\(\int \frac {(3+2 x) (1+x+x^3)^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx\) [768]

Optimal result

Integrand size = 35, antiderivative size = 59 \[ \int \frac {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx=-\text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{1+x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+2 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx=-\text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{1+x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+2 \text {$\#$1}^3}\&\right ] \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 4.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\frac {-\textit {\_R} x +\left (x^{3}+x +1\right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{3}-1}\right )\)	\(48\)
trager	\(\text {Expression too large to display}\)	\(1263\)

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((3+2*x)*(x^3+x+1)^(2/3)/(x^6+x^4+x^3+x^2+2*x+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 = 1.96 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \frac {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx=\int \frac {\left (2 x + 3\right ) \left (x^{3} + x + 1\right )^{\frac {2}{3}}}{x^{6} + x^{4} + x^{3} + x^{2} + 2 x + 1}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.59 \[ \int \frac {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx=\int { \frac {{\left (x^{3} + x + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{x^{6} + x^{4} + x^{3} + x^{2} + 2 \, x + 1} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.59 \[ \int \frac {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx=\int { \frac {{\left (x^{3} + x + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{x^{6} + x^{4} + x^{3} + x^{2} + 2 \, x + 1} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.59 \[ \int \frac {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx=\int \frac {\left (2\,x+3\right )\,{\left (x^3+x+1\right )}^{2/3}}{x^6+x^4+x^3+x^2+2\,x+1} \,d x \]

[In]

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

[Out]

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