3.114 \(\int \frac {x (1-x^3)^{2/3}}{1+x^3} \, dx\)

Optimal. Leaf size=250 \[ -\frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left ((1-x) (x+1)^2\right )}{6 \sqrt [3]{2}} \]

[Out]

-1/2*x^2*hypergeom([1/3, 2/3],[5/3],x^3)+1/12*ln((1-x)*(1+x)^2)*2^(2/3)+1/6*ln(1+2^(2/3)*(1-x)^2/(-x^3+1)^(2/3
)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(2/3)-1/3*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(2/3)-1/4*ln(-1+x+2^(2/3)*(-x
^3+1)^(1/3))*2^(2/3)+1/3*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)+1/6*arctan(1/3
*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)*2^(2/3)

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Rubi [C]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 0.10, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {510} \[ \frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[(x*(1 - x^3)^(2/3))/(1 + x^3),x]

[Out]

(x^2*AppellF1[2/3, -2/3, 1, 5/3, x^3, -x^3])/2

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 \left (1-x^3\right )^{2/3}}{1+x^3} \, dx &=\frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 26, normalized size = 0.10 \[ \frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^2*AppellF1[2/3, -2/3, 1, 5/3, x^3, -x^3])/2

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fricas [F]  time = 4.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}} x}{x^{3}+1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(1 - x^3)^(2/3))/(x^3 + 1),x)

[Out]

int((x*(1 - x^3)^(2/3))/(x^3 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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