3.28.41 \(\int \frac {(-1+x^3)^{2/3} (4+x^3)}{x^6 (-2-x^3+x^6)} \, dx\)

Optimal. Leaf size=254 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )}{6\ 2^{2/3}}-\frac {1}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-1}-2 x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (8-13 x^3\right )}{20 x^5}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1} x+\sqrt [3]{2} \left (x^3-1\right )^{2/3}+2 x^2\right )}{3 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{12\ 2^{2/3}} \]

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Rubi [F]  time = 0.74, 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 (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/4*(-1 + x^3)^(2/3)/x^2 - (2*(-1 + x^3)^(5/3))/(5*x^5) - (x*(-1 + x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, x^3
, x^3/2])/(4*(1 - x^3)^(2/3)) + ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) - Log[-x + (-1 + x^3)
^(1/3)]/4 - Defer[Int][(-1 + x^3)^(2/3)/(1 + x), x]/3 + ((1 + I*Sqrt[3])*Defer[Int][(-1 + x^3)^(2/3)/(-1 - I*S
qrt[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 (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx &=\int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{x^6}+\frac {\left (-1+x^3\right )^{2/3}}{2 x^3}-\frac {\left (-1+x^3\right )^{2/3}}{3 (1+x)}+\frac {(-2+x) \left (-1+x^3\right )^{2/3}}{3 \left (1-x+x^2\right )}+\frac {\left (-1+x^3\right )^{2/3}}{2 \left (-2+x^3\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+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx-2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \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 {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {\left (-1+x^3\right )^{2/3} \int \frac {\left (1-x^3\right )^{2/3}}{-2+x^3} \, dx}{2 \left (1-x^3\right )^{2/3}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{4 \left (1-x^3\right )^{2/3}}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \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 = 0.50, size = 236, normalized size = 0.93 \begin {gather*} \frac {2 \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{x^3-1}}\right )-8 \sqrt [3]{2} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}\right )+8 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {x^2}{\left (x^3-1\right )^{2/3}}+2^{2/3}\right )+4 \sqrt [3]{2} \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 )}{12\ 2^{2/3}}+\left (x^3-1\right )^{2/3} \left (\frac {2}{5 x^5}-\frac {13}{20 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^3)^(2/3)*(4 + x^3))/(x^6*(-2 - x^3 + x^6)),x]

[Out]

(2/(5*x^5) - 13/(20*x^2))*(-1 + x^3)^(2/3) + (8*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sq
rt[3]] - 2*Sqrt[3]*ArcTan[(1 + (2^(2/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] + 2*Log[2^(1/3) - x/(-1 + x^3)^(1/3)] -
8*2^(1/3)*Log[1 - (2^(1/3)*x)/(-1 + x^3)^(1/3)] - Log[2^(2/3) + x^2/(-1 + x^3)^(2/3) + (2^(1/3)*x)/(-1 + x^3)^
(1/3)] + 4*2^(1/3)*Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) + (2^(1/3)*x)/(-1 + x^3)^(1/3)])/(12*2^(2/3))

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IntegrateAlgebraic [A]  time = 0.62, size = 254, normalized size = 1.00 \begin {gather*} \frac {\left (8-13 x^3\right ) \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{6\ 2^{2/3}}-\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}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{12\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 + x^3))/(x^6*(-2 - x^3 + x^6)),x]

[Out]

((8 - 13*x^3)*(-1 + x^3)^(2/3))/(20*x^5) - ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))]/(2*2^(2/3)*Sqr
t[3]) + (2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 + x^3)^(1/3))])/Sqrt[3] + Log[-x + 2^(1/3)*(-1 + x^3)^(1/
3)]/(6*2^(2/3)) - (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)) - Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)]/(12*2
^(2/3))

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fricas [B]  time = 5.31, size = 521, normalized size = 2.05 \begin {gather*} -\frac {80 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{5} \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 ) - 20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, \sqrt {3} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) - 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) + 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 80 \, \left (-4\right )^{\frac {1}{3}} x^{5} \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 ) + 40 \, \left (-4\right )^{\frac {1}{3}} x^{5} \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 ) + 36 \, {\left (13 \, x^{3} - 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{720 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^3+4)/x^6/(x^6-x^3-2),x, algorithm="fricas")

[Out]

-1/720*(80*sqrt(3)*(-4)^(1/3)*x^5*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)) - 20*4^(1/6)*sqrt(3)*x^5*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(2*x^7 - 5*x^4 + 2*x)*(x
^3 - 1)^(2/3) + 4^(1/3)*sqrt(3)*(91*x^9 - 168*x^6 + 84*x^3 - 8) + 12*sqrt(3)*(19*x^8 - 22*x^5 + 4*x^2)*(x^3 -
1)^(1/3))/(53*x^9 - 48*x^6 - 12*x^3 + 8)) - 10*4^(2/3)*x^5*log((6*4^(1/3)*(x^3 - 1)^(1/3)*x^2 + 4^(2/3)*(x^3 -
2) - 12*(x^3 - 1)^(2/3)*x)/(x^3 - 2)) + 5*4^(2/3)*x^5*log((6*4^(2/3)*(2*x^4 - x)*(x^3 - 1)^(2/3) + 4^(1/3)*(1
9*x^6 - 22*x^3 + 4) + 6*(5*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 - 4*x^3 + 4)) - 80*(-4)^(1/3)*x^5*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)) + 40*(-4)^(1/3)*x^5*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)) + 36*(13*x^3 - 8)*(x^3 - 1)^(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^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^3+4)/x^6/(x^6-x^3-2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 42.03, size = 1854, normalized size = 7.30

method	result	size
risch	\(\text {Expression too large to display}\)	\(1854\)
trager	\(\text {Expression too large to display}\)	\(1899\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(2/3)*(x^3+4)/x^6/(x^6-x^3-2),x,method=_RETURNVERBOSE)

[Out]

-1/20*(13*x^6-21*x^3+8)/x^5/(x^3-1)^(1/3)-1/3*ln((-12*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*Ro
otOf(_Z^3+4)^3*x^3+9*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+24*(x^3-1)^(
2/3)*RootOf(_Z^3+4)^2*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*x+8*(x^3-1)^(1/3)*RootOf(_Z^3+4)^2
*x^2-48*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)*(x^3-1)^(1/3)*x^2-16*RootOf(_Z^3+
4)*x^3+12*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*x^3+16*x*(x^3-1)^(2/3)+16*RootOf(_Z^3+4)-12*Ro
otOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2))/(x^2-x+1)/(1+x))*RootOf(_Z^3+4)-1/2*ln((-12*RootOf(4*Root
Of(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^3*x^3+9*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+
9*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+24*(x^3-1)^(2/3)*RootOf(_Z^3+4)^2*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)
+9*_Z^2)*x+8*(x^3-1)^(1/3)*RootOf(_Z^3+4)^2*x^2-48*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootO
f(_Z^3+4)*(x^3-1)^(1/3)*x^2-16*RootOf(_Z^3+4)*x^3+12*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*x^3
+16*x*(x^3-1)^(2/3)+16*RootOf(_Z^3+4)-12*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2))/(x^2-x+1)/(1+x
))*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)+1/2*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_
Z^2)*ln(-(-18*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^3*x^3-9*RootOf(4*RootOf(_Z^
3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+24*(x^3-1)^(2/3)*RootOf(_Z^3+4)^2*RootOf(4*RootOf(_Z
^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*x-40*(x^3-1)^(1/3)*RootOf(_Z^3+4)^2*x^2-48*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*
RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)*(x^3-1)^(1/3)*x^2+72*RootOf(_Z^3+4)*x^3+36*RootOf(4*RootOf(_Z^3+4)^2+6*_
Z*RootOf(_Z^3+4)+9*_Z^2)*x^3-80*x*(x^3-1)^(2/3)-24*RootOf(_Z^3+4)-12*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^
3+4)+9*_Z^2))/(x^2-x+1)/(1+x))+1/16*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)*ln(-(
-6*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^4*x^3+9*RootOf(4*RootOf(_Z^3+4)^2+6*_Z
*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^3*x^3+48*RootOf(_Z^3+4)^2*x^3-72*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootO
f(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)*x^3-96*RootOf(_Z^3+4)*(x^3-1)^(1/3)*x^2+192*x*(x^3-1)^(2/3)-32*RootOf(_Z^3+4)
^2+48*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4))/(x^3-2))-1/24*ln(-(3*RootOf(4*Root
Of(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^4*x^3-9*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+
9*_Z^2)^2*RootOf(_Z^3+4)^3*x^3+60*(x^3-1)^(2/3)*RootOf(_Z^3+4)^2*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)
+9*_Z^2)*x+16*RootOf(_Z^3+4)^2*x^3-48*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)*x^3
+16*RootOf(_Z^3+4)*(x^3-1)^(1/3)*x^2+120*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*(x^3-1)^(1/3)*x
^2-32*x*(x^3-1)^(2/3)-16*RootOf(_Z^3+4)^2+48*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3
+4))/(x^3-2))*RootOf(_Z^3+4)^2-1/16*ln(-(3*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4
)^4*x^3-9*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^3*x^3+60*(x^3-1)^(2/3)*RootOf
(_Z^3+4)^2*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*x+16*RootOf(_Z^3+4)^2*x^3-48*RootOf(4*RootOf(
_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)*x^3+16*RootOf(_Z^3+4)*(x^3-1)^(1/3)*x^2+120*RootOf(4*Root
Of(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*(x^3-1)^(1/3)*x^2-32*x*(x^3-1)^(2/3)-16*RootOf(_Z^3+4)^2+48*RootOf(4*
RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4))/(x^3-2))*RootOf(4*RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z
^3+4)+9*_Z^2)*RootOf(_Z^3+4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^3+4)/x^6/(x^6-x^3-2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(-((x^3 - 1)^(2/3)*(x^3 + 4))/(x^6*(x^3 - x^6 + 2)), 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 (x^{3} + 4\right )}{x^{6} \left (x + 1\right ) \left (x^{3} - 2\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)*(x**3+4)/x**6/(x**6-x**3-2),x)

[Out]

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

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