\(\int \frac {(2+x^3) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx\) [769]

Optimal result

Integrand size = 43, antiderivative size = 59 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx=-\text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x+x^3-x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 2.78 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.64 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx=\frac {\sqrt [3]{x+x^3-x^4} \text {RootSum}\left [1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{-1-x^2+x^3}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{1+2 \text {$\#$1}^3}\&\right ]}{\sqrt [3]{x} \sqrt [3]{-1-x^2+x^3}} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 9.66 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +{\left (-x \left (x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{3}-1}\right )\)	\(53\)
trager	\(\text {Expression too large to display}\)	\(2045\)

[In]

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

Fricas [F(-2)]

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

[In]

integrate((x^3+2)*(-x^4+x^3+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 = 41, normalized size of antiderivative = 0.69 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx=\int \frac {\sqrt [3]{- x \left (x^{3} - x^{2} - 1\right )} \left (x^{3} + 2\right )}{x^{6} - x^{5} + x^{4} - 2 x^{3} + x^{2} + 1}\, dx \]

[In]

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

[Out]

Integral((-x*(x**3 - x**2 - 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 = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx=\int { \frac {{\left (-x^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{x^{6} - x^{5} + x^{4} - 2 \, x^{3} + x^{2} + 1} \,d x } \]

[In]

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

[Out]

integrate((-x^4 + x^3 + 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 = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx=\int { \frac {{\left (-x^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{x^{6} - x^{5} + x^{4} - 2 \, x^{3} + x^{2} + 1} \,d x } \]

[In]

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

[Out]

integrate((-x^4 + x^3 + 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 = 6.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx=\int \frac {\left (x^3+2\right )\,{\left (-x^4+x^3+x\right )}^{1/3}}{x^6-x^5+x^4-2\,x^3+x^2+1} \,d x \]

[In]

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

[Out]

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