\(\int \frac {x \sqrt [3]{2-x^3+x^8} (-6+5 x^8)}{4+x^6+4 x^8+x^{16}} \, dx\) [774]

Optimal result

Integrand size = 38, antiderivative size = 59 \[ \int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx=\frac {1}{2} \text {RootSum}\left [2+2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{2-x^3+x^8}-x \text {$\#$1}\right ) \text {$\#$1}}{1+\text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx=\int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx \]

[In]

Int[(x*(2 - x^3 + x^8)^(1/3)*(-6 + 5*x^8))/(4 + x^6 + 4*x^8 + x^16),x]

[Out]

-6*Defer[Int][(x*(2 - x^3 + x^8)^(1/3))/(4 + x^6 + 4*x^8 + x^16), x] + 5*Defer[Int][(x^9*(2 - x^3 + x^8)^(1/3)
)/(4 + x^6 + 4*x^8 + x^16), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {6 x \sqrt [3]{2-x^3+x^8}}{4+x^6+4 x^8+x^{16}}+\frac {5 x^9 \sqrt [3]{2-x^3+x^8}}{4+x^6+4 x^8+x^{16}}\right ) \, dx \\ & = 5 \int \frac {x^9 \sqrt [3]{2-x^3+x^8}}{4+x^6+4 x^8+x^{16}} \, dx-6 \int \frac {x \sqrt [3]{2-x^3+x^8}}{4+x^6+4 x^8+x^{16}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.39 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx=\frac {1}{2} \text {RootSum}\left [2+2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{2-x^3+x^8}-x \text {$\#$1}\right ) \text {$\#$1}}{1+\text {$\#$1}^3}\&\right ] \]

[In]

Integrate[(x*(2 - x^3 + x^8)^(1/3)*(-6 + 5*x^8))/(4 + x^6 + 4*x^8 + x^16),x]

[Out]

RootSum[2 + 2*#1^3 + #1^6 & , (-(Log[x]*#1) + Log[(2 - x^3 + x^8)^(1/3) - x*#1]*#1)/(1 + #1^3) & ]/2

Maple [F(-1)]

Timed out.

\[\int \frac {x \left (x^{8}-x^{3}+2\right )^{\frac {1}{3}} \left (5 x^{8}-6\right )}{x^{16}+4 x^{8}+x^{6}+4}d x\]

[In]

int(x*(x^8-x^3+2)^(1/3)*(5*x^8-6)/(x^16+4*x^8+x^6+4),x)

[Out]

int(x*(x^8-x^3+2)^(1/3)*(5*x^8-6)/(x^16+4*x^8+x^6+4),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*(x^8-x^3+2)^(1/3)*(5*x^8-6)/(x^16+4*x^8+x^6+4),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.48 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx=\int \frac {x \left (5 x^{8} - 6\right ) \sqrt [3]{x^{8} - x^{3} + 2}}{x^{16} + 4 x^{8} + x^{6} + 4}\, dx \]

[In]

integrate(x*(x**8-x**3+2)**(1/3)*(5*x**8-6)/(x**16+4*x**8+x**6+4),x)

[Out]

Integral(x*(5*x**8 - 6)*(x**8 - x**3 + 2)**(1/3)/(x**16 + 4*x**8 + x**6 + 4), x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64 \[ \int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx=\int { \frac {{\left (5 \, x^{8} - 6\right )} {\left (x^{8} - x^{3} + 2\right )}^{\frac {1}{3}} x}{x^{16} + 4 \, x^{8} + x^{6} + 4} \,d x } \]

[In]

integrate(x*(x^8-x^3+2)^(1/3)*(5*x^8-6)/(x^16+4*x^8+x^6+4),x, algorithm="maxima")

[Out]

integrate((5*x^8 - 6)*(x^8 - x^3 + 2)^(1/3)*x/(x^16 + 4*x^8 + x^6 + 4), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64 \[ \int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx=\int { \frac {{\left (5 \, x^{8} - 6\right )} {\left (x^{8} - x^{3} + 2\right )}^{\frac {1}{3}} x}{x^{16} + 4 \, x^{8} + x^{6} + 4} \,d x } \]

[In]

integrate(x*(x^8-x^3+2)^(1/3)*(5*x^8-6)/(x^16+4*x^8+x^6+4),x, algorithm="giac")

[Out]

integrate((5*x^8 - 6)*(x^8 - x^3 + 2)^(1/3)*x/(x^16 + 4*x^8 + x^6 + 4), x)

Mupad [N/A]

Not integrable

Time = 5.68 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64 \[ \int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx=\int \frac {x\,\left (5\,x^8-6\right )\,{\left (x^8-x^3+2\right )}^{1/3}}{x^{16}+4\,x^8+x^6+4} \,d x \]

[In]

int((x*(5*x^8 - 6)*(x^8 - x^3 + 2)^(1/3))/(x^6 + 4*x^8 + x^16 + 4),x)

[Out]

int((x*(5*x^8 - 6)*(x^8 - x^3 + 2)^(1/3))/(x^6 + 4*x^8 + x^16 + 4), x)