3.652 \(\int \frac{x^{10}}{(1-x^3)^{4/3} (1+x^3)} \, dx\)

Optimal. Leaf size=292 \[ -\frac{1}{2} x^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right )+\frac{x^5}{2 \sqrt [3]{1-x^3}}+\frac{3}{4} \left (1-x^3\right )^{2/3} x^2-\frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{12 \sqrt [3]{2}}+\frac{\log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6 \sqrt [3]{2}}+\frac{\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{8 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{4 \sqrt [3]{2} \sqrt{3}}-\frac{\log \left ((1-x) (x+1)^2\right )}{24 \sqrt [3]{2}} \]

[Out]

x^5/(2*(1 - x^3)^(1/3)) + (3*x^2*(1 - x^3)^(2/3))/4 - 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*Hyp
ergeometric2F1[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))

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Rubi [C]  time = 0.0188917, 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} \[ \frac{1}{11} x^{11} F_1\left (\frac{11}{3};\frac{4}{3},1;\frac{14}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[x^10/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

(x^11*AppellF1[11/3, 4/3, 1, 14/3, x^3, -x^3])/11

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{x^{10}}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=\frac{1}{11} x^{11} F_1\left (\frac{11}{3};\frac{4}{3},1;\frac{14}{3};x^3,-x^3\right )\\ \end{align*}

Mathematica [C]  time = 0.0729359, size = 71, normalized size = 0.24 \[ \frac{1}{20} x^2 \left (-4 x^3 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right )-15 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right )-\frac{5 \left (x^3-3\right )}{\sqrt [3]{1-x^3}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^10/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

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

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{10}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(-x^3+1)^(4/3)/(x^3+1),x)

[Out]

int(x^10/(-x^3+1)^(4/3)/(x^3+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{10}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="maxima")

[Out]

integrate(x^10/((x^3 + 1)*(-x^3 + 1)^(4/3)), 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^{10}}{x^{9} - x^{6} - x^{3} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="fricas")

[Out]

integral((-x^3 + 1)^(2/3)*x^10/(x^9 - x^6 - x^3 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{10}}{\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(x**10/(-x**3+1)**(4/3)/(x**3+1),x)

[Out]

Integral(x**10/((-(x - 1)*(x**2 + x + 1))**(4/3)*(x + 1)*(x**2 - x + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{10}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="giac")

[Out]

integrate(x^10/((x^3 + 1)*(-x^3 + 1)^(4/3)), x)