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

Optimal. Leaf size=423 \[ \frac {\sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}} \left (-3 x^4+x^2-x-1\right )}{x^4}+\frac {5}{3} \log \left (\sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}-1\right )-\sqrt [3]{6} \log \left (6^{2/3} \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}-3\right )-\frac {5}{6} \log \left (\left (\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}\right )^{2/3}+\sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}+1\right )+\frac {\sqrt [3]{3} \log \left (2 \sqrt [3]{6} \left (\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}\right )^{2/3}+6^{2/3} \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}+3\right )}{2^{2/3}}+\sqrt [3]{2} 3^{5/6} \tan ^{-1}\left (\frac {2\ 2^{2/3} \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{3^{5/6}}+\frac {1}{\sqrt {3}}\right )-\frac {5 \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]

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Rubi [F]  time = 10.03, 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 (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx &=\frac {\left (\sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{1+x-x^2+2 x^4}}{x^5 \left (-1-x+x^2+x^4\right ) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}\\ &=\frac {\left (\sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \left (-\frac {2 \sqrt [3]{1+x-x^2+2 x^4}}{(-1+x) \sqrt [3]{1+x-x^2+3 x^4}}+\frac {4 \sqrt [3]{1+x-x^2+2 x^4}}{x^5 \sqrt [3]{1+x-x^2+3 x^4}}+\frac {3 \sqrt [3]{1+x-x^2+2 x^4}}{x^4 \sqrt [3]{1+x-x^2+3 x^4}}-\frac {2 \sqrt [3]{1+x-x^2+2 x^4}}{x^3 \sqrt [3]{1+x-x^2+3 x^4}}+\frac {8 \sqrt [3]{1+x-x^2+2 x^4}}{x \sqrt [3]{1+x-x^2+3 x^4}}-\frac {2 \left (2+2 x+3 x^2\right ) \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}}\right ) \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}\\ &=-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{(-1+x) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^3 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\left (2+2 x+3 x^2\right ) \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (3 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^4 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (4 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^5 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (8 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}\\ &=-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{(-1+x) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^3 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \left (\frac {2 \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}}+\frac {2 x \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}}+\frac {3 x^2 \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}}\right ) \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (3 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^4 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (4 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^5 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (8 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}\\ &=-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{(-1+x) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^3 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (3 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^4 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (4 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^5 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (4 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (4 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {x \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (6 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {x^2 \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (8 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}\\ \end {align*}

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Mathematica [F]  time = 1.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.69, size = 423, normalized size = 1.00 \begin {gather*} \frac {\left (-1-x+x^2-3 x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^4}+\sqrt [3]{2} 3^{5/6} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2\ 2^{2/3} \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{3^{5/6}}\right )-\frac {5 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {5}{3} \log \left (-1+\sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}\right )-\sqrt [3]{6} \log \left (-3+6^{2/3} \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}\right )-\frac {5}{6} \log \left (1+\sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}+\left (\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}\right )^{2/3}\right )+\frac {\sqrt [3]{3} \log \left (3+6^{2/3} \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}+2 \sqrt [3]{6} \left (\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}\right )^{2/3}\right )}{2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 - x + x^2 - 3*x^4)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3))/x^4 + 2^(1/3)*3^(5/6)*ArcTan[1/Sq
rt[3] + (2*2^(2/3)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3))/3^(5/6)] - (5*ArcTan[1/Sqrt[3] + (2*((
1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3))/Sqrt[3]])/Sqrt[3] + (5*Log[-1 + ((1 + x - x^2 + 2*x^4)/(1 +
x - x^2 + 3*x^4))^(1/3)])/3 - 6^(1/3)*Log[-3 + 6^(2/3)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3)] -
(5*Log[1 + ((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3) + ((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4)
)^(2/3)])/6 + (3^(1/3)*Log[3 + 6^(2/3)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3) + 2*6^(1/3)*((1 + x
- x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(2/3)])/2^(2/3)

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fricas [B]  time = 70.10, size = 960, normalized size = 2.27

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*sqrt(3)*(-6)^(1/3)*x^4*arctan(1/3*(6*sqrt(3)*(-6)^(2/3)*(1947*x^12 - 2263*x^10 + 2263*x^9 + 3128*x^8 -
1730*x^7 - 974*x^6 + 2057*x^5 + 865*x^4 - 545*x^3 + 327*x + 109)*((2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1)
)^(1/3) + 24*sqrt(3)*(-6)^(1/3)*(39*x^12 + 11*x^10 - 11*x^9 - 34*x^8 + 46*x^7 + 28*x^6 - 61*x^5 - 23*x^4 + 25*
x^3 - 15*x - 5)*((2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1))^(2/3) + sqrt(3)*(16199*x^12 - 20631*x^10 + 20631
*x^9 + 29268*x^8 - 17274*x^7 - 9826*x^6 + 20841*x^5 + 8637*x^4 - 5945*x^3 + 3567*x + 1189))/(17497*x^12 - 2040
9*x^10 + 20409*x^9 + 28188*x^8 - 15558*x^7 - 8750*x^6 + 18471*x^5 + 7779*x^4 - 4855*x^3 + 2913*x + 971)) - 10*
sqrt(3)*x^4*arctan((26407150*sqrt(3)*(3*x^4 - x^2 + x + 1)*((2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1))^(2/3)
+ 15172108*sqrt(3)*(3*x^4 - x^2 + x + 1)*((2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1))^(1/3) + sqrt(3)*(47470
762*x^4 - 20789629*x^2 + 20789629*x + 20789629))/(29760814*x^4 - 16852563*x^2 + 16852563*x + 16852563)) + (-6)
^(1/3)*x^4*log((12*(-6)^(2/3)*(39*x^8 - 28*x^6 + 28*x^5 + 33*x^4 - 10*x^3 - 5*x^2 + 10*x + 5)*((2*x^4 - x^2 +
x + 1)/(3*x^4 - x^2 + x + 1))^(2/3) - (-6)^(1/3)*(649*x^8 - 538*x^6 + 538*x^5 + 647*x^4 - 218*x^3 - 109*x^2 +
218*x + 109) + 18*(75*x^8 - 58*x^6 + 58*x^5 + 69*x^4 - 22*x^3 - 11*x^2 + 22*x + 11)*((2*x^4 - x^2 + x + 1)/(3*
x^4 - x^2 + x + 1))^(1/3))/(x^8 + 2*x^6 - 2*x^5 - x^4 - 2*x^3 - x^2 + 2*x + 1)) - 2*(-6)^(1/3)*x^4*log(((-6)^(
2/3)*(x^4 + x^2 - x - 1) + 18*(-6)^(1/3)*(3*x^4 - x^2 + x + 1)*((2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1))^(
1/3) + 36*(3*x^4 - x^2 + x + 1)*((2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1))^(2/3))/(x^4 + x^2 - x - 1)) - 5*
x^4*log((x^4 + 3*(3*x^4 - x^2 + x + 1)*((2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1))^(2/3) - 3*(3*x^4 - x^2 +
x + 1)*((2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1))^(1/3))/x^4) + 6*(3*x^4 - x^2 + x + 1)*((2*x^4 - x^2 + x +
1)/(3*x^4 - x^2 + x + 1))^(1/3))/x^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (2 x^{2}-3 x -4\right ) \left (x^{4}-x^{2}+x +1\right ) \left (\frac {2 x^{4}-x^{2}+x +1}{3 x^{4}-x^{2}+x +1}\right )^{\frac {1}{3}}}{x^{5} \left (x^{4}+x^{2}-x -1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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