3.656 \(\int \frac {1}{x^2 (1-x^3)^{4/3} (1+x^3)} \, dx\)

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

[Out]

1/2/x/(-x^3+1)^(1/3)-3/2*(-x^3+1)^(2/3)/x-3/4*x^2*hypergeom([1/3, 2/3],[5/3],x^3)-1/48*ln((1-x)*(1+x)^2)*2^(2/
3)-1/24*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)+1/12*ln(1+2^(1/3)*(1-x)/(-x^
3+1)^(1/3))*2^(2/3)+1/16*ln(-1+x+2^(2/3)*(-x^3+1)^(1/3))*2^(2/3)-1/12*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^3+1)^(
1/3))*3^(1/2))*2^(2/3)*3^(1/2)-1/24*arctan(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)

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Rubi [C]  time = 0.02, antiderivative size = 24, 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} \[ -\frac {F_1\left (-\frac {1}{3};\frac {4}{3},1;\frac {2}{3};x^3,-x^3\right )}{x} \]

Warning: Unable to verify antiderivative.

[In]

Int[1/(x^2*(1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

-(AppellF1[-1/3, 4/3, 1, 2/3, x^3, -x^3]/x)

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^2 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=-\frac {F_1\left (-\frac {1}{3};\frac {4}{3},1;\frac {2}{3};x^3,-x^3\right )}{x}\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 76, normalized size = 0.26 \[ -\frac {3}{10} x^5 F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};x^3,-x^3\right )-x^2 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^3,-x^3\right )+\frac {3 x^3-2}{2 \sqrt [3]{1-x^3} x} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^2*(1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

(-2 + 3*x^3)/(2*x*(1 - x^3)^(1/3)) - x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3] - (3*x^5*AppellF1[5/3, 1/3, 1,
8/3, x^3, -x^3])/10

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fricas [F]  time = 2.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{11} - x^{8} - x^{5} + x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="giac")

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(4/3)*x^2), x)

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maple [F]  time = 0.57, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-x^{3}+1\right )^{\frac {4}{3}} \left (x^{3}+1\right ) x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(-x^3+1)^(4/3)/(x^3+1),x)

[Out]

int(1/x^2/(-x^3+1)^(4/3)/(x^3+1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="maxima")

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(4/3)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^2\,{\left (1-x^3\right )}^{4/3}\,\left (x^3+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(1 - x^3)^(4/3)*(x^3 + 1)),x)

[Out]

int(1/(x^2*(1 - x^3)^(4/3)*(x^3 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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