3.21.19 \(\int \frac {(1+x^3)^{2/3} (-1-2 x^3+2 x^6)}{x^9 (-1+x^3)} \, dx\)

Optimal. Leaf size=143 \[ -\frac {1}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3+1}-2 x\right )+\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+1}+x}\right )}{\sqrt {3}}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{2} \left (x^3+1\right )^{2/3}+2 x^2\right )}{3 \sqrt [3]{2}}+\frac {\left (x^3+1\right )^{2/3} \left (-41 x^6-26 x^3-5\right )}{40 x^8} \]

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Rubi [F]  time = 0.71, 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 (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^9 \left (-1+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^3)^(2/3)*(-1 - 2*x^3 + 2*x^6))/(x^9*(-1 + x^3)),x]

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 - (1 + x^3)^(5/3)/(8*x^8) - (21*(1 + x^3)^(5/3))/(40*x^5) + ArcTan[(1 + (2*x)/(1 + x^
3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-x + (1 + x^3)^(1/3)]/2 - Defer[Int][(1 + x^3)^(2/3)/(-1 + x), x]/3 + ((1 - I
*Sqrt[3])*Defer[Int][(1 + x^3)^(2/3)/(1 - I*Sqrt[3] + 2*x), x])/3 + ((1 + I*Sqrt[3])*Defer[Int][(1 + x^3)^(2/3
)/(1 + I*Sqrt[3] + 2*x), x])/3

Rubi steps

\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^9 \left (-1+x^3\right )} \, dx &=\int \left (-\frac {\left (1+x^3\right )^{2/3}}{3 (-1+x)}+\frac {\left (1+x^3\right )^{2/3}}{x^9}+\frac {3 \left (1+x^3\right )^{2/3}}{x^6}+\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {(2+x) \left (1+x^3\right )^{2/3}}{3 \left (1+x+x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx\right )+\frac {1}{3} \int \frac {(2+x) \left (1+x^3\right )^{2/3}}{1+x+x^2} \, dx+3 \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {\left (1+x^3\right )^{2/3}}{x^9} \, dx+\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\left (1+x^3\right )^{5/3}}{8 x^8}-\frac {3 \left (1+x^3\right )^{5/3}}{5 x^5}-\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{3} \int \left (\frac {\left (1-i \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x}\right ) \, dx-\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\left (1+x^3\right )^{5/3}}{8 x^8}-\frac {21 \left (1+x^3\right )^{5/3}}{40 x^5}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{3} \left (1-i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 131, normalized size = 0.92 \begin {gather*} \frac {-2 \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+\frac {2^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )}{3 \sqrt [3]{2}}-\frac {\left (x^3+1\right )^{2/3} \left (41 x^6+26 x^3+5\right )}{40 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)^(2/3)*(-1 - 2*x^3 + 2*x^6))/(x^9*(-1 + x^3)),x]

[Out]

-1/40*((1 + x^3)^(2/3)*(5 + 26*x^3 + 41*x^6))/x^8 + (2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt
[3]] - 2*Log[1 - (2^(1/3)*x)/(1 + x^3)^(1/3)] + Log[1 + (2^(2/3)*x^2)/(1 + x^3)^(2/3) + (2^(1/3)*x)/(1 + x^3)^
(1/3)])/(3*2^(1/3))

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IntegrateAlgebraic [A]  time = 0.35, size = 143, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (-5-26 x^3-41 x^6\right )}{40 x^8}+\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{2} \left (1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^3)^(2/3)*(-1 - 2*x^3 + 2*x^6))/(x^9*(-1 + x^3)),x]

[Out]

((1 + x^3)^(2/3)*(-5 - 26*x^3 - 41*x^6))/(40*x^8) + (2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x^3)^(1/3))]
)/Sqrt[3] - (2^(2/3)*Log[-2*x + 2^(2/3)*(1 + x^3)^(1/3)])/3 + Log[2*x^2 + 2^(2/3)*x*(1 + x^3)^(1/3) + 2^(1/3)*
(1 + x^3)^(2/3)]/(3*2^(1/3))

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fricas [B]  time = 2.74, size = 271, normalized size = 1.90 \begin {gather*} -\frac {40 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{8} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} - 4 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} + 16 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} + 111 \, x^{6} + 33 \, x^{3} + 1\right )}}{3 \, {\left (109 \, x^{9} + 105 \, x^{6} + 3 \, x^{3} - 1\right )}}\right ) - 40 \, \left (-4\right )^{\frac {1}{3}} x^{8} \log \left (\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) + 20 \, \left (-4\right )^{\frac {1}{3}} x^{8} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} + 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) + 9 \, {\left (41 \, x^{6} + 26 \, x^{3} + 5\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{360 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(2*x^6-2*x^3-1)/x^9/(x^3-1),x, algorithm="fricas")

[Out]

-1/360*(40*sqrt(3)*(-4)^(1/3)*x^8*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(5*x^7 - 4*x^4 - x)*(x^3 + 1)^(2/3) + 6*sqr
t(3)*(-4)^(1/3)*(19*x^8 + 16*x^5 + x^2)*(x^3 + 1)^(1/3) - sqrt(3)*(71*x^9 + 111*x^6 + 33*x^3 + 1))/(109*x^9 +
105*x^6 + 3*x^3 - 1)) - 40*(-4)^(1/3)*x^8*log((3*(-4)^(2/3)*(x^3 + 1)^(1/3)*x^2 - 6*(x^3 + 1)^(2/3)*x + (-4)^(
1/3)*(x^3 - 1))/(x^3 - 1)) + 20*(-4)^(1/3)*x^8*log(-(6*(-4)^(1/3)*(5*x^4 + x)*(x^3 + 1)^(2/3) - (-4)^(2/3)*(19
*x^6 + 16*x^3 + 1) - 24*(2*x^5 + x^2)*(x^3 + 1)^(1/3))/(x^6 - 2*x^3 + 1)) + 9*(41*x^6 + 26*x^3 + 5)*(x^3 + 1)^
(2/3))/x^8

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(2*x^6-2*x^3-1)/x^9/(x^3-1),x, algorithm="giac")

[Out]

integrate((2*x^6 - 2*x^3 - 1)*(x^3 + 1)^(2/3)/((x^3 - 1)*x^9), x)

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maple [C]  time = 17.04, size = 908, normalized size = 6.35

method	result	size
risch	\(-\frac {41 x^{9}+67 x^{6}+31 x^{3}+5}{40 x^{8} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {\RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+54 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}-12 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +5 \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+6 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}+\RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-2 x \left (x^{3}+1\right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{3}+4\right )+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}-\frac {\ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}-36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+12 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +\RootOf \left (\textit {\_Z}^{3}+4\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+30 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-10 x \left (x^{3}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{3}+4\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )}{3}-2 \ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}-36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+12 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +\RootOf \left (\textit {\_Z}^{3}+4\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+30 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-10 x \left (x^{3}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{3}+4\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )\)	\(908\)
trager	\(\text {Expression too large to display}\)	\(1121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(2*x^6-2*x^3-1)/x^9/(x^3-1),x,method=_RETURNVERBOSE)

[Out]

-1/40*(41*x^9+67*x^6+31*x^3+5)/x^8/(x^3+1)^(1/3)+1/3*RootOf(_Z^3+4)*ln((3*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(
_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3+54*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)
^2*x^3-12*(x^3+1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*x+5*RootOf(_Z^3+
4)^2*(x^3+1)^(1/3)*x^2+6*(x^3+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)*x^2
+RootOf(_Z^3+4)*x^3+18*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^3-2*x*(x^3+1)^(2/3)+RootOf(_Z^3+
4)+18*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(-1+x)/(x^2+x+1))-1/3*ln((3*RootOf(RootOf(_Z^3+4)^
2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3-36*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*
RootOf(_Z^3+4)^2*x^3+12*(x^3+1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*x+
RootOf(_Z^3+4)^2*(x^3+1)^(1/3)*x^2+30*(x^3+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootO
f(_Z^3+4)*x^2-3*RootOf(_Z^3+4)*x^3+36*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^3-10*x*(x^3+1)^(2
/3)-RootOf(_Z^3+4)+12*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(-1+x)/(x^2+x+1))*RootOf(_Z^3+4)-2
*ln((3*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3-36*RootOf(RootOf(_Z^3+4)^2+6*
_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+12*(x^3+1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)
+36*_Z^2)*RootOf(_Z^3+4)^2*x+RootOf(_Z^3+4)^2*(x^3+1)^(1/3)*x^2+30*(x^3+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*
RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)*x^2-3*RootOf(_Z^3+4)*x^3+36*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)
+36*_Z^2)*x^3-10*x*(x^3+1)^(2/3)-RootOf(_Z^3+4)+12*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(-1+x
)/(x^2+x+1))*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(2*x^6-2*x^3-1)/x^9/(x^3-1),x, algorithm="maxima")

[Out]

integrate((2*x^6 - 2*x^3 - 1)*(x^3 + 1)^(2/3)/((x^3 - 1)*x^9), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (-2\,x^6+2\,x^3+1\right )}{x^9\,\left (x^3-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^3 + 1)^(2/3)*(2*x^3 - 2*x^6 + 1))/(x^9*(x^3 - 1)),x)

[Out]

-int(((x^3 + 1)^(2/3)*(2*x^3 - 2*x^6 + 1))/(x^9*(x^3 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (2 x^{6} - 2 x^{3} - 1\right )}{x^{9} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+1)**(2/3)*(2*x**6-2*x**3-1)/x**9/(x**3-1),x)

[Out]

Integral(((x + 1)*(x**2 - x + 1))**(2/3)*(2*x**6 - 2*x**3 - 1)/(x**9*(x - 1)*(x**2 + x + 1)), x)

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