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

Optimal. Leaf size=151 \[ -\frac {7}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-1}-2 x\right )+\frac {7\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {7 \log \left (2^{2/3} \sqrt [3]{x^3-1} x+\sqrt [3]{2} \left (x^3-1\right )^{2/3}+2 x^2\right )}{3 \sqrt [3]{2}}+\frac {\left (x^3-1\right )^{2/3} \left (-22 x^6-15 x^3+2\right )}{5 x^5 \left (x^3+1\right )} \]

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Rubi [F]  time = 1.88, 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 (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-3*(-1 + x^3)^(2/3))/x^2 - (2*(-1 + x^3)^(5/3))/(5*x^5) + 2*Sqrt[3]*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[
3]] - 3*Log[-x + (-1 + x^3)^(1/3)] + (4*Defer[Int][(-1 + x^3)^(2/3)/(1 + I*Sqrt[3] - 2*x)^2, x])/3 - (2*(1 + I
*Sqrt[3])*Defer[Int][(-1 + x^3)^(2/3)/(1 + I*Sqrt[3] - 2*x)^2, x])/3 - (((2*I)/3)*Defer[Int][(-1 + x^3)^(2/3)/
(1 + I*Sqrt[3] - 2*x), x])/Sqrt[3] - Defer[Int][(-1 + x^3)^(2/3)/(1 + x)^2, x]/3 - (8*Defer[Int][(-1 + x^3)^(2
/3)/(1 + x), x])/3 + (2*(12 + (11*I)*Sqrt[3])*Defer[Int][(-1 + x^3)^(2/3)/(-1 - I*Sqrt[3] + 2*x), x])/9 + (4*D
efer[Int][(-1 + x^3)^(2/3)/(-1 + I*Sqrt[3] + 2*x)^2, x])/3 - (2*(1 - I*Sqrt[3])*Defer[Int][(-1 + x^3)^(2/3)/(-
1 + I*Sqrt[3] + 2*x)^2, x])/3 - (((2*I)/3)*Defer[Int][(-1 + x^3)^(2/3)/(-1 + I*Sqrt[3] + 2*x), x])/Sqrt[3] + (
2*(12 - (11*I)*Sqrt[3])*Defer[Int][(-1 + x^3)^(2/3)/(-1 + I*Sqrt[3] + 2*x), x])/9

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx &=\int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{x^6}+\frac {6 \left (-1+x^3\right )^{2/3}}{x^3}-\frac {\left (-1+x^3\right )^{2/3}}{3 (1+x)^2}-\frac {8 \left (-1+x^3\right )^{2/3}}{3 (1+x)}+\frac {(-1+x) \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2}+\frac {(-15+8 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)^2} \, dx\right )+\frac {1}{3} \int \frac {(-15+8 x) \left (-1+x^3\right )^{2/3}}{1-x+x^2} \, dx-2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+6 \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {(-1+x) \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{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)^2} \, dx+\frac {1}{3} \int \left (\frac {\left (8+\frac {22 i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{-1-i \sqrt {3}+2 x}+\frac {\left (8-\frac {22 i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x}\right ) \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+6 \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\int \left (-\frac {\left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2}+\frac {x \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2}\right ) \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-3 \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{(1+x)^2} \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+\frac {1}{9} \left (2 \left (12-11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (2 \left (12+11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1-i \sqrt {3}+2 x} \, dx-\int \frac {\left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx+\int \frac {x \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-3 \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{(1+x)^2} \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+\frac {1}{9} \left (2 \left (12-11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (2 \left (12+11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1-i \sqrt {3}+2 x} \, dx+\int \left (-\frac {2 \left (1+i \sqrt {3}\right ) \left (-1+x^3\right )^{2/3}}{3 \left (1+i \sqrt {3}-2 x\right )^2}+\frac {2 i \left (-1+x^3\right )^{2/3}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right )}-\frac {2 \left (1-i \sqrt {3}\right ) \left (-1+x^3\right )^{2/3}}{3 \left (-1+i \sqrt {3}+2 x\right )^2}+\frac {2 i \left (-1+x^3\right )^{2/3}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx-\int \left (-\frac {4 \left (-1+x^3\right )^{2/3}}{3 \left (1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \left (-1+x^3\right )^{2/3}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right )}-\frac {4 \left (-1+x^3\right )^{2/3}}{3 \left (-1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \left (-1+x^3\right )^{2/3}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-3 \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{(1+x)^2} \, dx+\frac {4}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{\left (1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {4}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{\left (-1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+\frac {(2 i) \int \frac {\left (-1+x^3\right )^{2/3}}{1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}+\frac {(2 i) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}-\frac {(4 i) \int \frac {\left (-1+x^3\right )^{2/3}}{1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}-\frac {(4 i) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}-\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{\left (-1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{\left (1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{9} \left (2 \left (12-11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (2 \left (12+11 i \sqrt {3}\right )\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 = 0.59, size = 146, normalized size = 0.97 \begin {gather*} \frac {7 \left (-2 \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac {2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )\right )}{3 \sqrt [3]{2}}+\left (x^3-1\right )^{2/3} \left (\frac {2}{5 x^5}-\frac {x}{x^3+1}-\frac {17}{5 x^2}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-1 + x^3)^(2/3)*(2/(5*x^5) - 17/(5*x^2) - x/(1 + x^3)) + (7*(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 = 0.48, size = 151, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (2-15 x^3-22 x^6\right )}{5 x^5 \left (1+x^3\right )}+\frac {7\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {7}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {7 \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)*(-2 + 2*x^3 + x^6))/(x^6*(1 + x^3)^2),x]

[Out]

((-1 + x^3)^(2/3)*(2 - 15*x^3 - 22*x^6))/(5*x^5*(1 + x^3)) + (7*2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 +
x^3)^(1/3))])/Sqrt[3] - (7*2^(2/3)*Log[-2*x + 2^(2/3)*(-1 + x^3)^(1/3)])/3 + (7*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 = 3.08, size = 294, normalized size = 1.95 \begin {gather*} -\frac {70 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \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 ) - 70 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \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 ) + 35 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \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 ) + 18 \, {\left (22 \, x^{6} + 15 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, {\left (x^{8} + x^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*(70*sqrt(3)*(-4)^(1/3)*(x^8 + x^5)*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(5*x^7 + 4*x^4 - x)*(x^3 - 1)^(2/3)
+ 6*sqrt(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)) - 70*(-4)^(1/3)*(x^8 + 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)) + 35*(-4)^(1/3)*(x^8 + 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)) + 18*(22*x^6
+ 15*x^3 - 2)*(x^3 - 1)^(2/3))/(x^8 + x^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 16.76, size = 635, normalized size = 4.21

method	result	size
risch	\(-\frac {22 x^{9}-7 x^{6}-17 x^{3}+2}{5 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}} \left (x^{3}+1\right )}+14 \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 ) \ln \left (\frac {9 \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 +4 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} 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}+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 ) x^{3}+2 x \left (x^{3}-1\right )^{\frac {2}{3}}-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 )+\frac {7 \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}+27 \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}+6 \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 +2 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}-3 \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}-27 \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}-5 x \left (x^{3}-1\right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{3}+4\right )+9 \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}\)	\(635\)
trager	\(\text {Expression too large to display}\)	\(1132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(22*x^9-7*x^6-17*x^3+2)/x^5/(x^3-1)^(1/3)/(x^3+1)+14*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)
*ln((9*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+4*(x^3-1)^(1/3)*RootOf(_Z^3+4)^2*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+12*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+
4)+36*_Z^2)*x^3+2*x*(x^3-1)^(2/3)-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))+7/3*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+27*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+6*(x^3-1)^(2/3)*RootOf(Roo
tOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*x+2*(x^3-1)^(1/3)*RootOf(_Z^3+4)^2*x^2-3*(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^3-27*RootOf(Ro
otOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^3-5*x*(x^3-1)^(2/3)+RootOf(_Z^3+4)+9*RootOf(RootOf(_Z^3+4)^2+6*_
Z*RootOf(_Z^3+4)+36*_Z^2))/(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^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )}^{2} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^6 + 2*x^3 - 2)*(x^3 - 1)^(2/3)/((x^3 + 1)^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^3-1\right )}^{2/3}\,\left (x^6+2\,x^3-2\right )}{x^6\,{\left (x^3+1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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