3.26.31 \(\int \frac {1-x^2+x^3}{(-1-x^2+x^3) \sqrt [3]{x^2+x^3}} \, dx\) [2531]

Optimal. Leaf size=212 \[ \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right )+2 \text {RootSum}\left [-3+4 \text {$\#$1}^3-3 \text {$\#$1}^6+\text {$\#$1}^9\& ,\frac {-\log (x)+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-2 \log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3-\log (x) \text {$\#$1}^6+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^6}{4 \text {$\#$1}-6 \text {$\#$1}^4+3 \text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 0.32, 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^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(Sqrt[3]*x^(2/3)*(1 + x)^(1/3)*ArcTan[(1 + (2*x^(1/3))/(1 + x)^(1/3))/Sqrt[3]])/(x^2 + x^3)^(1/3) - (3*x^(2/3)
*(1 + x)^(1/3)*Log[x^(1/3) - (1 + x)^(1/3)])/(2*(x^2 + x^3)^(1/3)) + (6*x^(2/3)*(1 + x)^(1/3)*Defer[Subst][Def
er[Int][1/((1 + x^3)^(1/3)*(-1 - x^6 + x^9)), x], x, x^(1/3)])/(x^2 + x^3)^(1/3)

Rubi steps

\begin {align*} \int \frac {1-x^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1-x^2+x^3}{x^{2/3} \sqrt [3]{1+x} \left (-1-x^2+x^3\right )} \, dx}{\sqrt [3]{x^2+x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1-x^6+x^9}{\sqrt [3]{1+x^3} \left (-1-x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^3}}+\frac {2}{\sqrt [3]{1+x^3} \left (-1-x^6+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (-1-x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}\\ &=\frac {\sqrt {3} x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{x^2+x^3}}-\frac {3 x^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{x}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{x^2+x^3}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (-1-x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 240, normalized size = 1.13 \begin {gather*} \frac {x^{2/3} \sqrt [3]{1+x} \left (6 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )-6 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )+3 \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )-4 \text {RootSum}\left [-3+4 \text {$\#$1}^3-3 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {\log (x)-3 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right )-2 \log (x) \text {$\#$1}^3+6 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3+\log (x) \text {$\#$1}^6-3 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^6}{4 \text {$\#$1}-6 \text {$\#$1}^4+3 \text {$\#$1}^7}\&\right ]\right )}{6 \sqrt [3]{x^2 (1+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(2/3)*(1 + x)^(1/3)*(6*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(1 + x)^(1/3))] - 6*Log[-x^(1/3) + (1
+ x)^(1/3)] + 3*Log[x^(2/3) + x^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)] - 4*RootSum[-3 + 4*#1^3 - 3*#1^6 + #1^9 &
, (Log[x] - 3*Log[(1 + x)^(1/3) - x^(1/3)*#1] - 2*Log[x]*#1^3 + 6*Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1^3 + Log[
x]*#1^6 - 3*Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1^6)/(4*#1 - 6*#1^4 + 3*#1^7) & ]))/(6*(x^2*(1 + x))^(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
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((x^3-x^2+1)/(x^3-x^2-1)/(x^3+x^2)^(1/3),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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