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

Optimal. Leaf size=106 \[ \frac {\left ((x+1)^2\right )^{2/3} \left (\frac {\log \left (3^{2/3} \sqrt [3]{x+1}-3\right )}{3^{2/3}}-\frac {\log \left (\sqrt [3]{3} (x+1)^{2/3}+3^{2/3} \sqrt [3]{x+1}+3\right )}{2\ 3^{2/3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{3^{5/6}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [6]{3}}\right )}{(x+1)^{4/3}} \]

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Rubi [A]  time = 0.05, antiderivative size = 127, normalized size of antiderivative = 1.20, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {646, 57, 617, 204, 31} \begin {gather*} -\frac {(x+1)^{2/3} \log (2-x)}{2\ 3^{2/3} \sqrt [3]{x^2+2 x+1}}+\frac {\sqrt [3]{3} (x+1)^{2/3} \log \left (\sqrt [3]{3}-\sqrt [3]{x+1}\right )}{2 \sqrt [3]{x^2+2 x+1}}-\frac {(x+1)^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}+\sqrt [3]{3}}{3^{5/6}}\right )}{\sqrt [6]{3} \sqrt [3]{x^2+2 x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 + x)^(2/3)*ArcTan[(3^(1/3) + 2*(1 + x)^(1/3))/3^(5/6)])/(3^(1/6)*(1 + 2*x + x^2)^(1/3))) - ((1 + x)^(2/3
)*Log[2 - x])/(2*3^(2/3)*(1 + 2*x + x^2)^(1/3)) + (3^(1/3)*(1 + x)^(2/3)*Log[3^(1/3) - (1 + x)^(1/3)])/(2*(1 +
2*x + x^2)^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {1}{(-2+x) \sqrt [3]{1+2 x+x^2}} \, dx &=\frac {(1+x)^{2/3} \int \frac {1}{(-2+x) (1+x)^{2/3}} \, dx}{\sqrt [3]{1+2 x+x^2}}\\ &=-\frac {(1+x)^{2/3} \log (2-x)}{2\ 3^{2/3} \sqrt [3]{1+2 x+x^2}}-\frac {\left (\sqrt [3]{3} (1+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{3}-x} \, dx,x,\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1+2 x+x^2}}-\frac {\left (3^{2/3} (1+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{3^{2/3}+\sqrt [3]{3} x+x^2} \, dx,x,\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1+2 x+x^2}}\\ &=-\frac {(1+x)^{2/3} \log (2-x)}{2\ 3^{2/3} \sqrt [3]{1+2 x+x^2}}+\frac {\sqrt [3]{3} (1+x)^{2/3} \log \left (\sqrt [3]{3}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1+2 x+x^2}}+\frac {\left (\sqrt [3]{3} (1+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{1+x}}{\sqrt [3]{3}}\right )}{\sqrt [3]{1+2 x+x^2}}\\ &=-\frac {(1+x)^{2/3} \tan ^{-1}\left (\frac {1}{3} \left (\sqrt {3}+2 \sqrt [6]{3} \sqrt [3]{1+x}\right )\right )}{\sqrt [6]{3} \sqrt [3]{1+2 x+x^2}}-\frac {(1+x)^{2/3} \log (2-x)}{2\ 3^{2/3} \sqrt [3]{1+2 x+x^2}}+\frac {\sqrt [3]{3} (1+x)^{2/3} \log \left (\sqrt [3]{3}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1+2 x+x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 100, normalized size = 0.94 \begin {gather*} -\frac {(x+1)^{2/3} \left (-2 \log \left (\sqrt [3]{3}-\sqrt [3]{x+1}\right )+\log \left ((x+1)^{2/3}+\sqrt [3]{3} \sqrt [3]{x+1}+3^{2/3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}+\sqrt [3]{3}}{3^{5/6}}\right )\right )}{2\ 3^{2/3} \sqrt [3]{(x+1)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((1 + x)^(2/3)*(2*Sqrt[3]*ArcTan[(3^(1/3) + 2*(1 + x)^(1/3))/3^(5/6)] - 2*Log[3^(1/3) - (1 + x)^(1/3)] +
Log[3^(2/3) + 3^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)]))/(3^(2/3)*((1 + x)^2)^(1/3))

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IntegrateAlgebraic [A]  time = 0.22, size = 153, normalized size = 1.44 \begin {gather*} \frac {\tan ^{-1}\left (\frac {3^{5/6} \sqrt [3]{1+2 x+x^2}}{2+2 x+\sqrt [3]{3} \sqrt [3]{1+2 x+x^2}}\right )}{\sqrt [6]{3}}-\frac {\log (1+x)}{3\ 3^{2/3}}+\frac {\log \left (3^{2/3}+3^{2/3} x-3 \sqrt [3]{1+2 x+x^2}\right )}{3^{2/3}}-\frac {\log \left (3^{2/3}+3^{2/3} x+3 \sqrt [3]{1+2 x+x^2}+\sqrt [3]{3} \left (1+2 x+x^2\right )^{2/3}\right )}{2\ 3^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(3^(5/6)*(1 + 2*x + x^2)^(1/3))/(2 + 2*x + 3^(1/3)*(1 + 2*x + x^2)^(1/3))]/3^(1/6) - Log[1 + x]/(3*3^(2
/3)) + Log[3^(2/3) + 3^(2/3)*x - 3*(1 + 2*x + x^2)^(1/3)]/3^(2/3) - Log[3^(2/3) + 3^(2/3)*x + 3*(1 + 2*x + x^2
)^(1/3) + 3^(1/3)*(1 + 2*x + x^2)^(2/3)]/(2*3^(2/3))

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fricas [A]  time = 0.67, size = 139, normalized size = 1.31 \begin {gather*} \frac {1}{3} \cdot 9^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {9^{\frac {1}{6}} {\left (9^{\frac {1}{3}} \sqrt {3} {\left (x + 1\right )} + 6 \, \sqrt {3} {\left (x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}}\right )}}{9 \, {\left (x + 1\right )}}\right ) - \frac {1}{18} \cdot 9^{\frac {2}{3}} \log \left (\frac {9^{\frac {2}{3}} {\left (x^{2} + 2 \, x + 1\right )} + 3 \cdot 9^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 9 \, {\left (x^{2} + 2 \, x + 1\right )}^{\frac {2}{3}}}{x^{2} + 2 \, x + 1}\right ) + \frac {1}{9} \cdot 9^{\frac {2}{3}} \log \left (-\frac {9^{\frac {1}{3}} {\left (x + 1\right )} - 3 \, {\left (x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}}}{x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*9^(1/6)*sqrt(3)*arctan(1/9*9^(1/6)*(9^(1/3)*sqrt(3)*(x + 1) + 6*sqrt(3)*(x^2 + 2*x + 1)^(1/3))/(x + 1)) -
1/18*9^(2/3)*log((9^(2/3)*(x^2 + 2*x + 1) + 3*9^(1/3)*(x^2 + 2*x + 1)^(1/3)*(x + 1) + 9*(x^2 + 2*x + 1)^(2/3))
/(x^2 + 2*x + 1)) + 1/9*9^(2/3)*log(-(9^(1/3)*(x + 1) - 3*(x^2 + 2*x + 1)^(1/3))/(x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.14, size = 942, normalized size = 8.89

method	result	size
trager	\(\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \ln \left (-\frac {63 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-3\right )^{2} x^{2}+6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-3\right )^{3} x^{2}+63 \left (x^{2}+2 x +1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+63 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-3\right )^{2} x +6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-3\right )^{3} x +72 \left (x^{2}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-3\right ) x -21 \left (x^{2}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-3\right )^{2} x +72 \left (x^{2}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-3\right )-21 \left (x^{2}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-3\right )^{2}-21 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{3}-3\right ) x^{2}+135 \left (x^{2}+2 x +1\right )^{\frac {2}{3}}-294 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) x -28 \RootOf \left (\textit {\_Z}^{3}-3\right ) x -273 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right )-26 \RootOf \left (\textit {\_Z}^{3}-3\right )}{\left (-2+x \right ) \left (1+x \right )}\right )+\frac {\RootOf \left (\textit {\_Z}^{3}-3\right ) \ln \left (\frac {18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-3\right )^{2} x^{2}+21 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-3\right )^{3} x^{2}+63 \left (x^{2}+2 x +1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-3\right )^{2} x +21 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-3\right )^{3} x -135 \left (x^{2}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-3\right ) x -21 \left (x^{2}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-3\right )^{2} x -135 \left (x^{2}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-3\right )-21 \left (x^{2}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) x^{2}+28 \RootOf \left (\textit {\_Z}^{3}-3\right ) x^{2}-72 \left (x^{2}+2 x +1\right )^{\frac {2}{3}}+102 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right ) x +119 \RootOf \left (\textit {\_Z}^{3}-3\right ) x +78 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-3\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-3\right )+9 \textit {\_Z}^{2}\right )+91 \RootOf \left (\textit {\_Z}^{3}-3\right )}{\left (-2+x \right ) \left (1+x \right )}\right )}{3}\)	\(942\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*ln(-(63*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2
)^2*RootOf(_Z^3-3)^2*x^2+6*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*RootOf(_Z^3-3)^3*x^2+63*(x^2+2*
x+1)^(2/3)*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*RootOf(_Z^3-3)^2+63*RootOf(RootOf(_Z^3-3)^2+3*_
Z*RootOf(_Z^3-3)+9*_Z^2)^2*RootOf(_Z^3-3)^2*x+6*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*RootOf(_Z^
3-3)^3*x+72*(x^2+2*x+1)^(1/3)*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*RootOf(_Z^3-3)*x-21*(x^2+2*x
+1)^(1/3)*RootOf(_Z^3-3)^2*x+72*(x^2+2*x+1)^(1/3)*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*RootOf(_
Z^3-3)-21*(x^2+2*x+1)^(1/3)*RootOf(_Z^3-3)^2-21*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*x^2-2*Root
Of(_Z^3-3)*x^2+135*(x^2+2*x+1)^(2/3)-294*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*x-28*RootOf(_Z^3-
3)*x-273*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)-26*RootOf(_Z^3-3))/(-2+x)/(1+x))+1/3*RootOf(_Z^3-
3)*ln((18*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)^2*RootOf(_Z^3-3)^2*x^2+21*RootOf(RootOf(_Z^3-3)^
2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*RootOf(_Z^3-3)^3*x^2+63*(x^2+2*x+1)^(2/3)*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z
^3-3)+9*_Z^2)*RootOf(_Z^3-3)^2+18*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)^2*RootOf(_Z^3-3)^2*x+21*
RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*RootOf(_Z^3-3)^3*x-135*(x^2+2*x+1)^(1/3)*RootOf(RootOf(_Z^
3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*RootOf(_Z^3-3)*x-21*(x^2+2*x+1)^(1/3)*RootOf(_Z^3-3)^2*x-135*(x^2+2*x+1)^(1
/3)*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*RootOf(_Z^3-3)-21*(x^2+2*x+1)^(1/3)*RootOf(_Z^3-3)^2+2
4*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*x^2+28*RootOf(_Z^3-3)*x^2-72*(x^2+2*x+1)^(2/3)+102*RootO
f(RootOf(_Z^3-3)^2+3*_Z*RootOf(_Z^3-3)+9*_Z^2)*x+119*RootOf(_Z^3-3)*x+78*RootOf(RootOf(_Z^3-3)^2+3*_Z*RootOf(_
Z^3-3)+9*_Z^2)+91*RootOf(_Z^3-3))/(-2+x)/(1+x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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