3.645 \(\int \frac{x^9}{(1-x^3)^{4/3} (1+x^3)} \, dx\)

Optimal. Leaf size=174 \[ \frac{x^4}{2 \sqrt [3]{1-x^3}}+\frac{5}{6} \left (1-x^3\right )^{2/3} x+\frac{\log \left (x^3+1\right )}{12 \sqrt [3]{2}}-\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}-\frac{1}{6} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}} \]

[Out]

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

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Rubi [C]  time = 0.0178236, antiderivative size = 26, normalized size of antiderivative = 0.15, 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} \[ \frac{1}{10} x^{10} F_1\left (\frac{10}{3};\frac{4}{3},1;\frac{13}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[x^9/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

(x^10*AppellF1[10/3, 4/3, 1, 13/3, x^3, -x^3])/10

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^9}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=\frac{1}{10} x^{10} F_1\left (\frac{10}{3};\frac{4}{3},1;\frac{13}{3};x^3,-x^3\right )\\ \end{align*}

Mathematica [C]  time = 0.172993, size = 152, normalized size = 0.87 \[ \frac{1}{72} \left (-6 x^4 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right )-\frac{12 \left (2 x^3-5\right ) x}{\sqrt [3]{1-x^3}}-5\ 2^{2/3} \left (-\log \left (\frac{2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}-\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )+2 \log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt{3}}\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^9/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

((-12*x*(-5 + 2*x^3))/(1 - x^3)^(1/3) - 6*x^4*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3] - 5*2^(2/3)*(2*Sqrt[3]*Arc
Tan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] - Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1
+ x^3)^(1/3)] + 2*Log[1 + (2^(1/3)*x)/(-1 + x^3)^(1/3)]))/72

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{9}}{{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^9/(-x^3+1)^(4/3)/(x^3+1),x)

[Out]

int(x^9/(-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^{9}}{{\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^9/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="maxima")

[Out]

integrate(x^9/((x^3 + 1)*(-x^3 + 1)^(4/3)), x)

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Fricas [B]  time = 1.93425, size = 776, normalized size = 4.46 \begin{align*} \frac{6 \, \sqrt{6} 2^{\frac{1}{6}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} - 1\right )} \arctan \left (\frac{2^{\frac{1}{6}}{\left (\sqrt{6} 2^{\frac{1}{3}} x + 2 \, \sqrt{6} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}}{6 \, x}\right ) + 6 \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} - 1\right )} \log \left (\frac{2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}} x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) - 3 \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} - 1\right )} \log \left (-\frac{2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} x^{2} + 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x -{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) + 8 \, \sqrt{3}{\left (x^{3} - 1\right )} \arctan \left (-\frac{\sqrt{3} x - 2 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right ) - 8 \,{\left (x^{3} - 1\right )} \log \left (\frac{x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) + 4 \,{\left (x^{3} - 1\right )} \log \left (\frac{x^{2} -{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) + 12 \,{\left (2 \, x^{4} - 5 \, x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{72 \,{\left (x^{3} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="fricas")

[Out]

1/72*(6*sqrt(6)*2^(1/6)*(-1)^(1/3)*(x^3 - 1)*arctan(1/6*2^(1/6)*(sqrt(6)*2^(1/3)*x + 2*sqrt(6)*(-1)^(1/3)*(-x^
3 + 1)^(1/3))/x) + 6*2^(2/3)*(-1)^(1/3)*(x^3 - 1)*log((2^(1/3)*(-1)^(2/3)*x + (-x^3 + 1)^(1/3))/x) - 3*2^(2/3)
*(-1)^(1/3)*(x^3 - 1)*log(-(2^(2/3)*(-1)^(1/3)*x^2 + 2^(1/3)*(-1)^(2/3)*(-x^3 + 1)^(1/3)*x - (-x^3 + 1)^(2/3))
/x^2) + 8*sqrt(3)*(x^3 - 1)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x) - 8*(x^3 - 1)*log((x + (-x
^3 + 1)^(1/3))/x) + 4*(x^3 - 1)*log((x^2 - (-x^3 + 1)^(1/3)*x + (-x^3 + 1)^(2/3))/x^2) + 12*(2*x^4 - 5*x)*(-x^
3 + 1)^(2/3))/(x^3 - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{9}}{\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**9/(-x**3+1)**(4/3)/(x**3+1),x)

[Out]

Integral(x**9/((-(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^{9}}{{\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^9/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="giac")

[Out]

integrate(x^9/((x^3 + 1)*(-x^3 + 1)^(4/3)), x)