\(\int \frac {(-3+x) (-2+x) (2-x+2 x^3)^{2/3}}{x^6 (-2+x+2 x^3)} \, dx\) [2211]

Optimal result

Integrand size = 34, antiderivative size = 164 \[ \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx=\frac {3 \left (2-x+2 x^3\right )^{2/3} \left (2-x+7 x^3\right )}{10 x^5}-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+\sqrt [3]{2} \sqrt [3]{2-x+2 x^3}}\right )+2 \sqrt [3]{2} \log \left (-2 x+\sqrt [3]{2} \sqrt [3]{2-x+2 x^3}\right )-\sqrt [3]{2} \log \left (4 x^2+2 \sqrt [3]{2} x \sqrt [3]{2-x+2 x^3}+2^{2/3} \left (2-x+2 x^3\right )^{2/3}\right ) \]

[Out]

3/10*(2*x^3-x+2)^(2/3)*(7*x^3-x+2)/x^5-2*2^(1/3)*3^(1/2)*arctan(3^(1/2)*x/(x+2^(1/3)*(2*x^3-x+2)^(1/3)))+2*2^(
1/3)*ln(-2*x+2^(1/3)*(2*x^3-x+2)^(1/3))-2^(1/3)*ln(4*x^2+2*2^(1/3)*x*(2*x^3-x+2)^(1/3)+2^(2/3)*(2*x^3-x+2)^(2/
3))

Rubi [F]

\[ \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx=\int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx \]

[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*} \text {integral}& = \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 \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.86 \[ \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx=\frac {3 \left (2-x+2 x^3\right )^{2/3} \left (2-x+7 x^3\right )}{10 x^5}-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+\sqrt [3]{4-2 x+4 x^3}}\right )+2 \sqrt [3]{2} \log \left (-2 x+\sqrt [3]{4-2 x+4 x^3}\right )-\sqrt [3]{2} \log \left (4 x^2+2 x \sqrt [3]{4-2 x+4 x^3}+\left (4-2 x+4 x^3\right )^{2/3}\right ) \]

[In]

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

Maple [A] (verified)

Time = 15.89 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\frac {20 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {1}{3}} \left (2 x^{3}-x +2\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}+20 \,2^{\frac {1}{3}} \ln \left (\frac {-2^{\frac {2}{3}} x +\left (2 x^{3}-x +2\right )^{\frac {1}{3}}}{x}\right ) x^{5}-10 \,2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} \left (2 x^{3}-x +2\right )^{\frac {1}{3}} x +2 \,2^{\frac {1}{3}} x^{2}+\left (2 x^{3}-x +2\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+21 \left (2 x^{3}-x +2\right )^{\frac {2}{3}} x^{3}-3 \left (2 x^{3}-x +2\right )^{\frac {2}{3}} x +6 \left (2 x^{3}-x +2\right )^{\frac {2}{3}}}{10 x^{5}}\)	\(174\)
risch	\(\text {Expression too large to display}\)	\(670\)
trager	\(\text {Expression too large to display}\)	\(1014\)

[In]

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

[Out]

1/10*(20*3^(1/2)*2^(1/3)*arctan(1/3*3^(1/2)/x*(x+2^(1/3)*(2*x^3-x+2)^(1/3)))*x^5+20*2^(1/3)*ln((-2^(2/3)*x+(2*
x^3-x+2)^(1/3))/x)*x^5-10*2^(1/3)*ln((2^(2/3)*(2*x^3-x+2)^(1/3)*x+2*2^(1/3)*x^2+(2*x^3-x+2)^(2/3))/x^2)*x^5+21
*(2*x^3-x+2)^(2/3)*x^3-3*(2*x^3-x+2)^(2/3)*x+6*(2*x^3-x+2)^(2/3))/x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (134) = 268\).

Time = 6.59 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.59 \[ \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx=-\frac {20 \, \sqrt {3} 2^{\frac {1}{3}} x^{5} \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (20 \, x^{7} + 8 \, x^{5} - 16 \, x^{4} - x^{3} + 4 \, x^{2} - 4 \, x\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} - 12 \, \sqrt {3} 2^{\frac {1}{3}} {\left (76 \, x^{8} - 32 \, x^{6} + 64 \, x^{5} + x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {1}{3}} - \sqrt {3} {\left (568 \, x^{9} - 444 \, x^{7} + 888 \, x^{6} + 66 \, x^{5} - 264 \, x^{4} + 263 \, x^{3} + 6 \, x^{2} - 12 \, x + 8\right )}}{3 \, {\left (872 \, x^{9} - 420 \, x^{7} + 840 \, x^{6} + 6 \, x^{5} - 24 \, x^{4} + 25 \, x^{3} - 6 \, x^{2} + 12 \, x - 8\right )}}\right ) - 20 \cdot 2^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 2^{\frac {2}{3}} {\left (2 \, x^{3} - x + 2\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} x - 2^{\frac {1}{3}} {\left (2 \, x^{3} + x - 2\right )}}{2 \, x^{3} + x - 2}\right ) + 10 \cdot 2^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (10 \, x^{4} - x^{2} + 2 \, x\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (76 \, x^{6} - 32 \, x^{4} + 64 \, x^{3} + x^{2} - 4 \, x + 4\right )} + 24 \, {\left (4 \, x^{5} - x^{3} + 2 \, x^{2}\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {1}{3}}}{4 \, x^{6} + 4 \, x^{4} - 8 \, x^{3} + x^{2} - 4 \, x + 4}\right ) - 9 \, {\left (7 \, x^{3} - x + 2\right )} {\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}}}{30 \, x^{5}} \]

[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

Sympy [F(-1)]

Timed out. \[ \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} {\left (x - 2\right )} {\left (x - 3\right )}}{{\left (2 \, x^{3} + x - 2\right )} x^{6}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - x + 2\right )}^{\frac {2}{3}} {\left (x - 2\right )} {\left (x - 3\right )}}{{\left (2 \, x^{3} + x - 2\right )} x^{6}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(-3+x) (-2+x) \left (2-x+2 x^3\right )^{2/3}}{x^6 \left (-2+x+2 x^3\right )} \, dx=\int \frac {\left (x-2\right )\,\left (x-3\right )\,{\left (2\,x^3-x+2\right )}^{2/3}}{x^6\,\left (2\,x^3+x-2\right )} \,d x \]

[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)