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

Optimal. Leaf size=151 \[ -\frac {5}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3+1}-2 x\right )+\frac {5\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+1}+x}\right )}{\sqrt {3}}+\frac {5 \log \left (2^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{2} \left (x^3+1\right )^{2/3}+2 x^2\right )}{3 \sqrt [3]{2}}+\frac {\left (x^3+1\right )^{2/3} \left (-17 x^6+10 x^3+2\right )}{5 x^5 \left (x^3-1\right )} \]

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Rubi [F]  time = 1.86, 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+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 + x^6))/(x^6*(-1 + x^3)^2),x]

[Out]

(-2*(1 + x^3)^(2/3))/x^2 - (2*(1 + x^3)^(5/3))/(5*x^5) + (4*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/Sqrt[
3] - 2*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*S
qrt[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 - 2*Defer[Int][(1 + x^3)^(2/3)/
(-1 + x), x] + (2*(9 - (8*I)*Sqrt[3])*Defer[Int][(1 + x^3)^(2/3)/(1 - I*Sqrt[3] + 2*x), x])/9 - (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] + (2*(9 + (8*I)*Sqr
t[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+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{3 (-1+x)^2}-\frac {2 \left (1+x^3\right )^{2/3}}{-1+x}+\frac {2 \left (1+x^3\right )^{2/3}}{x^6}+\frac {4 \left (1+x^3\right )^{2/3}}{x^3}+\frac {(1+x) \left (1+x^3\right )^{2/3}}{\left (1+x+x^2\right )^2}+\frac {(11+6 x) \left (1+x^3\right )^{2/3}}{3 \left (1+x+x^2\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{(-1+x)^2} \, dx+\frac {1}{3} \int \frac {(11+6 x) \left (1+x^3\right )^{2/3}}{1+x+x^2} \, dx-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+2 \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+4 \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 {2 \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 (6-\frac {16 i}{\sqrt {3}}\right ) \left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x}+\frac {\left (6+\frac {16 i}{\sqrt {3}}\right ) \left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x}\right ) \, dx-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+4 \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 {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {4 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-2 \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-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \left (2 \left (9-8 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 (9+8 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 {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {4 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-2 \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-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \left (2 \left (9-8 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 (9+8 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 {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {4 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-2 \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-2 \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 (9-8 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 (9+8 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 = 0.50, size = 138, normalized size = 0.91 \begin {gather*} \frac {5 \left (-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 )\right )}{3 \sqrt [3]{2}}+\left (x^3+1\right )^{2/3} \left (-\frac {2}{5 x^5}-\frac {x}{x^3-1}-\frac {12}{5 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x^3)^(2/3)*(-2/(5*x^5) - 12/(5*x^2) - x/(-1 + x^3)) + (5*(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+10 x^3-17 x^6\right )}{5 x^5 \left (-1+x^3\right )}+\frac {5\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {5}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+\frac {5 \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 + x^6))/(x^6*(-1 + x^3)^2),x]

[Out]

((1 + x^3)^(2/3)*(2 + 10*x^3 - 17*x^6))/(5*x^5*(-1 + x^3)) + (5*2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x
^3)^(1/3))])/Sqrt[3] - (5*2^(2/3)*Log[-2*x + 2^(2/3)*(1 + x^3)^(1/3)])/3 + (5*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 = 2.96, size = 297, normalized size = 1.97 \begin {gather*} -\frac {50 \, \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 ) - 50 \, \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 ) + 25 \, \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 (17 \, x^{6} - 10 \, 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^6/(x^3-1)^2,x, algorithm="fricas")

[Out]

-1/90*(50*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)) - 50*(-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)) + 25*(-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*(17*x^6
- 10*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\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^6/(x^3-1)^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 16.98, size = 768, normalized size = 5.09

method	result	size
trager	\(-\frac {\left (17 x^{6}-10 x^{3}-2\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{5 \left (x^{3}-1\right ) x^{5}}+2240 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \ln \left (\frac {-2544254963712 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}[phone] \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x [phone] \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}[phone] \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2}-16362409728 \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )-215807340672 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) x^{3}[phone] \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-148947870 x \left (x^{3}+1\right )^{\frac {2}{3}}-94652342400 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )[phone] \RootOf \left (\textit {\_Z}^{3}+4\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )+\frac {5 \RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (-\frac {1374442417152 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}-946523424 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}[phone] \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x [phone] \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}[phone] \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-10995539337216 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2}[phone] \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )-120672771744 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) x^{3}[phone] \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-144506961 x \left (x^{3}+1\right )^{\frac {2}{3}}-18407710944 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )[phone] \RootOf \left (\textit {\_Z}^{3}+4\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}\)	\(768\)
risch	\(-\frac {17 x^{9}+7 x^{6}-12 x^{3}-2}{5 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}} \left (x^{3}-1\right )}+\frac {5 \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 {5 \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}-10 \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 )\)	\(919\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(17*x^6-10*x^3-2)/(x^3-1)/x^5*(x^3+1)^(2/3)+2240*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_
Z^2)*ln((-2544254963712*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+20
45301216*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*RootOf(_Z^3+4)^3*x^3+147155162112*(x^3+1
)^(2/3)*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*RootOf(_Z^3+4)^2*x+194217355584*(x^3+1)^(
1/3)*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*RootOf(_Z^3+4)*x^2+218980896*RootOf(_Z^3+4)^
2*(x^3+1)^(1/3)*x^2+20354039709696*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)^2*RootOf(_Z^3+
4)^2-16362409728*RootOf(_Z^3+4)^3*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)-215807340672*Ro
otOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*x^3+173485371*RootOf(_Z^3+4)*x^3-148947870*x*(x^3+1
)^(2/3)-94652342400*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)[phone]*RootOf(_Z^3+4))/(-1+
x)/(x^2+x+1))+5/3*RootOf(_Z^3+4)*ln(-(1374442417152*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^
2)^2*RootOf(_Z^3+4)^2*x^3-946523424*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*RootOf(_Z^3+4
)^3*x^3+73577581056*(x^3+1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*RootOf(_Z^3+4)^
2*x+50046484320*(x^3+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*RootOf(_Z^3+4)*x^2+
109490448*RootOf(_Z^3+4)^2*(x^3+1)^(1/3)*x^2-10995539337216*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+180
6336*_Z^2)^2*RootOf(_Z^3+4)^2+7572187392*RootOf(_Z^3+4)^3*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+18063
36*_Z^2)-120672771744*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*x^3+83102503*RootOf(_Z^3+4)
*x^3-144506961*x*(x^3+1)^(2/3)-18407710944*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)+126766
53*RootOf(_Z^3+4))/(-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\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^6/(x^3-1)^2,x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

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