\(\int \frac {(2+x^3) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx\) [696]

Optimal result

Integrand size = 42, antiderivative size = 54 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=-\text {RootSum}\left [-1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x-x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{1+2 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx \]

[In]

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

[Out]

(6*(x - x^4)^(1/3)*Defer[Subst][Defer[Int][(x^3*(1 - x^9)^(1/3))/(1 + x^6 - 2*x^9 - x^12 - x^15 + x^18), x], x
, x^(1/3)])/(x^(1/3)*(1 - x^3)^(1/3)) + (3*(x - x^4)^(1/3)*Defer[Subst][Defer[Int][(x^12*(1 - x^9)^(1/3))/(1 +
x^6 - 2*x^9 - x^12 - x^15 + x^18), x], x, x^(1/3)])/(x^(1/3)*(1 - x^3)^(1/3))

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x-x^4} \int \frac {\sqrt [3]{x} \sqrt [3]{1-x^3} \left (2+x^3\right )}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{1-x^3}} \\ & = \frac {\left (3 \sqrt [3]{x-x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^9} \left (2+x^9\right )}{1+x^6-2 x^9-x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^3}} \\ & = \frac {\left (3 \sqrt [3]{x-x^4}\right ) \text {Subst}\left (\int \left (\frac {2 x^3 \sqrt [3]{1-x^9}}{1+x^6-2 x^9-x^{12}-x^{15}+x^{18}}+\frac {x^{12} \sqrt [3]{1-x^9}}{1+x^6-2 x^9-x^{12}-x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^3}} \\ & = \frac {\left (3 \sqrt [3]{x-x^4}\right ) \text {Subst}\left (\int \frac {x^{12} \sqrt [3]{1-x^9}}{1+x^6-2 x^9-x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^3}}+\frac {\left (6 \sqrt [3]{x-x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^9}}{1+x^6-2 x^9-x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=-\text {RootSum}\left [-1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x-x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{1+2 \text {$\#$1}^3}\&\right ] \]

[In]

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

[Out]

-RootSum[-1 + #1^3 + #1^6 & , (-(Log[x]*#1) + Log[(x - x^4)^(1/3) - x*#1]*#1)/(1 + 2*#1^3) & ]

Maple [N/A] (verified)

Time = 80.76 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}-1\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (-x^{4}+x \right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{3}+1}\right )\)	\(45\)
trager	\(\text {Expression too large to display}\)	\(12058\)

[In]

int((x^3+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),x,method=_RETURNVERBOSE)

[Out]

-sum(_R*ln((-_R*x+(-x^4+x)^(1/3))/x)/(2*_R^3+1),_R=RootOf(_Z^6+_Z^3-1))

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((x^3+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

Sympy [N/A]

Not integrable

Time = 1.84 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\int \frac {\sqrt [3]{- x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{3} + 2\right )}{x^{6} - x^{5} - x^{4} - 2 x^{3} + x^{2} + 1}\, dx \]

[In]

integrate((x**3+2)*(-x**4+x)**(1/3)/(x**6-x**5-x**4-2*x**3+x**2+1),x)

[Out]

Integral((-x*(x - 1)*(x**2 + x + 1))**(1/3)*(x**3 + 2)/(x**6 - x**5 - x**4 - 2*x**3 + x**2 + 1), x)

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\int { \frac {{\left (-x^{4} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{x^{6} - x^{5} - x^{4} - 2 \, x^{3} + x^{2} + 1} \,d x } \]

[In]

integrate((x^3+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),x, algorithm="maxima")

[Out]

integrate((-x^4 + x)^(1/3)*(x^3 + 2)/(x^6 - x^5 - x^4 - 2*x^3 + x^2 + 1), x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\int { \frac {{\left (-x^{4} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{x^{6} - x^{5} - x^{4} - 2 \, x^{3} + x^{2} + 1} \,d x } \]

[In]

integrate((x^3+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),x, algorithm="giac")

[Out]

integrate((-x^4 + x)^(1/3)*(x^3 + 2)/(x^6 - x^5 - x^4 - 2*x^3 + x^2 + 1), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\int \frac {{\left (x-x^4\right )}^{1/3}\,\left (x^3+2\right )}{x^6-x^5-x^4-2\,x^3+x^2+1} \,d x \]

[In]

int(((x - x^4)^(1/3)*(x^3 + 2))/(x^2 - 2*x^3 - x^4 - x^5 + x^6 + 1),x)

[Out]

int(((x - x^4)^(1/3)*(x^3 + 2))/(x^2 - 2*x^3 - x^4 - x^5 + x^6 + 1), x)