\(\int \frac {(-2+5 x^7) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx\) [811]

Optimal result

Integrand size = 41, antiderivative size = 61 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{2 x+x^3+2 x^8}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx \]

[In]

Int[((-2 + 5*x^7)*(2*x + x^3 + 2*x^8)^(1/3))/(4 + x^4 + 8*x^7 + 4*x^14),x]

[Out]

(-6*(2*x + x^3 + 2*x^8)^(1/3)*Defer[Subst][Defer[Int][(x^3*(2 + x^6 + 2*x^21)^(1/3))/(4 + x^12 + 8*x^21 + 4*x^
42), x], x, x^(1/3)])/(x^(1/3)*(2 + x^2 + 2*x^7)^(1/3)) + (15*(2*x + x^3 + 2*x^8)^(1/3)*Defer[Subst][Defer[Int
][(x^24*(2 + x^6 + 2*x^21)^(1/3))/(4 + x^12 + 8*x^21 + 4*x^42), x], x, x^(1/3)])/(x^(1/3)*(2 + x^2 + 2*x^7)^(1
/3))

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{2 x+x^3+2 x^8} \int \frac {\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7} \left (-2+5 x^7\right )}{4+x^4+8 x^7+4 x^{14}} \, dx}{\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}} \\ & = \frac {\left (3 \sqrt [3]{2 x+x^3+2 x^8}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{2+x^6+2 x^{21}} \left (-2+5 x^{21}\right )}{4+x^{12}+8 x^{21}+4 x^{42}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}} \\ & = \frac {\left (3 \sqrt [3]{2 x+x^3+2 x^8}\right ) \text {Subst}\left (\int \left (-\frac {2 x^3 \sqrt [3]{2+x^6+2 x^{21}}}{4+x^{12}+8 x^{21}+4 x^{42}}+\frac {5 x^{24} \sqrt [3]{2+x^6+2 x^{21}}}{4+x^{12}+8 x^{21}+4 x^{42}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}} \\ & = -\frac {\left (6 \sqrt [3]{2 x+x^3+2 x^8}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{2+x^6+2 x^{21}}}{4+x^{12}+8 x^{21}+4 x^{42}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}}+\frac {\left (15 \sqrt [3]{2 x+x^3+2 x^8}\right ) \text {Subst}\left (\int \frac {x^{24} \sqrt [3]{2+x^6+2 x^{21}}}{4+x^{12}+8 x^{21}+4 x^{42}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.67 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\frac {\sqrt [3]{x \left (2+x^2+2 x^7\right )} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{2+x^2+2 x^7}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]}{4 \sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}} \]

[In]

Integrate[((-2 + 5*x^7)*(2*x + x^3 + 2*x^8)^(1/3))/(4 + x^4 + 8*x^7 + 4*x^14),x]

[Out]

((x*(2 + x^2 + 2*x^7))^(1/3)*RootSum[2 - 2*#1^3 + #1^6 & , (-2*Log[x^(1/3)]*#1 + Log[(2 + x^2 + 2*x^7)^(1/3) -
x^(2/3)*#1]*#1)/(-1 + #1^3) & ])/(4*x^(1/3)*(2 + x^2 + 2*x^7)^(1/3))

Maple [F(-1)]

Timed out.

\[\int \frac {\left (5 x^{7}-2\right ) \left (2 x^{8}+x^{3}+2 x \right )^{\frac {1}{3}}}{4 x^{14}+8 x^{7}+x^{4}+4}d x\]

[In]

int((5*x^7-2)*(2*x^8+x^3+2*x)^(1/3)/(4*x^14+8*x^7+x^4+4),x)

[Out]

int((5*x^7-2)*(2*x^8+x^3+2*x)^(1/3)/(4*x^14+8*x^7+x^4+4),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((5*x^7-2)*(2*x^8+x^3+2*x)^(1/3)/(4*x^14+8*x^7+x^4+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 = 3.66 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.61 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\int \frac {\sqrt [3]{x \left (2 x^{7} + x^{2} + 2\right )} \left (5 x^{7} - 2\right )}{4 x^{14} + 8 x^{7} + x^{4} + 4}\, dx \]

[In]

integrate((5*x**7-2)*(2*x**8+x**3+2*x)**(1/3)/(4*x**14+8*x**7+x**4+4),x)

[Out]

Integral((x*(2*x**7 + x**2 + 2))**(1/3)*(5*x**7 - 2)/(4*x**14 + 8*x**7 + x**4 + 4), x)

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.67 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\int { \frac {{\left (2 \, x^{8} + x^{3} + 2 \, x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} - 2\right )}}{4 \, x^{14} + 8 \, x^{7} + x^{4} + 4} \,d x } \]

[In]

integrate((5*x^7-2)*(2*x^8+x^3+2*x)^(1/3)/(4*x^14+8*x^7+x^4+4),x, algorithm="maxima")

[Out]

integrate((2*x^8 + x^3 + 2*x)^(1/3)*(5*x^7 - 2)/(4*x^14 + 8*x^7 + x^4 + 4), x)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.67 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\int { \frac {{\left (2 \, x^{8} + x^{3} + 2 \, x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} - 2\right )}}{4 \, x^{14} + 8 \, x^{7} + x^{4} + 4} \,d x } \]

[In]

integrate((5*x^7-2)*(2*x^8+x^3+2*x)^(1/3)/(4*x^14+8*x^7+x^4+4),x, algorithm="giac")

[Out]

integrate((2*x^8 + x^3 + 2*x)^(1/3)*(5*x^7 - 2)/(4*x^14 + 8*x^7 + x^4 + 4), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.67 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\int \frac {\left (5\,x^7-2\right )\,{\left (2\,x^8+x^3+2\,x\right )}^{1/3}}{4\,x^{14}+8\,x^7+x^4+4} \,d x \]

[In]

int(((5*x^7 - 2)*(2*x + x^3 + 2*x^8)^(1/3))/(x^4 + 8*x^7 + 4*x^14 + 4),x)

[Out]

int(((5*x^7 - 2)*(2*x + x^3 + 2*x^8)^(1/3))/(x^4 + 8*x^7 + 4*x^14 + 4), x)