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

Optimal. Leaf size=59 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^6+2 \text {$\#$1}^3+2\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{x^8-x^3+2}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^3+1}\& \right ] \]

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Rubi [F]  time = 0.68, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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*}

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Mathematica [F]  time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.69, size = 59, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x \left (x^{8}-x^{3}+2\right )^{\frac {1}{3}} \left (5 x^{8}-6\right )}{x^{16}+4 x^{8}+x^{6}+4}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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