\(\int \frac {(-2+x^3) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx\) [716]

Optimal result

Integrand size = 39, antiderivative size = 55 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\&\right ] \]

[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} \left (-2+x^3\right ) \sqrt [3]{1+x^2+x^3}}{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 \left (-2+x^9\right ) \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 (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 = 0.00 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\frac {\sqrt [3]{x+x^3+x^4} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{1+x^2+x^3}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\&\right ]}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \]

[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 = 9.39 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{3}-1}\)	\(48\)
trager	\(\text {Expression too large to display}\)	\(2297\)

[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 = 1.76 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\int \frac {\sqrt [3]{x \left (x^{3} + x^{2} + 1\right )} \left (x^{3} - 2\right )}{x^{6} + x^{5} + x^{4} + 2 x^{3} + x^{2} + 1}\, dx \]

[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 = 39, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\int { \frac {{\left (x^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{x^{6} + x^{5} + x^{4} + 2 \, x^{3} + x^{2} + 1} \,d x } \]

[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 = 39, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\int { \frac {{\left (x^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{x^{6} + x^{5} + x^{4} + 2 \, x^{3} + x^{2} + 1} \,d x } \]

[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 = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\int \frac {\left (x^3-2\right )\,{\left (x^4+x^3+x\right )}^{1/3}}{x^6+x^5+x^4+2\,x^3+x^2+1} \,d x \]

[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)