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

Optimal. Leaf size=370 \[ \frac {\left (\frac {2 x^2-x+1}{3 x^2-x+1}\right )^{2/3} \left (-3 x^2+x-1\right )}{x^2}+\frac {7}{3} \log \left (\sqrt [3]{\frac {2 x^2-x+1}{3 x^2-x+1}}-1\right )-\frac {2\ 2^{2/3} \log \left (6^{2/3} \sqrt [3]{\frac {2 x^2-x+1}{3 x^2-x+1}}-3\right )}{\sqrt [3]{3}}-\frac {7}{6} \log \left (\left (\frac {2 x^2-x+1}{3 x^2-x+1}\right )^{2/3}+\sqrt [3]{\frac {2 x^2-x+1}{3 x^2-x+1}}+1\right )+\frac {2^{2/3} \log \left (2 \sqrt [3]{6} \left (\frac {2 x^2-x+1}{3 x^2-x+1}\right )^{2/3}+6^{2/3} \sqrt [3]{\frac {2 x^2-x+1}{3 x^2-x+1}}+3\right )}{\sqrt [3]{3}}-2\ 2^{2/3} \sqrt [6]{3} \tan ^{-1}\left (\frac {2\ 2^{2/3} \sqrt [3]{\frac {2 x^2-x+1}{3 x^2-x+1}}}{3^{5/6}}+\frac {1}{\sqrt {3}}\right )+\frac {7 \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {2 x^2-x+1}{3 x^2-x+1}}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]

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Rubi [F]  time = 4.93, 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 {(-2+x) \left (1-x+x^2\right )}{x^3 \left (-1+x+x^2\right ) \sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {(-2+x) \left (1-x+x^2\right )}{x^3 \left (-1+x+x^2\right ) \sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}} \, dx &=\frac {\sqrt [3]{1-x+2 x^2} \int \frac {(-2+x) \left (1-x+x^2\right ) \sqrt [3]{1-x+3 x^2}}{x^3 \left (-1+x+x^2\right ) \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}\\ &=\frac {\sqrt [3]{1-x+2 x^2} \int \left (\frac {2 \sqrt [3]{1-x+3 x^2}}{x^3 \sqrt [3]{1-x+2 x^2}}-\frac {\sqrt [3]{1-x+3 x^2}}{x^2 \sqrt [3]{1-x+2 x^2}}+\frac {4 \sqrt [3]{1-x+3 x^2}}{x \sqrt [3]{1-x+2 x^2}}-\frac {2 (1+2 x) \sqrt [3]{1-x+3 x^2}}{\left (-1+x+x^2\right ) \sqrt [3]{1-x+2 x^2}}\right ) \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}\\ &=-\frac {\sqrt [3]{1-x+2 x^2} \int \frac {\sqrt [3]{1-x+3 x^2}}{x^2 \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}+\frac {\left (2 \sqrt [3]{1-x+2 x^2}\right ) \int \frac {\sqrt [3]{1-x+3 x^2}}{x^3 \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}-\frac {\left (2 \sqrt [3]{1-x+2 x^2}\right ) \int \frac {(1+2 x) \sqrt [3]{1-x+3 x^2}}{\left (-1+x+x^2\right ) \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}+\frac {\left (4 \sqrt [3]{1-x+2 x^2}\right ) \int \frac {\sqrt [3]{1-x+3 x^2}}{x \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}\\ &=-\frac {\sqrt [3]{1-x+2 x^2} \int \frac {\sqrt [3]{1-x+3 x^2}}{x^2 \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}+\frac {\left (2 \sqrt [3]{1-x+2 x^2}\right ) \int \frac {\sqrt [3]{1-x+3 x^2}}{x^3 \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}-\frac {\left (2 \sqrt [3]{1-x+2 x^2}\right ) \int \left (\frac {2 \sqrt [3]{1-x+3 x^2}}{\left (1-\sqrt {5}+2 x\right ) \sqrt [3]{1-x+2 x^2}}+\frac {2 \sqrt [3]{1-x+3 x^2}}{\left (1+\sqrt {5}+2 x\right ) \sqrt [3]{1-x+2 x^2}}\right ) \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}+\frac {\left (4 \sqrt [3]{1-x+2 x^2}\right ) \int \frac {\sqrt [3]{1-x+3 x^2}}{x \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}\\ &=-\frac {\sqrt [3]{1-x+2 x^2} \int \frac {\sqrt [3]{1-x+3 x^2}}{x^2 \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}+\frac {\left (2 \sqrt [3]{1-x+2 x^2}\right ) \int \frac {\sqrt [3]{1-x+3 x^2}}{x^3 \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}+\frac {\left (4 \sqrt [3]{1-x+2 x^2}\right ) \int \frac {\sqrt [3]{1-x+3 x^2}}{x \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}-\frac {\left (4 \sqrt [3]{1-x+2 x^2}\right ) \int \frac {\sqrt [3]{1-x+3 x^2}}{\left (1-\sqrt {5}+2 x\right ) \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}-\frac {\left (4 \sqrt [3]{1-x+2 x^2}\right ) \int \frac {\sqrt [3]{1-x+3 x^2}}{\left (1+\sqrt {5}+2 x\right ) \sqrt [3]{1-x+2 x^2}} \, dx}{\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}}\\ \end {align*}

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Mathematica [F]  time = 1.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-2+x) \left (1-x+x^2\right )}{x^3 \left (-1+x+x^2\right ) \sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.32, size = 370, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x-3 x^2\right ) \left (\frac {1-x+2 x^2}{1-x+3 x^2}\right )^{2/3}}{x^2}-2\ 2^{2/3} \sqrt [6]{3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2\ 2^{2/3} \sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}}{3^{5/6}}\right )+\frac {7 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {7}{3} \log \left (-1+\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}\right )-\frac {2\ 2^{2/3} \log \left (-3+6^{2/3} \sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}\right )}{\sqrt [3]{3}}-\frac {7}{6} \log \left (1+\sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}+\left (\frac {1-x+2 x^2}{1-x+3 x^2}\right )^{2/3}\right )+\frac {2^{2/3} \log \left (3+6^{2/3} \sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}+2 \sqrt [3]{6} \left (\frac {1-x+2 x^2}{1-x+3 x^2}\right )^{2/3}\right )}{\sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x - 3*x^2)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(2/3))/x^2 - 2*2^(2/3)*3^(1/6)*ArcTan[1/Sqrt[3] + (2*2^(2/
3)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3))/3^(5/6)] + (7*ArcTan[1/Sqrt[3] + (2*((1 - x + 2*x^2)/(1 - x + 3*x^
2))^(1/3))/Sqrt[3]])/Sqrt[3] + (7*Log[-1 + ((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3)])/3 - (2*2^(2/3)*Log[-3 + 6
^(2/3)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3)])/3^(1/3) - (7*Log[1 + ((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3)
+ ((1 - x + 2*x^2)/(1 - x + 3*x^2))^(2/3)])/6 + (2^(2/3)*Log[3 + 6^(2/3)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/
3) + 2*6^(1/3)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(2/3)])/3^(1/3)

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fricas [B]  time = 18.52, size = 735, normalized size = 1.99 \begin {gather*} -\frac {2 \cdot 3^{\frac {2}{3}} \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {24 \cdot 3^{\frac {2}{3}} \left (-4\right )^{\frac {1}{3}} {\left (39 \, x^{4} - 28 \, x^{3} + 33 \, x^{2} - 10 \, x + 5\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {2}{3}} - 3^{\frac {1}{3}} \left (-4\right )^{\frac {2}{3}} {\left (649 \, x^{4} - 538 \, x^{3} + 647 \, x^{2} - 218 \, x + 109\right )} - 36 \, {\left (75 \, x^{4} - 58 \, x^{3} + 69 \, x^{2} - 22 \, x + 11\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {1}{3}}}{x^{4} + 2 \, x^{3} - x^{2} - 2 \, x + 1}\right ) - 4 \cdot 3^{\frac {2}{3}} \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {9 \cdot 3^{\frac {1}{3}} \left (-4\right )^{\frac {2}{3}} {\left (3 \, x^{2} - x + 1\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {1}{3}} + 3^{\frac {2}{3}} \left (-4\right )^{\frac {1}{3}} {\left (x^{2} + x - 1\right )} - 36 \, {\left (3 \, x^{2} - x + 1\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {2}{3}}}{x^{2} + x - 1}\right ) + 12 \cdot 3^{\frac {1}{6}} \left (-4\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {3^{\frac {1}{6}} {\left (12 \cdot 3^{\frac {2}{3}} \left (-4\right )^{\frac {2}{3}} {\left (39 \, x^{6} + 11 \, x^{5} - 34 \, x^{4} + 51 \, x^{3} - 38 \, x^{2} + 15 \, x - 5\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {2}{3}} + 18 \, \left (-4\right )^{\frac {1}{3}} {\left (1947 \, x^{6} - 2263 \, x^{5} + 3128 \, x^{4} - 1839 \, x^{3} + 1192 \, x^{2} - 327 \, x + 109\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {1}{3}} - 3^{\frac {1}{3}} {\left (16199 \, x^{6} - 20631 \, x^{5} + 29268 \, x^{4} - 18463 \, x^{3} + 12204 \, x^{2} - 3567 \, x + 1189\right )}\right )}}{3 \, {\left (17497 \, x^{6} - 20409 \, x^{5} + 28188 \, x^{4} - 16529 \, x^{3} + 10692 \, x^{2} - 2913 \, x + 971\right )}}\right ) + 42 \, \sqrt {3} x^{2} \arctan \left (\frac {26407150 \, \sqrt {3} {\left (3 \, x^{2} - x + 1\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {2}{3}} + 15172108 \, \sqrt {3} {\left (3 \, x^{2} - x + 1\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {1}{3}} + \sqrt {3} {\left (47470762 \, x^{2} - 20789629 \, x + 20789629\right )}}{29760814 \, x^{2} - 16852563 \, x + 16852563}\right ) - 21 \, x^{2} \log \left (\frac {x^{2} + 3 \, {\left (3 \, x^{2} - x + 1\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {2}{3}} - 3 \, {\left (3 \, x^{2} - x + 1\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {1}{3}}}{x^{2}}\right ) + 18 \, {\left (3 \, x^{2} - x + 1\right )} \left (\frac {2 \, x^{2} - x + 1}{3 \, x^{2} - x + 1}\right )^{\frac {2}{3}}}{18 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(2*3^(2/3)*(-4)^(1/3)*x^2*log(-(24*3^(2/3)*(-4)^(1/3)*(39*x^4 - 28*x^3 + 33*x^2 - 10*x + 5)*((2*x^2 - x
+ 1)/(3*x^2 - x + 1))^(2/3) - 3^(1/3)*(-4)^(2/3)*(649*x^4 - 538*x^3 + 647*x^2 - 218*x + 109) - 36*(75*x^4 - 58
*x^3 + 69*x^2 - 22*x + 11)*((2*x^2 - x + 1)/(3*x^2 - x + 1))^(1/3))/(x^4 + 2*x^3 - x^2 - 2*x + 1)) - 4*3^(2/3)
*(-4)^(1/3)*x^2*log(-(9*3^(1/3)*(-4)^(2/3)*(3*x^2 - x + 1)*((2*x^2 - x + 1)/(3*x^2 - x + 1))^(1/3) + 3^(2/3)*(
-4)^(1/3)*(x^2 + x - 1) - 36*(3*x^2 - x + 1)*((2*x^2 - x + 1)/(3*x^2 - x + 1))^(2/3))/(x^2 + x - 1)) + 12*3^(1
/6)*(-4)^(1/3)*x^2*arctan(1/3*3^(1/6)*(12*3^(2/3)*(-4)^(2/3)*(39*x^6 + 11*x^5 - 34*x^4 + 51*x^3 - 38*x^2 + 15*
x - 5)*((2*x^2 - x + 1)/(3*x^2 - x + 1))^(2/3) + 18*(-4)^(1/3)*(1947*x^6 - 2263*x^5 + 3128*x^4 - 1839*x^3 + 11
92*x^2 - 327*x + 109)*((2*x^2 - x + 1)/(3*x^2 - x + 1))^(1/3) - 3^(1/3)*(16199*x^6 - 20631*x^5 + 29268*x^4 - 1
8463*x^3 + 12204*x^2 - 3567*x + 1189))/(17497*x^6 - 20409*x^5 + 28188*x^4 - 16529*x^3 + 10692*x^2 - 2913*x + 9
71)) + 42*sqrt(3)*x^2*arctan((26407150*sqrt(3)*(3*x^2 - x + 1)*((2*x^2 - x + 1)/(3*x^2 - x + 1))^(2/3) + 15172
108*sqrt(3)*(3*x^2 - x + 1)*((2*x^2 - x + 1)/(3*x^2 - x + 1))^(1/3) + sqrt(3)*(47470762*x^2 - 20789629*x + 207
89629))/(29760814*x^2 - 16852563*x + 16852563)) - 21*x^2*log((x^2 + 3*(3*x^2 - x + 1)*((2*x^2 - x + 1)/(3*x^2
- x + 1))^(2/3) - 3*(3*x^2 - x + 1)*((2*x^2 - x + 1)/(3*x^2 - x + 1))^(1/3))/x^2) + 18*(3*x^2 - x + 1)*((2*x^2
- x + 1)/(3*x^2 - x + 1))^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-2+x \right ) \left (x^{2}-x +1\right )}{x^{3} \left (x^{2}+x -1\right ) \left (\frac {2 x^{2}-x +1}{3 x^{2}-x +1}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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