50 \sqrt {x^{6}-x^{2}+x}\, x -8 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right ) x +8 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5\right )}{5 x^{5} \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5 x^{3} \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-x^{5}-5 x \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}+2 x^{3}+5 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}+x -1}\right )}{5}-\RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right ) \ln \left (-\frac {225 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{5} x^{5}-225 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{5} x^{3}-175 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{3} x^{5}-225 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{5} x +130 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{3} x^{3}+34 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right ) x^{5}+225 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{5}+175 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{3} x -70 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2} \sqrt {x^{6}-x^{2}+x}\, x -17 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right ) x^{3}-175 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{3}-34 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right ) x +24 \sqrt {x^{6}-x^{2}+x}\, x +34 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )}{5 x^{5} \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5 x^{3} \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-5 x \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}-x^{3}+5 \RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )^{2}}\right )\)	\(943\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^5+2*x-3)*(x^6-x^2+x)^(1/2)/(x^10-x^8-3*x^6+2*x^5+x^4-x^3+x^2-2*x+1),x,method=_RETURNVERBOSE)

[Out]

-1/5*RootOf(_Z^2+25*RootOf(25*_Z^4-5*_Z^2-1)^2-5)*ln(-(225*RootOf(25*_Z^4-5*_Z^2-1)^4*RootOf(_Z^2+25*RootOf(25
*_Z^4-5*_Z^2-1)^2-5)*x^5-225*RootOf(25*_Z^4-5*_Z^2-1)^4*RootOf(_Z^2+25*RootOf(25*_Z^4-5*_Z^2-1)^2-5)*x^3+85*Ro
otOf(25*_Z^4-5*_Z^2-1)^2*RootOf(_Z^2+25*RootOf(25*_Z^4-5*_Z^2-1)^2-5)*x^5-225*RootOf(25*_Z^4-5*_Z^2-1)^4*RootO
f(_Z^2+25*RootOf(25*_Z^4-5*_Z^2-1)^2-5)*x-40*RootOf(25*_Z^4-5*_Z^2-1)^2*RootOf(_Z^2+25*RootOf(25*_Z^4-5*_Z^2-1
)^2-5)*x^3+8*RootOf(_Z^2+25*RootOf(25*_Z^4-5*_Z^2-1)^2-5)*x^5+225*RootOf(25*_Z^4-5*_Z^2-1)^4*RootOf(_Z^2+25*Ro
otOf(25*_Z^4-5*_Z^2-1)^2-5)+350*RootOf(25*_Z^4-5*_Z^2-1)^2*(x^6-x^2+x)^(1/2)*x-85*RootOf(25*_Z^4-5*_Z^2-1)^2*R
ootOf(_Z^2+25*RootOf(25*_Z^4-5*_Z^2-1)^2-5)*x+85*RootOf(25*_Z^4-5*_Z^2-1)^2*RootOf(_Z^2+25*RootOf(25*_Z^4-5*_Z
^2-1)^2-5)+50*(x^6-x^2+x)^(1/2)*x-8*RootOf(_Z^2+25*RootOf(25*_Z^4-5*_Z^2-1)^2-5)*x+8*RootOf(_Z^2+25*RootOf(25*
_Z^4-5*_Z^2-1)^2-5))/(5*x^5*RootOf(25*_Z^4-5*_Z^2-1)^2-5*x^3*RootOf(25*_Z^4-5*_Z^2-1)^2-x^5-5*x*RootOf(25*_Z^4
-5*_Z^2-1)^2+2*x^3+5*RootOf(25*_Z^4-5*_Z^2-1)^2+x-1))-RootOf(25*_Z^4-5*_Z^2-1)*ln(-(225*RootOf(25*_Z^4-5*_Z^2-
1)^5*x^5-225*RootOf(25*_Z^4-5*_Z^2-1)^5*x^3-175*RootOf(25*_Z^4-5*_Z^2-1)^3*x^5-225*RootOf(25*_Z^4-5*_Z^2-1)^5*
x+130*RootOf(25*_Z^4-5*_Z^2-1)^3*x^3+34*RootOf(25*_Z^4-5*_Z^2-1)*x^5+225*RootOf(25*_Z^4-5*_Z^2-1)^5+175*RootOf
(25*_Z^4-5*_Z^2-1)^3*x-70*RootOf(25*_Z^4-5*_Z^2-1)^2*(x^6-x^2+x)^(1/2)*x-17*RootOf(25*_Z^4-5*_Z^2-1)*x^3-175*R
ootOf(25*_Z^4-5*_Z^2-1)^3-34*RootOf(25*_Z^4-5*_Z^2-1)*x+24*(x^6-x^2+x)^(1/2)*x+34*RootOf(25*_Z^4-5*_Z^2-1))/(5
*x^5*RootOf(25*_Z^4-5*_Z^2-1)^2-5*x^3*RootOf(25*_Z^4-5*_Z^2-1)^2-5*x*RootOf(25*_Z^4-5*_Z^2-1)^2-x^3+5*RootOf(2
5*_Z^4-5*_Z^2-1)^2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - x^{2} + x} {\left (2 \, x^{5} + 2 \, x - 3\right )}}{x^{10} - x^{8} - 3 \, x^{6} + 2 \, x^{5} + x^{4} - x^{3} + x^{2} - 2 \, x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+2*x-3)*(x^6-x^2+x)^(1/2)/(x^10-x^8-3*x^6+2*x^5+x^4-x^3+x^2-2*x+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^6 - x^2 + x)*(2*x^5 + 2*x - 3)/(x^10 - x^8 - 3*x^6 + 2*x^5 + x^4 - x^3 + x^2 - 2*x + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^5+2\,x-3\right )\,\sqrt {x^6-x^2+x}}{x^{10}-x^8-3\,x^6+2\,x^5+x^4-x^3+x^2-2\,x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x + 2*x^5 - 3)*(x - x^2 + x^6)^(1/2))/(x^2 - 2*x - x^3 + x^4 + 2*x^5 - 3*x^6 - x^8 + x^10 + 1),x)

[Out]

int(((2*x + 2*x^5 - 3)*(x - x^2 + x^6)^(1/2))/(x^2 - 2*x - x^3 + x^4 + 2*x^5 - 3*x^6 - x^8 + x^10 + 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x^{5} - x + 1\right )} \left (2 x^{5} + 2 x - 3\right )}{x^{10} - x^{8} - 3 x^{6} + 2 x^{5} + x^{4} - x^{3} + x^{2} - 2 x + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**5+2*x-3)*(x**6-x**2+x)**(1/2)/(x**10-x**8-3*x**6+2*x**5+x**4-x**3+x**2-2*x+1),x)

[Out]

Integral(sqrt(x*(x**5 - x + 1))*(2*x**5 + 2*x - 3)/(x**10 - x**8 - 3*x**6 + 2*x**5 + x**4 - x**3 + x**2 - 2*x
+ 1), x)

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