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

Optimal. Leaf size=59 \[ -\frac {1}{4} \text {RootSum}\left [2 \text {$\#$1}^6-4 \text {$\#$1}^3+1\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{x^3-2 x-1}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^3-1}\& \right ] \]

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Rubi [F]  time = 0.46, 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 (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

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

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Mathematica [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x (3+4 x) \sqrt [3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 59, normalized size = 1.00 \begin {gather*} -\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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) & ]

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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*(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 43.21, size = 11032, normalized size = 186.98

method	result	size
trager	\(\text {Expression too large to display}\)	\(11032\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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