\(\int \frac {(3+x^2) (1+x^2+x^3)^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx\) [772]

Optimal result

Integrand size = 41, antiderivative size = 59 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx=\text {RootSum}\left [1-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{1+x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-3+2 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx=\text {RootSum}\left [1-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{1+x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-3+2 \text {$\#$1}^3}\&\right ] \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 31.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-3 \textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\frac {-\textit {\_R} x +\left (x^{3}+x^{2}+1\right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{3}-3}\)	\(48\)
trager	\(\text {Expression too large to display}\)	\(13345\)

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((x^2+3)*(x^3+x^2+1)^(2/3)/(x^6+x^5-x^4+x^3-2*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 = 2.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.63 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx=\int \frac {\left (x^{2} + 3\right ) \left (x^{3} + x^{2} + 1\right )^{\frac {2}{3}}}{x^{6} + x^{5} - x^{4} + x^{3} - 2 x^{2} - 1}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.69 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx=\int { \frac {{\left (x^{3} + x^{2} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 3\right )}}{x^{6} + x^{5} - x^{4} + x^{3} - 2 \, x^{2} - 1} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.69 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx=\int { \frac {{\left (x^{3} + x^{2} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 3\right )}}{x^{6} + x^{5} - x^{4} + x^{3} - 2 \, x^{2} - 1} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.78 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx=\int -\frac {\left (x^2+3\right )\,{\left (x^3+x^2+1\right )}^{2/3}}{-x^6-x^5+x^4-x^3+2\,x^2+1} \,d x \]

[In]

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

[Out]

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