3.31.20 \(\int \frac {(-4-3 x+2 x^2) (1+x-x^2+x^4) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 (-1-x+x^2+x^4)} \, dx\) [3020]

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

[Out]

(-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+2^(1/3)*3^(5/6)*arctan(1/3*3^(1/2)+2/3*2^(2/3)*((
2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3)*3^(1/6))-5/3*arctan(1/3*3^(1/2)+2/3*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1
/3)*3^(1/2))*3^(1/2)+5/3*ln(-1+((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3))-6^(1/3)*ln(-3+6^(2/3)*((2*x^4-x^2+x+1)
/(3*x^4-x^2+x+1))^(1/3))-5/6*ln(1+((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3)+((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(2
/3))+1/2*3^(1/3)*ln(3+6^(2/3)*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3)+2*6^(1/3)*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+
1))^(2/3))*2^(1/3)

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Rubi [F]
time = 7.07, 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 = 20.47, 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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Maple [F]
time = 0.01, 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*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (379) = 758\).
time = 26.56, size = 960, normalized size = 2.27 \begin {gather*} -\frac {2 \, \sqrt {3} \left (-6\right )^{\frac {1}{3}} x^{4} \arctan \left (\frac {6 \, \sqrt {3} \left (-6\right )^{\frac {2}{3}} {\left (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\right )} \left (\frac {2 \, x^{4} - x^{2} + x + 1}{3 \, x^{4} - x^{2} + x + 1}\right )^{\frac {1}{3}} + 24 \, \sqrt {3} \left (-6\right )^{\frac {1}{3}} {\left (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\right )} \left (\frac {2 \, x^{4} - x^{2} + x + 1}{3 \, x^{4} - x^{2} + x + 1}\right )^{\frac {2}{3}} + \sqrt {3} {\left (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\right )}}{3 \, {\left (17497 \, x^{12} - 20409 \, 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\right )}}\right ) - 10 \, \sqrt {3} x^{4} \arctan \left (\frac {26407150 \, \sqrt {3} {\left (3 \, 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 {2}{3}} + 15172108 \, \sqrt {3} {\left (3 \, 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}} + \sqrt {3} {\left (47470762 \, x^{4} - 20789629 \, x^{2} + 20789629 \, x + 20789629\right )}}{29760814 \, x^{4} - 16852563 \, x^{2} + 16852563 \, x + 16852563}\right ) + \left (-6\right )^{\frac {1}{3}} x^{4} \log \left (\frac {12 \, \left (-6\right )^{\frac {2}{3}} {\left (39 \, x^{8} - 28 \, x^{6} + 28 \, x^{5} + 33 \, x^{4} - 10 \, x^{3} - 5 \, x^{2} + 10 \, x + 5\right )} \left (\frac {2 \, x^{4} - x^{2} + x + 1}{3 \, x^{4} - x^{2} + x + 1}\right )^{\frac {2}{3}} - \left (-6\right )^{\frac {1}{3}} {\left (649 \, x^{8} - 538 \, x^{6} + 538 \, x^{5} + 647 \, x^{4} - 218 \, x^{3} - 109 \, x^{2} + 218 \, x + 109\right )} + 18 \, {\left (75 \, x^{8} - 58 \, x^{6} + 58 \, x^{5} + 69 \, x^{4} - 22 \, x^{3} - 11 \, x^{2} + 22 \, x + 11\right )} \left (\frac {2 \, x^{4} - x^{2} + x + 1}{3 \, x^{4} - x^{2} + x + 1}\right )^{\frac {1}{3}}}{x^{8} + 2 \, x^{6} - 2 \, x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1}\right ) - 2 \, \left (-6\right )^{\frac {1}{3}} x^{4} \log \left (\frac {\left (-6\right )^{\frac {2}{3}} {\left (x^{4} + x^{2} - x - 1\right )} + 18 \, \left (-6\right )^{\frac {1}{3}} {\left (3 \, 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}} + 36 \, {\left (3 \, 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 {2}{3}}}{x^{4} + x^{2} - x - 1}\right ) - 5 \, x^{4} \log \left (\frac {x^{4} + 3 \, {\left (3 \, 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 {2}{3}} - 3 \, {\left (3 \, 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^{4}}\right ) + 6 \, {\left (3 \, 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}}}{6 \, x^{4}} \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="
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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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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