3.622 \(\int \frac{1}{x^5 \sqrt [3]{1-x^3} (1+x^3)} \, dx\)

Optimal. Leaf size=289 \[ \frac{1}{4} x^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right )+\frac{\left (1-x^3\right )^{2/3}}{2 x}-\frac{\left (1-x^3\right )^{2/3}}{4 x^4}+\frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac{\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\log \left ((1-x) (x+1)^2\right )}{12 \sqrt [3]{2}} \]

[Out]

-(1 - x^3)^(2/3)/(4*x^4) + (1 - x^3)^(2/3)/(2*x) + ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]]/(
2^(1/3)*Sqrt[3]) + ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]]/(2*2^(1/3)*Sqrt[3]) + (x^2*Hypergeo
metric2F1[1/3, 2/3, 5/3, x^3])/4 + Log[(1 - x)*(1 + x)^2]/(12*2^(1/3)) + Log[1 + (2^(2/3)*(1 - x)^2)/(1 - x^3)
^(2/3) - (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(6*2^(1/3)) - Log[1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(3*2^(1/3
)) - Log[-1 + x + 2^(2/3)*(1 - x^3)^(1/3)]/(4*2^(1/3))

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Rubi [C]  time = 0.0181233, antiderivative size = 26, normalized size of antiderivative = 0.09, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {510} \[ -\frac{F_1\left (-\frac{4}{3};\frac{1}{3},1;-\frac{1}{3};x^3,-x^3\right )}{4 x^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-AppellF1[-4/3, 1/3, 1, -1/3, x^3, -x^3]/(4*x^4)

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^5 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=-\frac{F_1\left (-\frac{4}{3};\frac{1}{3},1;-\frac{1}{3};x^3,-x^3\right )}{4 x^4}\\ \end{align*}

Mathematica [C]  time = 0.0395003, size = 76, normalized size = 0.26 \[ \frac{2 x^9 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right )+15 x^6 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right )+5 \left (1-x^3\right )^{2/3} \left (2 x^3-1\right )}{20 x^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(5*(1 - x^3)^(2/3)*(-1 + 2*x^3) + 15*x^6*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3] + 2*x^9*AppellF1[5/3, 1/3, 1, 8
/3, x^3, -x^3])/(20*x^4)

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \left ({x}^{3}+1 \right ) }{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(-x^3+1)^(1/3)/(x^3+1),x)

[Out]

int(1/x^5/(-x^3+1)^(1/3)/(x^3+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{11} - x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="fricas")

[Out]

integral(-(-x^3 + 1)^(2/3)/(x^11 - x^5), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5} \sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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