\(\int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx\) [765]

Optimal result

Integrand size = 34, antiderivative size = 59 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=-\frac {1}{4} \text {RootSum}\left [1-4 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-1-2 x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx \]

[In]

Int[(x*(3 + 4*x)*(-1 - 2*x + x^3)^(1/3))/(-2 - 8*x - 8*x^2 + x^6),x]

[Out]

3*Defer[Int][(x*(-1 - 2*x + x^3)^(1/3))/(-2 - 8*x - 8*x^2 + x^6), x] + 4*Defer[Int][(x^2*(-1 - 2*x + x^3)^(1/3
))/(-2 - 8*x - 8*x^2 + x^6), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 x \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6}+\frac {4 x^2 \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6}\right ) \, dx \\ & = 3 \int \frac {x \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx+4 \int \frac {x^2 \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=-\frac {1}{4} \text {RootSum}\left [1-4 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-1-2 x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ] \]

[In]

Integrate[(x*(3 + 4*x)*(-1 - 2*x + x^3)^(1/3))/(-2 - 8*x - 8*x^2 + x^6),x]

[Out]

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

Maple [N/A] (verified)

Time = 37.51 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-4 \textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-2 x -1\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{3}-1}\right )}{4}\)	\(48\)
trager	\(\text {Expression too large to display}\)	\(11032\)

[In]

int(x*(4*x+3)*(x^3-2*x-1)^(1/3)/(x^6-8*x^2-8*x-2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*(3+4*x)*(x^3-2*x-1)^(1/3)/(x^6-8*x^2-8*x-2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.92 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\int \frac {x \sqrt [3]{\left (x + 1\right ) \left (x^{2} - x - 1\right )} \left (4 x + 3\right )}{x^{6} - 8 x^{2} - 8 x - 2}\, dx \]

[In]

integrate(x*(3+4*x)*(x**3-2*x-1)**(1/3)/(x**6-8*x**2-8*x-2),x)

[Out]

Integral(x*((x + 1)*(x**2 - x - 1))**(1/3)*(4*x + 3)/(x**6 - 8*x**2 - 8*x - 2), x)

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\int { \frac {{\left (x^{3} - 2 \, x - 1\right )}^{\frac {1}{3}} {\left (4 \, x + 3\right )} x}{x^{6} - 8 \, x^{2} - 8 \, x - 2} \,d x } \]

[In]

integrate(x*(3+4*x)*(x^3-2*x-1)^(1/3)/(x^6-8*x^2-8*x-2),x, algorithm="maxima")

[Out]

integrate((x^3 - 2*x - 1)^(1/3)*(4*x + 3)*x/(x^6 - 8*x^2 - 8*x - 2), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\int { \frac {{\left (x^{3} - 2 \, x - 1\right )}^{\frac {1}{3}} {\left (4 \, x + 3\right )} x}{x^{6} - 8 \, x^{2} - 8 \, x - 2} \,d x } \]

[In]

integrate(x*(3+4*x)*(x^3-2*x-1)^(1/3)/(x^6-8*x^2-8*x-2),x, algorithm="giac")

[Out]

integrate((x^3 - 2*x - 1)^(1/3)*(4*x + 3)*x/(x^6 - 8*x^2 - 8*x - 2), x)

Mupad [N/A]

Not integrable

Time = 5.64 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.63 \[ \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx=\int -\frac {x\,\left (4\,x+3\right )\,{\left (x^3-2\,x-1\right )}^{1/3}}{-x^6+8\,x^2+8\,x+2} \,d x \]

[In]

int(-(x*(4*x + 3)*(x^3 - 2*x - 1)^(1/3))/(8*x + 8*x^2 - x^6 + 2),x)

[Out]

int(-(x*(4*x + 3)*(x^3 - 2*x - 1)^(1/3))/(8*x + 8*x^2 - x^6 + 2), x)