3.14.45 \(\int \frac {1+x}{(-1-x+x^3) \sqrt [3]{-x^2+x^3}} \, dx\)

Optimal. Leaf size=97 \[ -\text {RootSum}\left [\text {$\#$1}^9-4 \text {$\#$1}^6+5 \text {$\#$1}^3-1\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+2 \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-2 \log (x)}{3 \text {$\#$1}^4-5 \text {$\#$1}}\& \right ] \]

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Rubi [F]  time = 0.61, 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 {1+x}{\left (-1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*(-1 + x)^(1/3)*x^(2/3)*Defer[Subst][Defer[Int][1/((-1 + x^3)^(1/3)*(-1 - x^3 + x^9)), x], x, x^(1/3)])/(-x^
2 + x^3)^(1/3) + (3*(-1 + x)^(1/3)*x^(2/3)*Defer[Subst][Defer[Int][x^3/((-1 + x^3)^(1/3)*(-1 - x^3 + x^9)), x]
, x, x^(1/3)])/(-x^2 + x^3)^(1/3)

Rubi steps

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

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Mathematica [B]  time = 0.10, size = 481, normalized size = 4.96 \begin {gather*} -\frac {\left (\frac {1}{x-1}+1\right )^{2/3} (x-1) \left (\text {RootSum}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+4 \text {$\#$1}-1\&,\frac {-\frac {2 \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {1}{x-1}+1}\right )}{\text {$\#$1}^{2/3}}+\frac {\log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {1}{x-1}+1}+\left (\frac {1}{x-1}+1\right )^{2/3}\right )}{\text {$\#$1}^{2/3}}+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {1}{x-1}+1}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{\text {$\#$1}^{2/3}}}{3 \text {$\#$1}^2-10 \text {$\#$1}+4}\&\right ]-3 \text {RootSum}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+4 \text {$\#$1}-1\&,\frac {\sqrt [3]{\text {$\#$1}} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {1}{x-1}+1}+\left (\frac {1}{x-1}+1\right )^{2/3}\right )-2 \sqrt [3]{\text {$\#$1}} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {1}{x-1}+1}\right )+2 \sqrt {3} \sqrt [3]{\text {$\#$1}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {1}{x-1}+1}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{3 \text {$\#$1}^2-10 \text {$\#$1}+4}\&\right ]+2 \text {RootSum}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+4 \text {$\#$1}-1\&,\frac {-2 \text {$\#$1}^{4/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {1}{x-1}+1}\right )+\text {$\#$1}^{4/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {1}{x-1}+1}+\left (\frac {1}{x-1}+1\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{4/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {1}{x-1}+1}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{3 \text {$\#$1}^2-10 \text {$\#$1}+4}\&\right ]\right )}{2 \sqrt [3]{(x-1) x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((1 + (-1 + x)^(-1))^(2/3)*(-1 + x)*(RootSum[-1 + 4*#1 - 5*#1^2 + #1^3 & , ((2*Sqrt[3]*ArcTan[(1 + (2*(1
+ (-1 + x)^(-1))^(1/3))/#1^(1/3))/Sqrt[3]])/#1^(2/3) - (2*Log[-(1 + (-1 + x)^(-1))^(1/3) + #1^(1/3)])/#1^(2/3)
+ Log[(1 + (-1 + x)^(-1))^(2/3) + (1 + (-1 + x)^(-1))^(1/3)*#1^(1/3) + #1^(2/3)]/#1^(2/3))/(4 - 10*#1 + 3*#1^
2) & ] - 3*RootSum[-1 + 4*#1 - 5*#1^2 + #1^3 & , (2*Sqrt[3]*ArcTan[(1 + (2*(1 + (-1 + x)^(-1))^(1/3))/#1^(1/3)
)/Sqrt[3]]*#1^(1/3) - 2*Log[-(1 + (-1 + x)^(-1))^(1/3) + #1^(1/3)]*#1^(1/3) + Log[(1 + (-1 + x)^(-1))^(2/3) +
(1 + (-1 + x)^(-1))^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(1/3))/(4 - 10*#1 + 3*#1^2) & ] + 2*RootSum[-1 + 4*#1 - 5*#1
^2 + #1^3 & , (2*Sqrt[3]*ArcTan[(1 + (2*(1 + (-1 + x)^(-1))^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(4/3) - 2*Log[-(1 + (
-1 + x)^(-1))^(1/3) + #1^(1/3)]*#1^(4/3) + Log[(1 + (-1 + x)^(-1))^(2/3) + (1 + (-1 + x)^(-1))^(1/3)*#1^(1/3)
+ #1^(2/3)]*#1^(4/3))/(4 - 10*#1 + 3*#1^2) & ]))/((-1 + x)*x^2)^(1/3)

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IntegrateAlgebraic [A]  time = 0.00, size = 97, normalized size = 1.00 \begin {gather*} -\text {RootSum}\left [-1+5 \text {$\#$1}^3-4 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+3 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-RootSum[-1 + 5*#1^3 - 4*#1^6 + #1^9 & , (-2*Log[x] + 2*Log[(-x^2 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-x
^2 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 3*#1^4) & ]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^3-x-1)/(x^3-x^2)^(1/3),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 163.86, size = 190624, normalized size = 1965.20

method	result	size
trager	\(\text {Expression too large to display}\)	\(190624\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {x+1}{{\left (x^3-x^2\right )}^{1/3}\,\left (-x^3+x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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