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

Optimal. Leaf size=181 \[ \frac {\log \left (2 \sqrt [3]{x^2-3 x+2}+\sqrt [3]{2} x-2 \sqrt [3]{2}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2^{2/3} x^2+4 \left (x^2-3 x+2\right )^{2/3}+\left (4 \sqrt [3]{2}-2 \sqrt [3]{2} x\right ) \sqrt [3]{x^2-3 x+2}-4\ 2^{2/3} x+4\ 2^{2/3}\right )}{4 \sqrt [3]{2}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2-3 x+2}}{\sqrt [3]{x^2-3 x+2}-\sqrt [3]{2} x+2 \sqrt [3]{2}}\right )}{2 \sqrt [3]{2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 176, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {755, 123} \begin {gather*} \frac {3 \sqrt [3]{x-2} \sqrt [3]{x-1} \log \left (-\frac {(x-2)^{2/3}}{\sqrt [3]{2}}-\sqrt [3]{2} \sqrt [3]{x-1}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}}-\frac {\sqrt [3]{x-2} \sqrt [3]{x-1} \log (x)}{2 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}}-\frac {\sqrt {3} \sqrt [3]{x-2} \sqrt [3]{x-1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} (x-2)^{2/3}}{\sqrt {3} \sqrt [3]{x-1}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(2 - 3*x + x^2)^(1/3)),x]

[Out]

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

Rule 123

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[
(b*(b*e - a*f))/(b*c - a*d)^2, 3]}, -Simp[Log[a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[(Sqrt[3]*ArcTan[1/Sqrt[3
] + (2*q*(c + d*x)^(2/3))/(Sqrt[3]*(e + f*x)^(1/3))])/(2*q*(b*c - a*d)), x] + Simp[(3*Log[q*(c + d*x)^(2/3) -
(e + f*x)^(1/3)])/(4*q*(b*c - a*d)), x])] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]

Rule 755

Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[((b + q + 2*c*x)^(1/3)*(b - q + 2*c*x)^(1/3))/(a + b*x + c*x^2)^(1/3), Int[1/((d + e*x)*(b + q + 2*c
*x)^(1/3)*(b - q + 2*c*x)^(1/3)), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c^2*d^2 -
b*c*d*e - 2*b^2*e^2 + 9*a*c*e^2, 0]

Rubi steps

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

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Mathematica [C]  time = 0.03, size = 59, normalized size = 0.33 \begin {gather*} -\frac {3 \sqrt [3]{1-\frac {2}{x}} \sqrt [3]{1-\frac {1}{x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {1}{x},\frac {2}{x}\right )}{2 \sqrt [3]{x^2-3 x+2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(1 - 2/x)^(1/3)*(1 - x^(-1))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, x^(-1), 2/x])/(2*(2 - 3*x + x^2)^(1/3))

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IntegrateAlgebraic [A]  time = 0.23, size = 181, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{2-3 x+x^2}}{2 \sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{2-3 x+x^2}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2 \sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{2-3 x+x^2}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (4\ 2^{2/3}-4\ 2^{2/3} x+2^{2/3} x^2+\left (4 \sqrt [3]{2}-2 \sqrt [3]{2} x\right ) \sqrt [3]{2-3 x+x^2}+4 \left (2-3 x+x^2\right )^{2/3}\right )}{4 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.85, size = 277, normalized size = 1.53 \begin {gather*} -\frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{6} + 36 \, x^{5} - 612 \, x^{4} + 2880 \, x^{3} - 5760 \, x^{2} + 5184 \, x - 1728\right )} + 12 \, \sqrt {2} {\left (x^{5} - 38 \, x^{4} + 252 \, x^{3} - 648 \, x^{2} + 720 \, x - 288\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}} + 48 \cdot 2^{\frac {1}{6}} {\left (x^{4} - 6 \, x^{3} + 6 \, x^{2}\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (x^{6} - 108 \, x^{5} + 972 \, x^{4} - 3456 \, x^{3} + 6048 \, x^{2} - 5184 \, x + 1728\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 6 \cdot 2^{\frac {1}{3}} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}} {\left (x - 2\right )} + 12 \, {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {2}{3}} {\left (x^{2} - 6 \, x + 6\right )} + 2^{\frac {1}{3}} {\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )} - 6 \, {\left (x^{3} - 14 \, x^{2} + 36 \, x - 24\right )} {\left (x^{2} - 3 \, x + 2\right )}^{\frac {1}{3}}}{x^{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^6 + 36*x^5 - 612*x^4 + 2880*x^3 - 5760*x^2 + 5184
*x - 1728) + 12*sqrt(2)*(x^5 - 38*x^4 + 252*x^3 - 648*x^2 + 720*x - 288)*(x^2 - 3*x + 2)^(1/3) + 48*2^(1/6)*(x
^4 - 6*x^3 + 6*x^2)*(x^2 - 3*x + 2)^(2/3))/(x^6 - 108*x^5 + 972*x^4 - 3456*x^3 + 6048*x^2 - 5184*x + 1728)) +
1/12*2^(2/3)*log((2^(2/3)*x^2 + 6*2^(1/3)*(x^2 - 3*x + 2)^(1/3)*(x - 2) + 12*(x^2 - 3*x + 2)^(2/3))/x^2) - 1/2
4*2^(2/3)*log((12*2^(2/3)*(x^2 - 3*x + 2)^(2/3)*(x^2 - 6*x + 6) + 2^(1/3)*(x^4 - 36*x^3 + 180*x^2 - 288*x + 14
4) - 6*(x^3 - 14*x^2 + 36*x - 24)*(x^2 - 3*x + 2)^(1/3))/x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 8.57, size = 1069, normalized size = 5.91

method	result	size
trager	\(\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \ln \left (-\frac {-28 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{2}-68 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}+108 \left (x^{2}-3 x +2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+126 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +306 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x +54 \left (x^{2}-3 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x +237 \left (x^{2}-3 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x -126 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3}-306 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-108 \left (x^{2}-3 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-474 \left (x^{2}-3 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )-7 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-17 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{2}-258 \left (x^{2}-3 x +2\right )^{\frac {2}{3}}-168 \RootOf \left (\textit {\_Z}^{3}-4\right ) x -408 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x +168 \RootOf \left (\textit {\_Z}^{3}-4\right )+408 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )}{x^{2}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{3}-4\right ) \ln \left (\frac {-68 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{2}-112 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}+216 \left (x^{2}-3 x +2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+306 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +504 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x +108 \left (x^{2}-3 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -258 \left (x^{2}-3 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x -306 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3}-504 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-216 \left (x^{2}-3 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+516 \left (x^{2}-3 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )-119 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-196 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{2}+948 \left (x^{2}-3 x +2\right )^{\frac {2}{3}}+1020 \RootOf \left (\textit {\_Z}^{3}-4\right ) x +1680 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x -1020 \RootOf \left (\textit {\_Z}^{3}-4\right )-1680 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )}{x^{2}}\right )}{4}\)	\(1069\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*ln(-(-28*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4
*_Z^2)*RootOf(_Z^3-4)^3*x^2-68*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+108*
(x^2-3*x+2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2+126*RootOf(RootOf(_Z^3-
4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x+306*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*
RootOf(_Z^3-4)^2*x+54*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)^2*x+237*(x^2-3*x+2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*
RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x-126*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)
^3-306*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2-108*(x^2-3*x+2)^(1/3)*RootOf(_Z^
3-4)^2-474*(x^2-3*x+2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)-7*RootOf(_Z^3-
4)*x^2-17*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2-258*(x^2-3*x+2)^(2/3)-168*RootOf(_Z^3-4)*x-4
08*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x+168*RootOf(_Z^3-4)+408*RootOf(RootOf(_Z^3-4)^2+2*_Z*R
ootOf(_Z^3-4)+4*_Z^2))/x^2)+1/4*RootOf(_Z^3-4)*ln((-68*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*Roo
tOf(_Z^3-4)^3*x^2-112*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+216*(x^2-3*x+
2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2+306*RootOf(RootOf(_Z^3-4)^2+2*_Z
*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x+504*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z
^3-4)^2*x+108*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)^2*x-258*(x^2-3*x+2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_
Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x-306*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3-504*R
ootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2-216*(x^2-3*x+2)^(1/3)*RootOf(_Z^3-4)^2+5
16*(x^2-3*x+2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)-119*RootOf(_Z^3-4)*x^2
-196*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2+948*(x^2-3*x+2)^(2/3)+1020*RootOf(_Z^3-4)*x+1680*
RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x-1020*RootOf(_Z^3-4)-1680*RootOf(RootOf(_Z^3-4)^2+2*_Z*Ro
otOf(_Z^3-4)+4*_Z^2))/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^2 - 3*x + 2)^(1/3)),x)

[Out]

int(1/(x*(x^2 - 3*x + 2)^(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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