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

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 = 0.21, size = 146, normalized size = 1.00 \begin {gather*} \frac {3 \left (2+x-x^3-x^4\right )^{2/3} \left (-4-2 x-3 x^3+2 x^4\right )}{10 x^5}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2+x-x^3-x^4}}\right )-\log \left (-x+\sqrt [3]{2+x-x^3-x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{2+x-x^3-x^4}+\left (2+x-x^3-x^4\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 = 3.52, 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 = 10.32, size = 572, normalized size = 3.92

method	result	size
trager	\(\frac {3 \left (-2+x \right ) \left (2 x^{3}+x^{2}+2 x +2\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}}}{10 x^{5}}-\ln \left (\frac {-95643648 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{4}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}+3100896 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{4}+3680064 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x -1046304 \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{2}+81024 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}-15890 x^{4}-27435 \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x +38334 \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x -6356 x^{3}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-3100896 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x -6201792 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+15890 x +31780}{\left (2+x \right ) \left (-1+x \right ) \left (x^{2}+x +1\right )}\right )+96 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \ln \left (-\frac {-58576896 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{4}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}+81024 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{4}+3680064 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x -2633760 \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{2}-3019872 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}+15567 x^{4}-10899 \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x +38334 \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x +25945 x^{3}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-81024 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x -162048 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )-15567 x -31134}{\left (2+x \right ) \left (-1+x \right ) \left (x^{2}+x +1\right )}\right )\)	\(572\)
risch	\(-\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 (-1-x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x \right ) \left (-1+x \right ) \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +1\right ) \left (2+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 (-1-x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x \right ) \left (-1+x \right ) \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +1\right ) \left (2+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 (-1-x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x \right ) \left (-1+x \right ) \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +1\right ) \left (2+x \right )}\right )\)	\(698\)

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,method=_RETURNVERBOSE)

[Out]

3/10*(-2+x)*(2*x^3+x^2+2*x+2)/x^5*(-x^4-x^3+x+2)^(2/3)-ln((-95643648*RootOf(9216*_Z^2-96*_Z+1)^2*x^4+143465472
*RootOf(9216*_Z^2-96*_Z+1)^2*x^3+3100896*RootOf(9216*_Z^2-96*_Z+1)*x^4+3680064*RootOf(9216*_Z^2-96*_Z+1)*(-x^4
-x^3+x+2)^(2/3)*x-1046304*(-x^4-x^3+x+2)^(1/3)*RootOf(9216*_Z^2-96*_Z+1)*x^2+81024*RootOf(9216*_Z^2-96*_Z+1)*x
^3-15890*x^4-27435*(-x^4-x^3+x+2)^(2/3)*x+38334*(-x^4-x^3+x+2)^(1/3)*x^2+95643648*RootOf(9216*_Z^2-96*_Z+1)^2*
x-6356*x^3+191287296*RootOf(9216*_Z^2-96*_Z+1)^2-3100896*RootOf(9216*_Z^2-96*_Z+1)*x-6201792*RootOf(9216*_Z^2-
96*_Z+1)+15890*x+31780)/(2+x)/(-1+x)/(x^2+x+1))+96*RootOf(9216*_Z^2-96*_Z+1)*ln(-(-58576896*RootOf(9216*_Z^2-9
6*_Z+1)^2*x^4+87865344*RootOf(9216*_Z^2-96*_Z+1)^2*x^3+81024*RootOf(9216*_Z^2-96*_Z+1)*x^4+3680064*RootOf(9216
*_Z^2-96*_Z+1)*(-x^4-x^3+x+2)^(2/3)*x-2633760*(-x^4-x^3+x+2)^(1/3)*RootOf(9216*_Z^2-96*_Z+1)*x^2-3019872*RootO
f(9216*_Z^2-96*_Z+1)*x^3+15567*x^4-10899*(-x^4-x^3+x+2)^(2/3)*x+38334*(-x^4-x^3+x+2)^(1/3)*x^2+58576896*RootOf
(9216*_Z^2-96*_Z+1)^2*x+25945*x^3+117153792*RootOf(9216*_Z^2-96*_Z+1)^2-81024*RootOf(9216*_Z^2-96*_Z+1)*x-1620
48*RootOf(9216*_Z^2-96*_Z+1)-15567*x-31134)/(2+x)/(-1+x)/(x^2+x+1))

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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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