3.650 \(\int \frac{1}{x^6 (1-x^3)^{4/3} (1+x^3)} \, dx\)

Optimal. Leaf size=144 \[ -\frac{4 \left (1-x^3\right )^{2/3}}{5 x^2}-\frac{7 \left (1-x^3\right )^{2/3}}{10 x^5}+\frac{1}{2 x^5 \sqrt [3]{1-x^3}}-\frac{\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}} \]

[Out]

1/(2*x^5*(1 - x^3)^(1/3)) - (7*(1 - x^3)^(2/3))/(10*x^5) - (4*(1 - x^3)^(2/3))/(5*x^2) - 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))

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Rubi [C]  time = 8.31832, antiderivative size = 397, normalized size of antiderivative = 2.76, 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} \[ -\frac{54 \left (x^3+1\right )^2 \left (6 x^3+1\right ) x^6 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+54 \left (x^3+1\right )^3 x^6 \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+567 x^{15} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+594 x^{15} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-378 x^{12} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+972 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-819 x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+342 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+476 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-36 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+182 x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-28 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-567 x^{15}+378 x^{12}+819 x^9-476 x^6-182 x^3+28}{70 x^8 \left (1-x^3\right )^{7/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[1/(x^6*(1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

-(28 - 182*x^3 - 476*x^6 + 819*x^9 + 378*x^12 - 567*x^15 - 28*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3
)] + 182*x^3*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)] + 476*x^6*Hypergeometric2F1[1/3, 1, 4/3, (-2*x
^3)/(1 - x^3)] - 819*x^9*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)] - 378*x^12*Hypergeometric2F1[1/3,
1, 4/3, (-2*x^3)/(1 - x^3)] + 567*x^15*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)] - 36*x^6*Hypergeomet
ric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] + 342*x^9*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] + 972*x
^12*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] + 594*x^15*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1
- x^3)] + 54*x^6*(1 + x^3)^2*(1 + 6*x^3)*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, (-2*x^3)/(1 - x^3)] + 54*x
^6*(1 + x^3)^3*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, (-2*x^3)/(1 - x^3)])/(70*x^8*(1 - x^3)^(7/3))

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^6 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=-\frac{28-182 x^3-476 x^6+819 x^9+378 x^{12}-567 x^{15}-28 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+182 x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+476 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-819 x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-378 x^{12} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+567 x^{15} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-36 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+342 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+972 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+594 x^{15} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+54 x^6 \left (1+x^3\right )^2 \left (1+6 x^3\right ) \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+54 x^6 \left (1+x^3\right )^3 \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )}{70 x^8 \left (1-x^3\right )^{7/3}}\\ \end{align*}

Mathematica [C]  time = 5.01472, size = 373, normalized size = 2.59 \[ \frac{-54 \left (x^3+1\right )^2 \left (6 x^3+1\right ) x^6 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{10}{3}\right \},\frac{2 x^3}{x^3-1}\right )-54 \left (x^3+1\right )^3 x^6 \text{HypergeometricPFQ}\left (\left \{2,2,2,\frac{7}{3}\right \},\left \{1,1,\frac{10}{3}\right \},\frac{2 x^3}{x^3-1}\right )-567 x^{15} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right )-594 x^{15} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right )+378 x^{12} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right )-972 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right )+819 x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right )-342 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right )-476 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right )+36 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right )-182 x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right )+28 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right )+567 x^{15}-378 x^{12}-819 x^9+476 x^6+182 x^3-28}{70 x^8 \left (1-x^3\right )^{7/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^6*(1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

(-28 + 182*x^3 + 476*x^6 - 819*x^9 - 378*x^12 + 567*x^15 + 28*Hypergeometric2F1[1/3, 1, 4/3, (2*x^3)/(-1 + x^3
)] - 182*x^3*Hypergeometric2F1[1/3, 1, 4/3, (2*x^3)/(-1 + x^3)] - 476*x^6*Hypergeometric2F1[1/3, 1, 4/3, (2*x^
3)/(-1 + x^3)] + 819*x^9*Hypergeometric2F1[1/3, 1, 4/3, (2*x^3)/(-1 + x^3)] + 378*x^12*Hypergeometric2F1[1/3,
1, 4/3, (2*x^3)/(-1 + x^3)] - 567*x^15*Hypergeometric2F1[1/3, 1, 4/3, (2*x^3)/(-1 + x^3)] + 36*x^6*Hypergeomet
ric2F1[2, 7/3, 10/3, (2*x^3)/(-1 + x^3)] - 342*x^9*Hypergeometric2F1[2, 7/3, 10/3, (2*x^3)/(-1 + x^3)] - 972*x
^12*Hypergeometric2F1[2, 7/3, 10/3, (2*x^3)/(-1 + x^3)] - 594*x^15*Hypergeometric2F1[2, 7/3, 10/3, (2*x^3)/(-1
+ x^3)] - 54*x^6*(1 + x^3)^2*(1 + 6*x^3)*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, (2*x^3)/(-1 + x^3)] - 54*x
^6*(1 + x^3)^3*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, (2*x^3)/(-1 + x^3)])/(70*x^8*(1 - x^3)^(7/3))

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({x}^{3}+1 \right ){x}^{6}} \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^6/(-x^3+1)^(4/3)/(x^3+1),x)

[Out]

int(1/x^6/(-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{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{4}{3}} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="maxima")

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(4/3)*x^6), x)

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Fricas [B]  time = 22.1061, size = 780, normalized size = 5.42 \begin{align*} -\frac{10 \, \sqrt{6} 2^{\frac{1}{6}}{\left (x^{8} - x^{5}\right )} \arctan \left (\frac{2^{\frac{1}{6}}{\left (6 \, \sqrt{6} 2^{\frac{2}{3}}{\left (5 \, x^{7} + 4 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - \sqrt{6} 2^{\frac{1}{3}}{\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )} + 12 \, \sqrt{6}{\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}}{6 \,{\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \cdot 2^{\frac{2}{3}}{\left (x^{8} - x^{5}\right )} \log \left (\frac{6 \cdot 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 2^{\frac{2}{3}}{\left (x^{3} + 1\right )} + 6 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x}{x^{3} + 1}\right ) + 5 \cdot 2^{\frac{2}{3}}{\left (x^{8} - x^{5}\right )} \log \left (\frac{3 \cdot 2^{\frac{2}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \,{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 36 \,{\left (8 \, x^{6} - x^{3} - 2\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{360 \,{\left (x^{8} - x^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="fricas")

[Out]

-1/360*(10*sqrt(6)*2^(1/6)*(x^8 - x^5)*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/3)*(5*x^7 + 4*x^4 - x)*(-x^3 + 1)^(2
/3) - sqrt(6)*2^(1/3)*(71*x^9 - 111*x^6 + 33*x^3 - 1) + 12*sqrt(6)*(19*x^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3))/(
109*x^9 - 105*x^6 + 3*x^3 + 1)) - 10*2^(2/3)*(x^8 - x^5)*log((6*2^(1/3)*(-x^3 + 1)^(1/3)*x^2 + 2^(2/3)*(x^3 +
1) + 6*(-x^3 + 1)^(2/3)*x)/(x^3 + 1)) + 5*2^(2/3)*(x^8 - x^5)*log((3*2^(2/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) + 2^
(1/3)*(19*x^6 - 16*x^3 + 1) - 12*(2*x^5 - x^2)*(-x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) + 36*(8*x^6 - x^3 - 2)*(-x
^3 + 1)^(2/3))/(x^8 - x^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{6} \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**6/(-x**3+1)**(4/3)/(x**3+1),x)

[Out]

Integral(1/(x**6*(-(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{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{4}{3}} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="giac")

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(4/3)*x^6), x)