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

Optimal. Leaf size=175 \[ -\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3-x+1}-3 x\right )+\frac {3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-x+1}+\sqrt [3]{3} x}\right )}{2\ 2^{2/3}}-\frac {3 \left (x^3-x+1\right )^{2/3}}{4 x^2}+\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3-x+1} x+2^{2/3} \sqrt [3]{3} \left (x^3-x+1\right )^{2/3}+3 x^2\right ) \]

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Rubi [F]  time = 2.14, 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 {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(27*2^(1/3)*(1 - x + x^3)^(2/3)*Defer[Int][((((18 - 2*Sqrt[69])^(1/3) + (2*(9 + Sqrt[69]))^(1/3))/6^(2/3) + x)
^(2/3)*((-6 + 2^(1/3)*(27 - 3*Sqrt[69])^(2/3) + 6*3^(1/3)*(2/(9 - Sqrt[69]))^(2/3))/18 - (((9 - Sqrt[69])^(1/3
) + (9 + Sqrt[69])^(1/3))*x)/(2^(1/3)*3^(2/3)) + x^2)^(2/3))/x^3, x])/((6^(1/3)*((18 - 2*Sqrt[69])^(1/3) + (2*
(9 + Sqrt[69]))^(1/3)) + 6*x)^(2/3)*(-6 + 2^(1/3)*(27 - 3*Sqrt[69])^(2/3) + 6*3^(1/3)*(2/(9 - Sqrt[69]))^(2/3)
- 3*2^(2/3)*3^(1/3)*((9 - Sqrt[69])^(1/3) + (9 + Sqrt[69])^(1/3))*x + 18*x^2)^(2/3)) + (9*2^(1/3)*(1 - x + x^
3)^(2/3)*Defer[Int][((((18 - 2*Sqrt[69])^(1/3) + (2*(9 + Sqrt[69]))^(1/3))/6^(2/3) + x)^(2/3)*((-6 + 2^(1/3)*(
27 - 3*Sqrt[69])^(2/3) + 6*3^(1/3)*(2/(9 - Sqrt[69]))^(2/3))/18 - (((9 - Sqrt[69])^(1/3) + (9 + Sqrt[69])^(1/3
))*x)/(2^(1/3)*3^(2/3)) + x^2)^(2/3))/x^2, x])/((6^(1/3)*((18 - 2*Sqrt[69])^(1/3) + (2*(9 + Sqrt[69]))^(1/3))
+ 6*x)^(2/3)*(-6 + 2^(1/3)*(27 - 3*Sqrt[69])^(2/3) + 6*3^(1/3)*(2/(9 - Sqrt[69]))^(2/3) - 3*2^(2/3)*3^(1/3)*((
9 - Sqrt[69])^(1/3) + (9 + Sqrt[69])^(1/3))*x + 18*x^2)^(2/3)) + (9*2^(1/3)*(1 - x + x^3)^(2/3)*Defer[Int][(((
(18 - 2*Sqrt[69])^(1/3) + (2*(9 + Sqrt[69]))^(1/3))/6^(2/3) + x)^(2/3)*((-6 + 2^(1/3)*(27 - 3*Sqrt[69])^(2/3)
+ 6*3^(1/3)*(2/(9 - Sqrt[69]))^(2/3))/18 - (((9 - Sqrt[69])^(1/3) + (9 + Sqrt[69])^(1/3))*x)/(2^(1/3)*3^(2/3))
+ x^2)^(2/3))/x, x])/((6^(1/3)*((18 - 2*Sqrt[69])^(1/3) + (2*(9 + Sqrt[69]))^(1/3)) + 6*x)^(2/3)*(-6 + 2^(1/3
)*(27 - 3*Sqrt[69])^(2/3) + 6*3^(1/3)*(2/(9 - Sqrt[69]))^(2/3) - 3*2^(2/3)*3^(1/3)*((9 - Sqrt[69])^(1/3) + (9
+ Sqrt[69])^(1/3))*x + 18*x^2)^(2/3)) - (5*Defer[Int][(1 - x + x^3)^(2/3)/(-2 + 2*x + x^3), x])/2 - Defer[Int]
[(x*(1 - x + x^3)^(2/3))/(-2 + 2*x + x^3), x]/2 - Defer[Int][(x^2*(1 - x + x^3)^(2/3))/(-2 + 2*x + x^3), x]/2

Rubi steps

\begin {align*} \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx &=\int \left (\frac {3 \left (1-x+x^3\right )^{2/3}}{2 x^3}+\frac {\left (1-x+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1-x+x^3\right )^{2/3}}{2 x}+\frac {\left (-5-x-x^2\right ) \left (1-x+x^3\right )^{2/3}}{2 \left (-2+2 x+x^3\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{x^2} \, dx+\frac {1}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{x} \, dx+\frac {1}{2} \int \frac {\left (-5-x-x^2\right ) \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx+\frac {3}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{x^3} \, dx\\ &=\frac {1}{2} \int \left (-\frac {5 \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3}-\frac {x \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3}-\frac {x^2 \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3}\right ) \, dx+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^2} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (3 \left (1-x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^3} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}\\ &=-\left (\frac {1}{2} \int \frac {x \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx\right )-\frac {1}{2} \int \frac {x^2 \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx-\frac {5}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^2} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (3 \left (1-x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^3} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.58, size = 175, normalized size = 1.00 \begin {gather*} -\frac {3 \left (1-x+x^3\right )^{2/3}}{4 x^2}+\frac {3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1-x+x^3}}\right )}{2\ 2^{2/3}}-\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1-x+x^3}\right )+\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1-x+x^3}+2^{2/3} \sqrt [3]{3} \left (1-x+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(1 - x + x^3)^(2/3))/(4*x^2) + (3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(1 - x + x^3)^(1/3))])
/(2*2^(2/3)) - ((3/2)^(2/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(1 - x + x^3)^(1/3)])/2 + ((3/2)^(2/3)*Log[3*x^2 + 2^(1
/3)*3^(2/3)*x*(1 - x + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(1 - x + x^3)^(2/3)])/4

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fricas [B]  time = 15.74, size = 426, normalized size = 2.43 \begin {gather*} -\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-9\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (4 \cdot 4^{\frac {2}{3}} \left (-9\right )^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{5} - 7 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} - 2 \, x\right )} {\left (x^{3} - x + 1\right )}^{\frac {2}{3}} + 12 \, \left (-9\right )^{\frac {1}{3}} {\left (55 \, x^{8} - 50 \, x^{6} + 50 \, x^{5} + 4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2}\right )} {\left (x^{3} - x + 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{7} + 600 \, x^{6} + 204 \, x^{5} - 408 \, x^{4} + 196 \, x^{3} + 24 \, x^{2} - 24 \, x + 8\right )}\right )}}{6 \, {\left (487 \, x^{9} - 480 \, x^{7} + 480 \, x^{6} + 12 \, x^{5} - 24 \, x^{4} + 20 \, x^{3} - 24 \, x^{2} + 24 \, x - 8\right )}}\right ) - 2 \cdot 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} \left (-9\right )^{\frac {2}{3}} {\left (x^{3} - x + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} {\left (x^{3} + 2 \, x - 2\right )} - 36 \, {\left (x^{3} - x + 1\right )}^{\frac {2}{3}} x}{x^{3} + 2 \, x - 2}\right ) + 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {18 \cdot 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} {\left (4 \, x^{4} - x^{2} + x\right )} {\left (x^{3} - x + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-9\right )^{\frac {2}{3}} {\left (55 \, x^{6} - 50 \, x^{4} + 50 \, x^{3} + 4 \, x^{2} - 8 \, x + 4\right )} - 54 \, {\left (7 \, x^{5} - 4 \, x^{3} + 4 \, x^{2}\right )} {\left (x^{3} - x + 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} - 8 \, x + 4}\right ) + 36 \, {\left (x^{3} - x + 1\right )}^{\frac {2}{3}}}{48 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(4*4^(1/6)*sqrt(3)*(-9)^(1/3)*x^2*arctan(1/6*4^(1/6)*sqrt(3)*(4*4^(2/3)*(-9)^(2/3)*(4*x^7 + 7*x^5 - 7*x^
4 - 2*x^3 + 4*x^2 - 2*x)*(x^3 - x + 1)^(2/3) + 12*(-9)^(1/3)*(55*x^8 - 50*x^6 + 50*x^5 + 4*x^4 - 8*x^3 + 4*x^2
)*(x^3 - x + 1)^(1/3) - 4^(1/3)*(377*x^9 - 600*x^7 + 600*x^6 + 204*x^5 - 408*x^4 + 196*x^3 + 24*x^2 - 24*x + 8
))/(487*x^9 - 480*x^7 + 480*x^6 + 12*x^5 - 24*x^4 + 20*x^3 - 24*x^2 + 24*x - 8)) - 2*4^(2/3)*(-9)^(1/3)*x^2*lo
g(-(6*4^(1/3)*(-9)^(2/3)*(x^3 - x + 1)^(1/3)*x^2 + 4^(2/3)*(-9)^(1/3)*(x^3 + 2*x - 2) - 36*(x^3 - x + 1)^(2/3)
*x)/(x^3 + 2*x - 2)) + 4^(2/3)*(-9)^(1/3)*x^2*log(-(18*4^(2/3)*(-9)^(1/3)*(4*x^4 - x^2 + x)*(x^3 - x + 1)^(2/3
) - 4^(1/3)*(-9)^(2/3)*(55*x^6 - 50*x^4 + 50*x^3 + 4*x^2 - 8*x + 4) - 54*(7*x^5 - 4*x^3 + 4*x^2)*(x^3 - x + 1)
^(1/3))/(x^6 + 4*x^4 - 4*x^3 + 4*x^2 - 8*x + 4)) + 36*(x^3 - x + 1)^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 18.88, size = 635, normalized size = 3.63

method	result	size
risch	\(-\frac {3 \left (x^{3}-x +1\right )^{\frac {2}{3}}}{4 x^{2}}+\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \ln \left (-\frac {5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+14 \left (x^{3}-x +1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x +7 \left (x^{3}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}+48 \left (x^{3}-x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{2}+10 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}+36 x^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )+6 \left (x^{3}-x +1\right )^{\frac {2}{3}} x -10 \RootOf \left (\textit {\_Z}^{3}+18\right ) x -36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) x +10 \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )}{x^{3}+2 x -2}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{3}+18\right ) \ln \left (-\frac {-\RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}-3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+9 \left (x^{3}-x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{2}+5 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}+15 x^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )+9 \left (x^{3}-x +1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}+18\right ) x -6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) x +2 \RootOf \left (\textit {\_Z}^{3}+18\right )+6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )}{x^{3}+2 x -2}\right )}{4}\)	\(635\)
trager	\(\text {Expression too large to display}\)	\(1234\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(x^3-x+1)^(2/3)/x^2+3/2*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)*ln(-(5*RootOf(RootOf(_Z^3+
18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)*RootOf(_Z^3+18)^3*x^3+18*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*
_Z^2)^2*RootOf(_Z^3+18)^2*x^3+14*(x^3-x+1)^(2/3)*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)*RootOf
(_Z^3+18)^2*x+7*(x^3-x+1)^(1/3)*RootOf(_Z^3+18)^2*x^2+48*(x^3-x+1)^(1/3)*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(
_Z^3+18)+36*_Z^2)*RootOf(_Z^3+18)*x^2+10*RootOf(_Z^3+18)*x^3+36*x^3*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+
18)+36*_Z^2)+6*(x^3-x+1)^(2/3)*x-10*RootOf(_Z^3+18)*x-36*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2
)*x+10*RootOf(_Z^3+18)+36*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2))/(x^3+2*x-2))+1/4*RootOf(_Z^3
+18)*ln(-(-RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)*RootOf(_Z^3+18)^3*x^3-3*RootOf(RootOf(_Z^3+1
8)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)^2*RootOf(_Z^3+18)^2*x^3+9*(x^3-x+1)^(1/3)*RootOf(RootOf(_Z^3+18)^2+6*_Z*Roo
tOf(_Z^3+18)+36*_Z^2)*RootOf(_Z^3+18)*x^2+5*RootOf(_Z^3+18)*x^3+15*x^3*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z
^3+18)+36*_Z^2)+9*(x^3-x+1)^(2/3)*x-2*RootOf(_Z^3+18)*x-6*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^
2)*x+2*RootOf(_Z^3+18)+6*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2))/(x^3+2*x-2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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