3.16.60 \(\int \frac {(-1-2 x+x^2) (-1+2 x+x^2)}{(1-x+2 x^2+x^3+x^4) \sqrt [3]{-x+x^5}} \, dx\)

Optimal. Leaf size=107 \[ \log \left (\sqrt [3]{x^5-x}+x^2+1\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^5-x}}{\sqrt [3]{x^5-x}-2 x^2-2}\right )-\frac {1}{2} \log \left (\left (x^5-x\right )^{2/3}+x^4+2 x^2+\left (-x^2-1\right ) \sqrt [3]{x^5-x}+1\right ) \]

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Rubi [F]  time = 2.10, 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 (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*x*(1 - x^4)^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, x^4])/(2*(-x + x^5)^(1/3)) + (3*x^(1/3)*(-1 + x^4)^(1/3)
*Defer[Subst][Defer[Int][x^4/((-1 + x^12)^(1/3)*(1 - x^3 + 2*x^6 + x^9 + x^12)), x], x, x^(1/3)])/(-x + x^5)^(
1/3) - (24*x^(1/3)*(-1 + x^4)^(1/3)*Defer[Subst][Defer[Int][x^7/((-1 + x^12)^(1/3)*(1 - x^3 + 2*x^6 + x^9 + x^
12)), x], x, x^(1/3)])/(-x + x^5)^(1/3) - (3*x^(1/3)*(-1 + x^4)^(1/3)*Defer[Subst][Defer[Int][x^10/((-1 + x^12
)^(1/3)*(1 - x^3 + 2*x^6 + x^9 + x^12)), x], x, x^(1/3)])/(-x + x^5)^(1/3)

Rubi steps

\begin {align*} \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^4} \left (1-x+2 x^2+x^3+x^4\right )} \, dx}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {1-6 x^2+x^4}{\sqrt [3]{x} \sqrt [3]{-1+x^4} \left (1-x+2 x^2+x^3+x^4\right )} \, dx}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x \left (1-6 x^6+x^{12}\right )}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {x}{\sqrt [3]{-1+x^{12}}}+\frac {x^4 \left (1-8 x^3-x^6\right )}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^{12}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (1-8 x^3-x^6\right )}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {x^4}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )}-\frac {8 x^7}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )}-\frac {x^{10}}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}-\frac {\left (24 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}\\ &=\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}-\frac {\left (24 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.18, size = 107, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-x+x^5}}{-2-2 x^2+\sqrt [3]{-x+x^5}}\right )+\log \left (1+x^2+\sqrt [3]{-x+x^5}\right )-\frac {1}{2} \log \left (1+2 x^2+x^4+\left (-1-x^2\right ) \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*(-x + x^5)^(1/3))/(-2 - 2*x^2 + (-x + x^5)^(1/3))] + Log[1 + x^2 + (-x + x^5)^(1/3)] -
Log[1 + 2*x^2 + x^4 + (-1 - x^2)*(-x + x^5)^(1/3) + (-x + x^5)^(2/3)]/2

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fricas [A]  time = 2.22, size = 154, normalized size = 1.44 \begin {gather*} -\sqrt {3} \arctan \left (\frac {541310 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + \sqrt {3} {\left (311575 \, x^{4} + 193471 \, x^{3} + 623150 \, x^{2} - 193471 \, x + 311575\right )} + 777518 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {2}{3}}}{3 \, {\left (166375 \, x^{4} - 493039 \, x^{3} + 332750 \, x^{2} + 493039 \, x + 166375\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} + x^{3} + 2 \, x^{2} + 3 \, {\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} - x + 3 \, {\left (x^{5} - x\right )}^{\frac {2}{3}} + 1}{x^{4} + x^{3} + 2 \, x^{2} - x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*arctan(1/3*(541310*sqrt(3)*(x^5 - x)^(1/3)*(x^2 + 1) + sqrt(3)*(311575*x^4 + 193471*x^3 + 623150*x^2
- 193471*x + 311575) + 777518*sqrt(3)*(x^5 - x)^(2/3))/(166375*x^4 - 493039*x^3 + 332750*x^2 + 493039*x + 1663
75)) + 1/2*log((x^4 + x^3 + 2*x^2 + 3*(x^5 - x)^(1/3)*(x^2 + 1) - x + 3*(x^5 - x)^(2/3) + 1)/(x^4 + x^3 + 2*x^
2 - x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 7.78, size = 517, normalized size = 4.83

method	result	size
trager	\(\ln \left (-\frac {130149507345815310 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}-542289613940897125 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-66362067078692233 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+260299014691630620 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-1421624873572223196 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x^{2}-1570891338664105235 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-1094963791801900343 x^{4}+542289613940897125 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -2253715023870923763 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}}-132724134157384466 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-2253715023870923763 \left (x^{5}-x \right )^{\frac {1}{3}} x^{2}-345778039516389582 x^{3}+130149507345815310 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-1421624873572223196 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}}+1570891338664105235 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -832090150298700567 \left (x^{5}-x \right )^{\frac {2}{3}}-2189927583603800686 x^{2}-66362067078692233 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2253715023870923763 \left (x^{5}-x \right )^{\frac {1}{3}}+345778039516389582 x -1094963791801900343}{x^{4}+x^{3}+2 x^{2}-x +1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {345778039516389582 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}-1440741831318289925 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+1570891338664105235 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+691556079032779164 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-832090150298700567 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x^{2}-1637253405742797468 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+542289613940897125 x^{4}+1440741831318289925 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -2253715023870923763 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}}+3141782677328210470 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-2253715023870923763 \left (x^{5}-x \right )^{\frac {1}{3}} x^{2}-412140106595081815 x^{3}+345778039516389582 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-832090150298700567 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}}+1637253405742797468 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -1421624873572223196 \left (x^{5}-x \right )^{\frac {2}{3}}+1084579227881794250 x^{2}+1570891338664105235 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2253715023870923763 \left (x^{5}-x \right )^{\frac {1}{3}}+412140106595081815 x +542289613940897125}{x^{4}+x^{3}+2 x^{2}-x +1}\right )\)	\(517\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(-(130149507345815310*RootOf(_Z^2+_Z+1)^2*x^4-542289613940897125*RootOf(_Z^2+_Z+1)^2*x^3-66362067078692233*R
ootOf(_Z^2+_Z+1)*x^4+260299014691630620*RootOf(_Z^2+_Z+1)^2*x^2-1421624873572223196*RootOf(_Z^2+_Z+1)*(x^5-x)^
(1/3)*x^2-1570891338664105235*RootOf(_Z^2+_Z+1)*x^3-1094963791801900343*x^4+542289613940897125*RootOf(_Z^2+_Z+
1)^2*x-2253715023870923763*RootOf(_Z^2+_Z+1)*(x^5-x)^(2/3)-132724134157384466*RootOf(_Z^2+_Z+1)*x^2-2253715023
870923763*(x^5-x)^(1/3)*x^2-345778039516389582*x^3+130149507345815310*RootOf(_Z^2+_Z+1)^2-1421624873572223196*
RootOf(_Z^2+_Z+1)*(x^5-x)^(1/3)+1570891338664105235*RootOf(_Z^2+_Z+1)*x-832090150298700567*(x^5-x)^(2/3)-21899
27583603800686*x^2-66362067078692233*RootOf(_Z^2+_Z+1)-2253715023870923763*(x^5-x)^(1/3)+345778039516389582*x-
1094963791801900343)/(x^4+x^3+2*x^2-x+1))+RootOf(_Z^2+_Z+1)*ln((345778039516389582*RootOf(_Z^2+_Z+1)^2*x^4-144
0741831318289925*RootOf(_Z^2+_Z+1)^2*x^3+1570891338664105235*RootOf(_Z^2+_Z+1)*x^4+691556079032779164*RootOf(_
Z^2+_Z+1)^2*x^2-832090150298700567*RootOf(_Z^2+_Z+1)*(x^5-x)^(1/3)*x^2-1637253405742797468*RootOf(_Z^2+_Z+1)*x
^3+542289613940897125*x^4+1440741831318289925*RootOf(_Z^2+_Z+1)^2*x-2253715023870923763*RootOf(_Z^2+_Z+1)*(x^5
-x)^(2/3)+3141782677328210470*RootOf(_Z^2+_Z+1)*x^2-2253715023870923763*(x^5-x)^(1/3)*x^2-412140106595081815*x
^3+345778039516389582*RootOf(_Z^2+_Z+1)^2-832090150298700567*RootOf(_Z^2+_Z+1)*(x^5-x)^(1/3)+16372534057427974
68*RootOf(_Z^2+_Z+1)*x-1421624873572223196*(x^5-x)^(2/3)+1084579227881794250*x^2+1570891338664105235*RootOf(_Z
^2+_Z+1)-2253715023870923763*(x^5-x)^(1/3)+412140106595081815*x+542289613940897125)/(x^4+x^3+2*x^2-x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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