3.18.1 \(\int \frac {(2+x-x^3-x^4)^{2/3} (6+2 x+x^4) (-2-x+x^3+x^4)}{x^6 (-2-x+2 x^3+x^4)} \, dx\)

Optimal. Leaf size=146 \[ -\log \left (\sqrt [3]{-x^4-x^3+x+2}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{-x^4-x^3+x+2}+x}\right )+\frac {3 \left (-x^4-x^3+x+2\right )^{2/3} \left (2 x^4-3 x^3-2 x-4\right )}{10 x^5}+\frac {1}{2} \log \left (x^2+\sqrt [3]{-x^4-x^3+x+2} x+\left (-x^4-x^3+x+2\right )^{2/3}\right ) \]

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Rubi [F]  time = 2.68, 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 (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][(2 + x - x^3 - x^4)^(2/3)/(1 - x), x] + 6*Defer[Int][(2 + x - x^3 - x^4)^(2/3)/x^6, x] + 2*Defer[In
t][(2 + x - x^3 - x^4)^(2/3)/x^5, x] + 3*Defer[Int][(2 + x - x^3 - x^4)^(2/3)/x^3, x] + Defer[Int][(2 + x - x^
3 - x^4)^(2/3)/x^2, x]/2 + Defer[Int][(2 + x - x^3 - x^4)^(2/3)/x, x]/4 - Defer[Int][(2 + x - x^3 - x^4)^(2/3)
/(2 + x), x]/4 + (1 - I*Sqrt[3])*Defer[Int][(2 + x - x^3 - x^4)^(2/3)/(1 - I*Sqrt[3] + 2*x), x] + (1 + I*Sqrt[
3])*Defer[Int][(2 + x - x^3 - x^4)^(2/3)/(1 + I*Sqrt[3] + 2*x), x]

Rubi steps

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

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Mathematica [F]  time = 1.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((2 + x - x^3 - x^4)^(2/3)*(6 + 2*x + x^4)*(-2 - x + x^3 + x^4))/(x^6*(-2 - x + 2*x^3 + x^4)),x]

[Out]

Integrate[((2 + x - x^3 - x^4)^(2/3)*(6 + 2*x + x^4)*(-2 - x + x^3 + x^4))/(x^6*(-2 - x + 2*x^3 + x^4)), x]

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IntegrateAlgebraic [A]  time = 0.23, size = 146, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{-x^4-x^3+x+2}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{-x^4-x^3+x+2}+x}\right )+\frac {3 \left (-x^4-x^3+x+2\right )^{2/3} \left (2 x^4-3 x^3-2 x-4\right )}{10 x^5}+\frac {1}{2} \log \left (x^2+\sqrt [3]{-x^4-x^3+x+2} x+\left (-x^4-x^3+x+2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 + x - x^3 - x^4)^(2/3)*(6 + 2*x + x^4)*(-2 - x + x^3 + x^4))/(x^6*(-2 - x + 2*x^3 + x^4
)),x]

[Out]

(3*(2 + x - x^3 - x^4)^(2/3)*(-4 - 2*x - 3*x^3 + 2*x^4))/(10*x^5) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(2 + x -
x^3 - x^4)^(1/3))] - Log[-x + (2 + x - x^3 - x^4)^(1/3)] + Log[x^2 + x*(2 + x - x^3 - x^4)^(1/3) + (2 + x - x
^3 - x^4)^(2/3)]/2

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fricas [A]  time = 3.23, size = 204, normalized size = 1.40 \begin {gather*} -\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {49772 \, \sqrt {3} {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {1}{3}} x^{2} - 31378 \, \sqrt {3} {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (17661 \, x^{4} + 26125 \, x^{3} - 17661 \, x - 35322\right )}}{24389 \, x^{4} - 72947 \, x^{3} - 24389 \, x - 48778}\right ) + 5 \, x^{5} \log \left (\frac {x^{4} + 2 \, x^{3} - 3 \, {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {2}{3}} x - x - 2}{x^{4} + 2 \, x^{3} - x - 2}\right ) - 3 \, {\left (2 \, x^{4} - 3 \, x^{3} - 2 \, x - 4\right )} {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {2}{3}}}{10 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4-x^3+x+2)^(2/3)*(x^4+2*x+6)*(x^4+x^3-x-2)/x^6/(x^4+2*x^3-x-2),x, algorithm="fricas")

[Out]

-1/10*(10*sqrt(3)*x^5*arctan(-(49772*sqrt(3)*(-x^4 - x^3 + x + 2)^(1/3)*x^2 - 31378*sqrt(3)*(-x^4 - x^3 + x +
2)^(2/3)*x - sqrt(3)*(17661*x^4 + 26125*x^3 - 17661*x - 35322))/(24389*x^4 - 72947*x^3 - 24389*x - 48778)) + 5
*x^5*log((x^4 + 2*x^3 - 3*(-x^4 - x^3 + x + 2)^(1/3)*x^2 + 3*(-x^4 - x^3 + x + 2)^(2/3)*x - x - 2)/(x^4 + 2*x^
3 - x - 2)) - 3*(2*x^4 - 3*x^3 - 2*x - 4)*(-x^4 - x^3 + x + 2)^(2/3))/x^5

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3} - x - 2\right )} {\left (x^{4} + 2 \, x + 6\right )} {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {2}{3}}}{{\left (x^{4} + 2 \, x^{3} - x - 2\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4-x^3+x+2)^(2/3)*(x^4+2*x+6)*(x^4+x^3-x-2)/x^6/(x^4+2*x^3-x-2),x, algorithm="giac")

[Out]

integrate((x^4 + x^3 - x - 2)*(x^4 + 2*x + 6)*(-x^4 - x^3 + x + 2)^(2/3)/((x^4 + 2*x^3 - x - 2)*x^6), x)

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maple [C]  time = 3.80, size = 698, normalized size = 4.78 \begin {gather*} -\frac {3 \left (2 x^{8}-x^{7}-3 x^{6}-4 x^{5}-7 x^{4}+2 x^{3}+2 x^{2}+8 x +8\right )}{10 x^{5} \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}+\left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x +\left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\left (2+x \right ) \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +1\right ) \left (-1-x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x \right ) \left (-1+x \right )}\right )-\ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{4}-2 \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x -2 \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-x -2}{\left (2+x \right ) \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +1\right ) \left (-1-x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x \right ) \left (-1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{4}-2 \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x -2 \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-x -2}{\left (2+x \right ) \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +1\right ) \left (-1-x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x \right ) \left (-1+x \right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4-x^3+x+2)^(2/3)*(x^4+2*x+6)*(x^4+x^3-x-2)/x^6/(x^4+2*x^3-x-2),x)

[Out]

-3/10*(2*x^8-x^7-3*x^6-4*x^5-7*x^4+2*x^3+2*x^2+8*x+8)/x^5/(-x^4-x^3+x+2)^(1/3)+RootOf(_Z^2-_Z+1)*ln(-(-RootOf(
_Z^2-_Z+1)^2*x^3-RootOf(_Z^2-_Z+1)*x^4+RootOf(_Z^2-_Z+1)*(-x^4-x^3+x+2)^(2/3)*x+RootOf(_Z^2-_Z+1)*(-x^4-x^3+x+
2)^(1/3)*x^2+(-x^4-x^3+x+2)^(2/3)*x+(-x^4-x^3+x+2)^(1/3)*x^2+RootOf(_Z^2-_Z+1)*x+2*RootOf(_Z^2-_Z+1))/(2+x)/(R
ootOf(_Z^2-_Z+1)*x+1)/(-1-x+RootOf(_Z^2-_Z+1)*x)/(-1+x))-ln((RootOf(_Z^2-_Z+1)^2*x^3-RootOf(_Z^2-_Z+1)*x^4+Roo
tOf(_Z^2-_Z+1)*(-x^4-x^3+x+2)^(2/3)*x+RootOf(_Z^2-_Z+1)*(-x^4-x^3+x+2)^(1/3)*x^2-2*RootOf(_Z^2-_Z+1)*x^3+x^4-2
*(-x^4-x^3+x+2)^(2/3)*x-2*(-x^4-x^3+x+2)^(1/3)*x^2+x^3+RootOf(_Z^2-_Z+1)*x+2*RootOf(_Z^2-_Z+1)-x-2)/(2+x)/(Roo
tOf(_Z^2-_Z+1)*x+1)/(-1-x+RootOf(_Z^2-_Z+1)*x)/(-1+x))*RootOf(_Z^2-_Z+1)+ln((RootOf(_Z^2-_Z+1)^2*x^3-RootOf(_Z
^2-_Z+1)*x^4+RootOf(_Z^2-_Z+1)*(-x^4-x^3+x+2)^(2/3)*x+RootOf(_Z^2-_Z+1)*(-x^4-x^3+x+2)^(1/3)*x^2-2*RootOf(_Z^2
-_Z+1)*x^3+x^4-2*(-x^4-x^3+x+2)^(2/3)*x-2*(-x^4-x^3+x+2)^(1/3)*x^2+x^3+RootOf(_Z^2-_Z+1)*x+2*RootOf(_Z^2-_Z+1)
-x-2)/(2+x)/(RootOf(_Z^2-_Z+1)*x+1)/(-1-x+RootOf(_Z^2-_Z+1)*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^{3} - x - 2\right )} {\left (x^{4} + 2 \, x + 6\right )} {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {2}{3}}}{{\left (x^{4} + 2 \, x^{3} - x - 2\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4-x^3+x+2)^(2/3)*(x^4+2*x+6)*(x^4+x^3-x-2)/x^6/(x^4+2*x^3-x-2),x, algorithm="maxima")

[Out]

integrate((x^4 + x^3 - x - 2)*(x^4 + 2*x + 6)*(-x^4 - x^3 + x + 2)^(2/3)/((x^4 + 2*x^3 - x - 2)*x^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^4+2\,x+6\right )\,{\left (-x^4-x^3+x+2\right )}^{5/3}}{x^6\,\left (-x^4-2\,x^3+x+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x + x^4 + 6)*(x - x^3 - x^4 + 2)^(5/3))/(x^6*(x - 2*x^3 - x^4 + 2)),x)

[Out]

int(((2*x + x^4 + 6)*(x - x^3 - x^4 + 2)^(5/3))/(x^6*(x - 2*x^3 - x^4 + 2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{4} + 2 x + 6\right ) \left (- x^{4} - x^{3} + x + 2\right )^{\frac {2}{3}} \left (x^{4} + x^{3} - x - 2\right )}{x^{6} \left (x - 1\right ) \left (x + 2\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**4-x**3+x+2)**(2/3)*(x**4+2*x+6)*(x**4+x**3-x-2)/x**6/(x**4+2*x**3-x-2),x)

[Out]

Integral((x**4 + 2*x + 6)*(-x**4 - x**3 + x + 2)**(2/3)*(x**4 + x**3 - x - 2)/(x**6*(x - 1)*(x + 2)*(x**2 + x
+ 1)), x)

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