3.7.96 \(\int \frac {(2+x^3) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx\)

Optimal. Leaf size=54 \[ -\text {RootSum}\left [\text {$\#$1}^6+\text {$\#$1}^3-1\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{x-x^4}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{2 \text {$\#$1}^3+1}\& \right ] \]

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Rubi [F]  time = 1.08, 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 {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(6*(x - x^4)^(1/3)*Defer[Subst][Defer[Int][(x^3*(1 - x^9)^(1/3))/(1 + x^6 - 2*x^9 - x^12 - x^15 + x^18), x], x
, x^(1/3)])/(x^(1/3)*(1 - x^3)^(1/3)) + (3*(x - x^4)^(1/3)*Defer[Subst][Defer[Int][(x^12*(1 - x^9)^(1/3))/(1 +
x^6 - 2*x^9 - x^12 - x^15 + x^18), x], x, x^(1/3)])/(x^(1/3)*(1 - x^3)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx &=\frac {\sqrt [3]{x-x^4} \int \frac {\sqrt [3]{x} \sqrt [3]{1-x^3} \left (2+x^3\right )}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{1-x^3}}\\ &=\frac {\left (3 \sqrt [3]{x-x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^9} \left (2+x^9\right )}{1+x^6-2 x^9-x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^3}}\\ &=\frac {\left (3 \sqrt [3]{x-x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {2 x^3 \sqrt [3]{1-x^9}}{1+x^6-2 x^9-x^{12}-x^{15}+x^{18}}+\frac {x^{12} \sqrt [3]{1-x^9}}{1+x^6-2 x^9-x^{12}-x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^3}}\\ &=\frac {\left (3 \sqrt [3]{x-x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt [3]{1-x^9}}{1+x^6-2 x^9-x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^3}}+\frac {\left (6 \sqrt [3]{x-x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^9}}{1+x^6-2 x^9-x^{12}-x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^3}}\\ \end {align*}

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Mathematica [F]  time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 54, normalized size = 1.00 \begin {gather*} -\text {RootSum}\left [-1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x-x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{1+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-RootSum[-1 + #1^3 + #1^6 & , (-(Log[x]*#1) + Log[(x - x^4)^(1/3) - x*#1]*#1)/(1 + 2*#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+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),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^{4} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{x^{6} - x^{5} - x^{4} - 2 \, x^{3} + x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 119.27, size = 15547, normalized size = 287.91

method	result	size
trager	\(\text {Expression too large to display}\)	\(15547\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),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^{4} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{x^{6} - x^{5} - x^{4} - 2 \, x^{3} + x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (x-x^4\right )}^{1/3}\,\left (x^3+2\right )}{x^6-x^5-x^4-2\,x^3+x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{- x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{3} + 2\right )}{x^{6} - x^{5} - x^{4} - 2 x^{3} + x^{2} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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