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

Optimal. Leaf size=164 \[ 2 \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{2 x^3-x+2}-2 x\right )-2 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt [3]{2} \sqrt [3]{2 x^3-x+2}+x}\right )+\frac {3 \left (2 x^3-x+2\right )^{2/3} \left (7 x^3-x+2\right )}{10 x^5}-\sqrt [3]{2} \log \left (2 \sqrt [3]{2} \sqrt [3]{2 x^3-x+2} x+2^{2/3} \left (2 x^3-x+2\right )^{2/3}+4 x^2\right ) \]

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Rubi [F]  time = 4.04, 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+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-27*(2 - x + 2*x^3)^(2/3)*Defer[Int][(((6^(2/3) + 6^(1/3)*(18 - Sqrt[318])^(2/3))/(3*(18 - Sqrt[318])^(1/3))
+ 2*x)^(2/3)*((-6 + (108 - 6*Sqrt[318])^(2/3) + (6*6^(1/3))/(18 - Sqrt[318])^(2/3))/9 - (2*(18 + Sqrt[318])^(1
/3)*(6^(1/3) + (18 - Sqrt[318])^(2/3))*x)/3 + 4*x^2)^(2/3))/x^6, x])/(((108 - 6*Sqrt[318])^(1/3) + 6^(2/3)/(18
- Sqrt[318])^(1/3) + 6*x)^(2/3)*(-6 + (108 - 6*Sqrt[318])^(2/3) + (6*6^(1/3))/(18 - Sqrt[318])^(2/3) - 6*(18
+ Sqrt[318])^(1/3)*(6^(1/3) + (18 - Sqrt[318])^(2/3))*x + 36*x^2)^(2/3)) + (9*(2 - x + 2*x^3)^(2/3)*Defer[Int]
[(((6^(2/3) + 6^(1/3)*(18 - Sqrt[318])^(2/3))/(3*(18 - Sqrt[318])^(1/3)) + 2*x)^(2/3)*((-6 + (108 - 6*Sqrt[318
])^(2/3) + (6*6^(1/3))/(18 - Sqrt[318])^(2/3))/9 - (2*(18 + Sqrt[318])^(1/3)*(6^(1/3) + (18 - Sqrt[318])^(2/3)
)*x)/3 + 4*x^2)^(2/3))/x^5, x])/(((108 - 6*Sqrt[318])^(1/3) + 6^(2/3)/(18 - Sqrt[318])^(1/3) + 6*x)^(2/3)*(-6
+ (108 - 6*Sqrt[318])^(2/3) + (6*6^(1/3))/(18 - Sqrt[318])^(2/3) - 6*(18 + Sqrt[318])^(1/3)*(6^(1/3) + (18 - S
qrt[318])^(2/3))*x + 36*x^2)^(2/3)) - (27*(2 - x + 2*x^3)^(2/3)*Defer[Int][(((6^(2/3) + 6^(1/3)*(18 - Sqrt[318
])^(2/3))/(3*(18 - Sqrt[318])^(1/3)) + 2*x)^(2/3)*((-6 + (108 - 6*Sqrt[318])^(2/3) + (6*6^(1/3))/(18 - Sqrt[31
8])^(2/3))/9 - (2*(18 + Sqrt[318])^(1/3)*(6^(1/3) + (18 - Sqrt[318])^(2/3))*x)/3 + 4*x^2)^(2/3))/x^3, x])/(((1
08 - 6*Sqrt[318])^(1/3) + 6^(2/3)/(18 - Sqrt[318])^(1/3) + 6*x)^(2/3)*(-6 + (108 - 6*Sqrt[318])^(2/3) + (6*6^(
1/3))/(18 - Sqrt[318])^(2/3) - 6*(18 + Sqrt[318])^(1/3)*(6^(1/3) + (18 - Sqrt[318])^(2/3))*x + 36*x^2)^(2/3))
- (9*(2 - x + 2*x^3)^(2/3)*Defer[Int][(((6^(2/3) + 6^(1/3)*(18 - Sqrt[318])^(2/3))/(3*(18 - Sqrt[318])^(1/3))
+ 2*x)^(2/3)*((-6 + (108 - 6*Sqrt[318])^(2/3) + (6*6^(1/3))/(18 - Sqrt[318])^(2/3))/9 - (2*(18 + Sqrt[318])^(1
/3)*(6^(1/3) + (18 - Sqrt[318])^(2/3))*x)/3 + 4*x^2)^(2/3))/x^2, x])/(2*((108 - 6*Sqrt[318])^(1/3) + 6^(2/3)/(
18 - Sqrt[318])^(1/3) + 6*x)^(2/3)*(-6 + (108 - 6*Sqrt[318])^(2/3) + (6*6^(1/3))/(18 - Sqrt[318])^(2/3) - 6*(1
8 + Sqrt[318])^(1/3)*(6^(1/3) + (18 - Sqrt[318])^(2/3))*x + 36*x^2)^(2/3)) - (9*(2 - x + 2*x^3)^(2/3)*Defer[In
t][(((6^(2/3) + 6^(1/3)*(18 - Sqrt[318])^(2/3))/(3*(18 - Sqrt[318])^(1/3)) + 2*x)^(2/3)*((-6 + (108 - 6*Sqrt[3
18])^(2/3) + (6*6^(1/3))/(18 - Sqrt[318])^(2/3))/9 - (2*(18 + Sqrt[318])^(1/3)*(6^(1/3) + (18 - Sqrt[318])^(2/
3))*x)/3 + 4*x^2)^(2/3))/x, x])/(4*((108 - 6*Sqrt[318])^(1/3) + 6^(2/3)/(18 - Sqrt[318])^(1/3) + 6*x)^(2/3)*(-
6 + (108 - 6*Sqrt[318])^(2/3) + (6*6^(1/3))/(18 - Sqrt[318])^(2/3) - 6*(18 + Sqrt[318])^(1/3)*(6^(1/3) + (18 -
Sqrt[318])^(2/3))*x + 36*x^2)^(2/3)) + (25*Defer[Int][(2 - x + 2*x^3)^(2/3)/(-2 + x + 2*x^3), x])/4 + Defer[I
nt][(x*(2 - x + 2*x^3)^(2/3))/(-2 + x + 2*x^3), x] + Defer[Int][(x^2*(2 - x + 2*x^3)^(2/3))/(-2 + x + 2*x^3),
x]/2

Rubi steps

\begin {align*} \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx &=\int \left (-\frac {3 \left (2-x+2 x^3\right )^{2/3}}{x^6}+\frac {\left (2-x+2 x^3\right )^{2/3}}{x^5}-\frac {3 \left (2-x+2 x^3\right )^{2/3}}{x^3}-\frac {\left (2-x+2 x^3\right )^{2/3}}{2 x^2}-\frac {\left (2-x+2 x^3\right )^{2/3}}{4 x}+\frac {\left (25+4 x+2 x^2\right ) \left (2-x+2 x^3\right )^{2/3}}{4 \left (-2+x+2 x^3\right )}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\left (2-x+2 x^3\right )^{2/3}}{x} \, dx\right )+\frac {1}{4} \int \frac {\left (25+4 x+2 x^2\right ) \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3} \, dx-\frac {1}{2} \int \frac {\left (2-x+2 x^3\right )^{2/3}}{x^2} \, dx-3 \int \frac {\left (2-x+2 x^3\right )^{2/3}}{x^6} \, dx-3 \int \frac {\left (2-x+2 x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (2-x+2 x^3\right )^{2/3}}{x^5} \, dx\\ &=\frac {1}{4} \int \left (\frac {25 \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3}+\frac {4 x \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3}+\frac {2 x^2 \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3}\right ) \, dx-\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x} \, dx}{4 \left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^2} \, dx}{2 \left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}+\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^5} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (3 \left (2-x+2 x^3\right )^{2/3}\right ) \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^6} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (3 \left (2-x+2 x^3\right )^{2/3}\right ) \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^3} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}\\ &=\frac {1}{2} \int \frac {x^2 \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3} \, dx+\frac {25}{4} \int \frac {\left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3} \, dx-\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x} \, dx}{4 \left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^2} \, dx}{2 \left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}+\frac {\left (2-x+2 x^3\right )^{2/3} \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^5} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (3 \left (2-x+2 x^3\right )^{2/3}\right ) \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^6} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}-\frac {\left (3 \left (2-x+2 x^3\right )^{2/3}\right ) \int \frac {\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}{x^3} \, dx}{\left (\frac {6^{2/3}+\sqrt [3]{6} \left (18-\sqrt {318}\right )^{2/3}}{3 \sqrt [3]{18-\sqrt {318}}}+2 x\right )^{2/3} \left (\frac {1}{9} \left (-6+\left (108-6 \sqrt {318}\right )^{2/3}+\frac {6 \sqrt [3]{6}}{\left (18-\sqrt {318}\right )^{2/3}}\right )-\frac {2}{3} \sqrt [3]{18+\sqrt {318}} \left (\sqrt [3]{6}+\left (18-\sqrt {318}\right )^{2/3}\right ) x+4 x^2\right )^{2/3}}+\int \frac {x \left (2-x+2 x^3\right )^{2/3}}{-2+x+2 x^3} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.57, size = 164, normalized size = 1.00 \begin {gather*} \frac {3 \left (2-x+2 x^3\right )^{2/3} \left (2-x+7 x^3\right )}{10 x^5}-2 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+\sqrt [3]{2} \sqrt [3]{2-x+2 x^3}}\right )+2 \sqrt [3]{2} \log \left (-2 x+\sqrt [3]{2} \sqrt [3]{2-x+2 x^3}\right )-\sqrt [3]{2} \log \left (4 x^2+2 \sqrt [3]{2} x \sqrt [3]{2-x+2 x^3}+2^{2/3} \left (2-x+2 x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(2 - x + 2*x^3)^(2/3)*(2 - x + 7*x^3))/(10*x^5) - 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(1/3)*(2 - x
+ 2*x^3)^(1/3))] + 2*2^(1/3)*Log[-2*x + 2^(1/3)*(2 - x + 2*x^3)^(1/3)] - 2^(1/3)*Log[4*x^2 + 2*2^(1/3)*x*(2 -
x + 2*x^3)^(1/3) + 2^(2/3)*(2 - x + 2*x^3)^(2/3)]

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fricas [B]  time = 12.52, size = 424, normalized size = 2.59 \begin {gather*} -\frac {20 \, \sqrt {3} 2^{\frac {1}{3}} x^{5} \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (20 \, x^{7} + 8 \, x^{5} - 16 \, x^{4} - x^{3} + 4 \, x^{2} - 4 \, x\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} - 12 \, \sqrt {3} 2^{\frac {1}{3}} {\left (76 \, x^{8} - 32 \, x^{6} + 64 \, x^{5} + x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {1}{3}} - \sqrt {3} {\left (568 \, x^{9} - 444 \, x^{7} + 888 \, x^{6} + 66 \, x^{5} - 264 \, x^{4} + 263 \, x^{3} + 6 \, x^{2} - 12 \, x + 8\right )}}{3 \, {\left (872 \, x^{9} - 420 \, x^{7} + 840 \, x^{6} + 6 \, x^{5} - 24 \, x^{4} + 25 \, x^{3} - 6 \, x^{2} + 12 \, x - 8\right )}}\right ) - 20 \cdot 2^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 2^{\frac {2}{3}} {\left (2 \, x^{3} - x + 2\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} x - 2^{\frac {1}{3}} {\left (2 \, x^{3} + x - 2\right )}}{2 \, x^{3} + x - 2}\right ) + 10 \cdot 2^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (10 \, x^{4} - x^{2} + 2 \, x\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (76 \, x^{6} - 32 \, x^{4} + 64 \, x^{3} + x^{2} - 4 \, x + 4\right )} + 24 \, {\left (4 \, x^{5} - x^{3} + 2 \, x^{2}\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {1}{3}}}{4 \, x^{6} + 4 \, x^{4} - 8 \, x^{3} + x^{2} - 4 \, x + 4}\right ) - 9 \, {\left (7 \, x^{3} - x + 2\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}}}{30 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(20*sqrt(3)*2^(1/3)*x^5*arctan(1/3*(6*sqrt(3)*2^(2/3)*(20*x^7 + 8*x^5 - 16*x^4 - x^3 + 4*x^2 - 4*x)*(2*x
^3 - x + 2)^(2/3) - 12*sqrt(3)*2^(1/3)*(76*x^8 - 32*x^6 + 64*x^5 + x^4 - 4*x^3 + 4*x^2)*(2*x^3 - x + 2)^(1/3)
- sqrt(3)*(568*x^9 - 444*x^7 + 888*x^6 + 66*x^5 - 264*x^4 + 263*x^3 + 6*x^2 - 12*x + 8))/(872*x^9 - 420*x^7 +
840*x^6 + 6*x^5 - 24*x^4 + 25*x^3 - 6*x^2 + 12*x - 8)) - 20*2^(1/3)*x^5*log(-(6*2^(2/3)*(2*x^3 - x + 2)^(1/3)*
x^2 - 6*(2*x^3 - x + 2)^(2/3)*x - 2^(1/3)*(2*x^3 + x - 2))/(2*x^3 + x - 2)) + 10*2^(1/3)*x^5*log((6*2^(1/3)*(1
0*x^4 - x^2 + 2*x)*(2*x^3 - x + 2)^(2/3) + 2^(2/3)*(76*x^6 - 32*x^4 + 64*x^3 + x^2 - 4*x + 4) + 24*(4*x^5 - x^
3 + 2*x^2)*(2*x^3 - x + 2)^(1/3))/(4*x^6 + 4*x^4 - 8*x^3 + x^2 - 4*x + 4)) - 9*(7*x^3 - x + 2)*(2*x^3 - x + 2)
^(2/3))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 19.08, size = 671, normalized size = 4.09

method	result	size
risch	\(\frac {-\frac {6}{5} x +\frac {6}{5}+\frac {21}{5} x^{6}+\frac {3}{10} x^{2}+\frac {27}{5} x^{3}-\frac {27}{10} x^{4}}{x^{5} \left (2 x^{3}-x +2\right )^{\frac {1}{3}}}+8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \ln \left (-\frac {24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+64 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-16 \left (2 x^{3}-x +2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +8 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (2 x^{3}-x +2\right )^{\frac {1}{3}} x^{2}+40 \left (2 x^{3}-x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}-6 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-16 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) x^{3}+2 \left (2 x^{3}-x +2\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{3}-2\right ) x +8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) x -6 \RootOf \left (\textit {\_Z}^{3}-2\right )-16 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right )}{2 x^{3}+x -2}\right )+2 \RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (\frac {8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+16 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-12 \left (2 x^{3}-x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) x^{3}-3 \left (2 x^{3}-x +2\right )^{\frac {2}{3}} x -\RootOf \left (\textit {\_Z}^{3}-2\right ) x -2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right ) x +2 \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+16 \textit {\_Z}^{2}\right )}{2 x^{3}+x -2}\right )\)	\(671\)
trager	\(\text {Expression too large to display}\)	\(1499\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(14*x^6-9*x^4+18*x^3+x^2-4*x+4)/x^5/(2*x^3-x+2)^(1/3)+8*RootOf(RootOf(_Z^3-2)^2+4*_Z*RootOf(_Z^3-2)+16*_Z
^2)*ln(-(24*RootOf(RootOf(_Z^3-2)^2+4*_Z*RootOf(_Z^3-2)+16*_Z^2)*RootOf(_Z^3-2)^3*x^3+64*RootOf(RootOf(_Z^3-2)
^2+4*_Z*RootOf(_Z^3-2)+16*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-16*(2*x^3-x+2)^(2/3)*RootOf(RootOf(_Z^3-2)^2+4*_Z*RootO
f(_Z^3-2)+16*_Z^2)*RootOf(_Z^3-2)^2*x+8*RootOf(_Z^3-2)^2*(2*x^3-x+2)^(1/3)*x^2+40*(2*x^3-x+2)^(1/3)*RootOf(Roo
tOf(_Z^3-2)^2+4*_Z*RootOf(_Z^3-2)+16*_Z^2)*RootOf(_Z^3-2)*x^2-6*RootOf(_Z^3-2)*x^3-16*RootOf(RootOf(_Z^3-2)^2+
4*_Z*RootOf(_Z^3-2)+16*_Z^2)*x^3+2*(2*x^3-x+2)^(2/3)*x+3*RootOf(_Z^3-2)*x+8*RootOf(RootOf(_Z^3-2)^2+4*_Z*RootO
f(_Z^3-2)+16*_Z^2)*x-6*RootOf(_Z^3-2)-16*RootOf(RootOf(_Z^3-2)^2+4*_Z*RootOf(_Z^3-2)+16*_Z^2))/(2*x^3+x-2))+2*
RootOf(_Z^3-2)*ln((8*RootOf(RootOf(_Z^3-2)^2+4*_Z*RootOf(_Z^3-2)+16*_Z^2)*RootOf(_Z^3-2)^3*x^3+16*RootOf(RootO
f(_Z^3-2)^2+4*_Z*RootOf(_Z^3-2)+16*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-12*(2*x^3-x+2)^(1/3)*RootOf(RootOf(_Z^3-2)^2+4
*_Z*RootOf(_Z^3-2)+16*_Z^2)*RootOf(_Z^3-2)*x^2+6*RootOf(_Z^3-2)*x^3+12*RootOf(RootOf(_Z^3-2)^2+4*_Z*RootOf(_Z^
3-2)+16*_Z^2)*x^3-3*(2*x^3-x+2)^(2/3)*x-RootOf(_Z^3-2)*x-2*RootOf(RootOf(_Z^3-2)^2+4*_Z*RootOf(_Z^3-2)+16*_Z^2
)*x+2*RootOf(_Z^3-2)+4*RootOf(RootOf(_Z^3-2)^2+4*_Z*RootOf(_Z^3-2)+16*_Z^2))/(2*x^3+x-2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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