\(\int \frac {(2+x-x^3-x^4)^{2/3} (6+2 x+x^4) (-2-x+x^3+x^4)}{x^6 (-2-x+2 x^3+x^4)} \, dx\) [2041]

Optimal result

Integrand size = 55, antiderivative size = 146 \[ \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx=\frac {3 \left (2+x-x^3-x^4\right )^{2/3} \left (-4-2 x-3 x^3+2 x^4\right )}{10 x^5}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2+x-x^3-x^4}}\right )-\log \left (-x+\sqrt [3]{2+x-x^3-x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{2+x-x^3-x^4}+\left (2+x-x^3-x^4\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx=\int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx \]

[In]

Int[((2 + x - x^3 - x^4)^(2/3)*(6 + 2*x + x^4)*(-2 - x + x^3 + x^4))/(x^6*(-2 - x + 2*x^3 + x^4)),x]

[Out]

Defer[Int][(2 + x - x^3 - x^4)^(2/3)/(1 - x), x] + 6*Defer[Int][(2 + x - x^3 - x^4)^(2/3)/x^6, x] + 2*Defer[In
t][(2 + x - x^3 - x^4)^(2/3)/x^5, x] + 3*Defer[Int][(2 + x - x^3 - x^4)^(2/3)/x^3, x] + Defer[Int][(2 + x - x^
3 - x^4)^(2/3)/x^2, x]/2 + Defer[Int][(2 + x - x^3 - x^4)^(2/3)/x, x]/4 - Defer[Int][(2 + x - x^3 - x^4)^(2/3)
/(2 + x), x]/4 + (1 - I*Sqrt[3])*Defer[Int][(2 + x - x^3 - x^4)^(2/3)/(1 - I*Sqrt[3] + 2*x), x] + (1 + I*Sqrt[
3])*Defer[Int][(2 + x - x^3 - x^4)^(2/3)/(1 + I*Sqrt[3] + 2*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (2+x-x^3-x^4\right )^{2/3}}{1-x}+\frac {6 \left (2+x-x^3-x^4\right )^{2/3}}{x^6}+\frac {2 \left (2+x-x^3-x^4\right )^{2/3}}{x^5}+\frac {3 \left (2+x-x^3-x^4\right )^{2/3}}{x^3}+\frac {\left (2+x-x^3-x^4\right )^{2/3}}{2 x^2}+\frac {\left (2+x-x^3-x^4\right )^{2/3}}{4 x}-\frac {\left (2+x-x^3-x^4\right )^{2/3}}{4 (2+x)}+\frac {(2+x) \left (2+x-x^3-x^4\right )^{2/3}}{1+x+x^2}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x} \, dx-\frac {1}{4} \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{2+x} \, dx+\frac {1}{2} \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^2} \, dx+2 \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^5} \, dx+3 \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^3} \, dx+6 \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^6} \, dx+\int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{1-x} \, dx+\int \frac {(2+x) \left (2+x-x^3-x^4\right )^{2/3}}{1+x+x^2} \, dx \\ & = \frac {1}{4} \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x} \, dx-\frac {1}{4} \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{2+x} \, dx+\frac {1}{2} \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^2} \, dx+2 \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^5} \, dx+3 \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^3} \, dx+6 \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^6} \, dx+\int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{1-x} \, dx+\int \left (\frac {\left (1-i \sqrt {3}\right ) \left (2+x-x^3-x^4\right )^{2/3}}{1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \left (2+x-x^3-x^4\right )^{2/3}}{1+i \sqrt {3}+2 x}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x} \, dx-\frac {1}{4} \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{2+x} \, dx+\frac {1}{2} \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^2} \, dx+2 \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^5} \, dx+3 \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^3} \, dx+6 \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{x^6} \, dx+\left (1-i \sqrt {3}\right ) \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{1-i \sqrt {3}+2 x} \, dx+\left (1+i \sqrt {3}\right ) \int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{1+i \sqrt {3}+2 x} \, dx+\int \frac {\left (2+x-x^3-x^4\right )^{2/3}}{1-x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx=\frac {3 \left (2+x-x^3-x^4\right )^{2/3} \left (-4-2 x-3 x^3+2 x^4\right )}{10 x^5}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2+x-x^3-x^4}}\right )-\log \left (-x+\sqrt [3]{2+x-x^3-x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{2+x-x^3-x^4}+\left (2+x-x^3-x^4\right )^{2/3}\right ) \]

[In]

Integrate[((2 + x - x^3 - x^4)^(2/3)*(6 + 2*x + x^4)*(-2 - x + x^3 + x^4))/(x^6*(-2 - x + 2*x^3 + x^4)),x]

[Out]

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

Maple [A] (verified)

Time = 6.64 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(\frac {5 \ln \left (\frac {x^{2}+x \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}}+\left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-10 \ln \left (\frac {-x +\left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (6 x^{4}-9 x^{3}-6 x -12\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}}}{10 x^{5}}\)	\(149\)
risch	\(-\frac {3 \left (2 x^{8}-x^{7}-3 x^{6}-4 x^{5}-7 x^{4}+2 x^{3}+2 x^{2}+8 x +8\right )}{10 x^{5} \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}}}-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{4}+\left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x +\left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}-x -2}{\left (2+x \right ) \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +1\right ) \left (-1-x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x \right ) \left (-1+x \right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2 \left (-x^{4}-x^{3}+x +2\right )^{\frac {2}{3}} x +\left (-x^{4}-x^{3}+x +2\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\left (2+x \right ) \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +1\right ) \left (-1-x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x \right ) \left (-1+x \right )}\right )\)	\(434\)
trager	\(\text {Expression too large to display}\)	\(843\)

[In]

int((-x^4-x^3+x+2)^(2/3)*(x^4+2*x+6)*(x^4+x^3-x-2)/x^6/(x^4+2*x^3-x-2),x,method=_RETURNVERBOSE)

[Out]

1/10*(5*ln((x^2+x*(-x^4-x^3+x+2)^(1/3)+(-x^4-x^3+x+2)^(2/3))/x^2)*x^5-10*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(-x
^4-x^3+x+2)^(1/3)))*x^5-10*ln((-x+(-x^4-x^3+x+2)^(1/3))/x)*x^5+(6*x^4-9*x^3-6*x-12)*(-x^4-x^3+x+2)^(2/3))/x^5

Fricas [A] (verification not implemented)

none

Time = 1.99 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.40 \[ \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {49772 \, \sqrt {3} {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {1}{3}} x^{2} - 31378 \, \sqrt {3} {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (17661 \, x^{4} + 26125 \, x^{3} - 17661 \, x - 35322\right )}}{24389 \, x^{4} - 72947 \, x^{3} - 24389 \, x - 48778}\right ) + 5 \, x^{5} \log \left (\frac {x^{4} + 2 \, x^{3} - 3 \, {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {2}{3}} x - x - 2}{x^{4} + 2 \, x^{3} - x - 2}\right ) - 3 \, {\left (2 \, x^{4} - 3 \, x^{3} - 2 \, x - 4\right )} {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {2}{3}}}{10 \, x^{5}} \]

[In]

integrate((-x^4-x^3+x+2)^(2/3)*(x^4+2*x+6)*(x^4+x^3-x-2)/x^6/(x^4+2*x^3-x-2),x, algorithm="fricas")

[Out]

-1/10*(10*sqrt(3)*x^5*arctan(-(49772*sqrt(3)*(-x^4 - x^3 + x + 2)^(1/3)*x^2 - 31378*sqrt(3)*(-x^4 - x^3 + x +
2)^(2/3)*x - sqrt(3)*(17661*x^4 + 26125*x^3 - 17661*x - 35322))/(24389*x^4 - 72947*x^3 - 24389*x - 48778)) + 5
*x^5*log((x^4 + 2*x^3 - 3*(-x^4 - x^3 + x + 2)^(1/3)*x^2 + 3*(-x^4 - x^3 + x + 2)^(2/3)*x - x - 2)/(x^4 + 2*x^
3 - x - 2)) - 3*(2*x^4 - 3*x^3 - 2*x - 4)*(-x^4 - x^3 + x + 2)^(2/3))/x^5

Sympy [F]

\[ \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx=\int \frac {\left (x^{4} + 2 x + 6\right ) \left (- x^{4} - x^{3} + x + 2\right )^{\frac {2}{3}} \left (x^{4} + x^{3} - x - 2\right )}{x^{6} \left (x - 1\right ) \left (x + 2\right ) \left (x^{2} + x + 1\right )}\, dx \]

[In]

integrate((-x**4-x**3+x+2)**(2/3)*(x**4+2*x+6)*(x**4+x**3-x-2)/x**6/(x**4+2*x**3-x-2),x)

[Out]

Integral((x**4 + 2*x + 6)*(-x**4 - x**3 + x + 2)**(2/3)*(x**4 + x**3 - x - 2)/(x**6*(x - 1)*(x + 2)*(x**2 + x
+ 1)), x)

Maxima [F]

\[ \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3} - x - 2\right )} {\left (x^{4} + 2 \, x + 6\right )} {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {2}{3}}}{{\left (x^{4} + 2 \, x^{3} - x - 2\right )} x^{6}} \,d x } \]

[In]

integrate((-x^4-x^3+x+2)^(2/3)*(x^4+2*x+6)*(x^4+x^3-x-2)/x^6/(x^4+2*x^3-x-2),x, algorithm="maxima")

[Out]

integrate((x^4 + x^3 - x - 2)*(x^4 + 2*x + 6)*(-x^4 - x^3 + x + 2)^(2/3)/((x^4 + 2*x^3 - x - 2)*x^6), x)

Giac [F]

\[ \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3} - x - 2\right )} {\left (x^{4} + 2 \, x + 6\right )} {\left (-x^{4} - x^{3} + x + 2\right )}^{\frac {2}{3}}}{{\left (x^{4} + 2 \, x^{3} - x - 2\right )} x^{6}} \,d x } \]

[In]

integrate((-x^4-x^3+x+2)^(2/3)*(x^4+2*x+6)*(x^4+x^3-x-2)/x^6/(x^4+2*x^3-x-2),x, algorithm="giac")

[Out]

integrate((x^4 + x^3 - x - 2)*(x^4 + 2*x + 6)*(-x^4 - x^3 + x + 2)^(2/3)/((x^4 + 2*x^3 - x - 2)*x^6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (2+x-x^3-x^4\right )^{2/3} \left (6+2 x+x^4\right ) \left (-2-x+x^3+x^4\right )}{x^6 \left (-2-x+2 x^3+x^4\right )} \, dx=\int \frac {\left (x^4+2\,x+6\right )\,{\left (-x^4-x^3+x+2\right )}^{5/3}}{x^6\,\left (-x^4-2\,x^3+x+2\right )} \,d x \]

[In]

int(((2*x + x^4 + 6)*(x - x^3 - x^4 + 2)^(5/3))/(x^6*(x - 2*x^3 - x^4 + 2)),x)

[Out]

int(((2*x + x^4 + 6)*(x - x^3 - x^4 + 2)^(5/3))/(x^6*(x - 2*x^3 - x^4 + 2)), x)