3.657 \(\int \frac{1}{x^5 (1-x^3)^{4/3} (1+x^3)} \, dx\)

Optimal. Leaf size=308 \[ -\frac{1}{2} x^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right )-\frac{\left (1-x^3\right )^{2/3}}{x}-\frac{3 \left (1-x^3\right )^{2/3}}{4 x^4}+\frac{1}{2 \sqrt [3]{1-x^3} x^4}+\frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{12 \sqrt [3]{2}}-\frac{\log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{8 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{4 \sqrt [3]{2} \sqrt{3}}+\frac{\log \left ((1-x) (x+1)^2\right )}{24 \sqrt [3]{2}} \]

[Out]

1/(2*x^4*(1 - x^3)^(1/3)) - (3*(1 - x^3)^(2/3))/(4*x^4) - (1 - x^3)^(2/3)/x + ArcTan[(1 - (2*2^(1/3)*(1 - x))/
(1 - x^3)^(1/3))/Sqrt[3]]/(2*2^(1/3)*Sqrt[3]) + ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]]/(4*2^(
1/3)*Sqrt[3]) - (x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3])/2 + Log[(1 - x)*(1 + x)^2]/(24*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)]/(12*2^(1/3)) - Log[1 + (2^(1/3)*(1 -
x))/(1 - x^3)^(1/3)]/(6*2^(1/3)) - Log[-1 + x + 2^(2/3)*(1 - x^3)^(1/3)]/(8*2^(1/3))

________________________________________________________________________________________

Rubi [C]  time = 0.0196007, antiderivative size = 26, normalized size of antiderivative = 0.08, 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} \[ -\frac{F_1\left (-\frac{4}{3};\frac{4}{3},1;-\frac{1}{3};x^3,-x^3\right )}{4 x^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-AppellF1[-4/3, 4/3, 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 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=-\frac{F_1\left (-\frac{4}{3};\frac{4}{3},1;-\frac{1}{3};x^3,-x^3\right )}{4 x^4}\\ \end{align*}

Mathematica [C]  time = 0.0811564, size = 79, normalized size = 0.26 \[ -\frac{4 x^9 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right )+5 x^6 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right )+\frac{5 \left (-4 x^6+x^3+1\right )}{\sqrt [3]{1-x^3}}}{20 x^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({x}^{3}+1 \right ){x}^{5}} \left ( -{x}^{3}+1 \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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{4}{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)^(4/3)/(x^3+1),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{14} - x^{11} - x^{8} + x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="fricas")

[Out]

integral((-x^3 + 1)^(2/3)/(x^14 - x^11 - x^8 + x^5), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{4}{3}} \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)**(4/3)/(x**3+1),x)

[Out]

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

________________________________________________________________________________________

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{4}{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)^(4/3)/(x^3+1),x, algorithm="giac")

[Out]

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