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

Optimal. Leaf size=162 \[ -\frac{49 \left (1-x^3\right )^{2/3}}{40 x^2}-\frac{13 \left (1-x^3\right )^{2/3}}{20 x^5}-\frac{5 \left (1-x^3\right )^{2/3}}{8 x^8}+\frac{1}{2 x^8 \sqrt [3]{1-x^3}}+\frac{\log \left (x^3+1\right )}{12 \sqrt [3]{2}}-\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}} \]

[Out]

1/(2*x^8*(1 - x^3)^(1/3)) - (5*(1 - x^3)^(2/3))/(8*x^8) - (13*(1 - x^3)^(2/3))/(20*x^5) - (49*(1 - x^3)^(2/3))
/(40*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))

________________________________________________________________________________________

Rubi [C]  time = 10.5945, antiderivative size = 643, normalized size of antiderivative = 3.97, 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} \[ -\frac{-81 x^{18} \, _5F_4\left (2,2,2,2,\frac{7}{3};1,1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-324 x^{15} \, _5F_4\left (2,2,2,2,\frac{7}{3};1,1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-486 x^{12} \, _5F_4\left (2,2,2,2,\frac{7}{3};1,1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-324 x^9 \, _5F_4\left (2,2,2,2,\frac{7}{3};1,1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+27 \left (x^3+1\right )^2 \left (-105 x^6-18 x^3+7\right ) x^6 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+54 \left (1-15 x^3\right ) \left (x^3+1\right )^3 x^6 \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-81 x^6 \, _5F_4\left (2,2,2,2,\frac{7}{3};1,1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-3402 x^{18} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-4050 x^{18} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+2268 x^{15} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-6696 x^{15} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+4914 x^{12} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-2268 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-2856 x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+312 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-1162 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-66 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+308 x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-70 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+3402 x^{18}-2268 x^{15}-4914 x^{12}+2856 x^9+1162 x^6-308 x^3+70}{280 x^{11} \left (1-x^3\right )^{7/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(70 - 308*x^3 + 1162*x^6 + 2856*x^9 - 4914*x^12 - 2268*x^15 + 3402*x^18 - 70*Hypergeometric2F1[1/3, 1, 4/3, (
-2*x^3)/(1 - x^3)] + 308*x^3*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)] - 1162*x^6*Hypergeometric2F1[1
/3, 1, 4/3, (-2*x^3)/(1 - x^3)] - 2856*x^9*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)] + 4914*x^12*Hype
rgeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)] + 2268*x^15*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)]
- 3402*x^18*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)] - 66*x^6*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^
3)/(1 - x^3)] + 312*x^9*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] - 2268*x^12*Hypergeometric2F1[2, 7
/3, 10/3, (-2*x^3)/(1 - x^3)] - 6696*x^15*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] - 4050*x^18*Hype
rgeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] + 27*x^6*(1 + x^3)^2*(7 - 18*x^3 - 105*x^6)*HypergeometricPFQ[
{2, 2, 7/3}, {1, 10/3}, (-2*x^3)/(1 - x^3)] + 54*x^6*(1 - 15*x^3)*(1 + x^3)^3*HypergeometricPFQ[{2, 2, 2, 7/3}
, {1, 1, 10/3}, (-2*x^3)/(1 - x^3)] - 81*x^6*HypergeometricPFQ[{2, 2, 2, 2, 7/3}, {1, 1, 1, 10/3}, (-2*x^3)/(1
- x^3)] - 324*x^9*HypergeometricPFQ[{2, 2, 2, 2, 7/3}, {1, 1, 1, 10/3}, (-2*x^3)/(1 - x^3)] - 486*x^12*Hyperg
eometricPFQ[{2, 2, 2, 2, 7/3}, {1, 1, 1, 10/3}, (-2*x^3)/(1 - x^3)] - 324*x^15*HypergeometricPFQ[{2, 2, 2, 2,
7/3}, {1, 1, 1, 10/3}, (-2*x^3)/(1 - x^3)] - 81*x^18*HypergeometricPFQ[{2, 2, 2, 2, 7/3}, {1, 1, 1, 10/3}, (-2
*x^3)/(1 - x^3)])/(280*x^11*(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^9 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=-\frac{70-308 x^3+1162 x^6+2856 x^9-4914 x^{12}-2268 x^{15}+3402 x^{18}-70 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+308 x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-1162 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-2856 x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+4914 x^{12} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+2268 x^{15} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-3402 x^{18} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-66 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+312 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-2268 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-6696 x^{15} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-4050 x^{18} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )+27 x^6 \left (1+x^3\right )^2 \left (7-18 x^3-105 x^6\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 (1-15 x^3\right ) \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 )-81 x^6 \, _5F_4\left (2,2,2,2,\frac{7}{3};1,1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-324 x^9 \, _5F_4\left (2,2,2,2,\frac{7}{3};1,1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-486 x^{12} \, _5F_4\left (2,2,2,2,\frac{7}{3};1,1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-324 x^{15} \, _5F_4\left (2,2,2,2,\frac{7}{3};1,1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-81 x^{18} \, _5F_4\left (2,2,2,2,\frac{7}{3};1,1,1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )}{280 x^{11} \left (1-x^3\right )^{7/3}}\\ \end{align*}

Mathematica [A]  time = 5.16572, size = 136, normalized size = 0.84 \[ \frac{1}{120} \left (5\ 2^{2/3} \left (\log \left (\frac{2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}-\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )-2 \log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt{3}}\right )\right )-\frac{3 \left (-49 x^9+23 x^6+x^3+5\right )}{x^8 \sqrt [3]{1-x^3}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-3*(5 + x^3 + 23*x^6 - 49*x^9))/(x^8*(1 - x^3)^(1/3)) + 5*2^(2/3)*(-2*Sqrt[3]*ArcTan[(-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)]))/120

________________________________________________________________________________________

Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({x}^{3}+1 \right ){x}^{9}} \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^9/(-x^3+1)^(4/3)/(x^3+1),x)

[Out]

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

________________________________________________________________________________________

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^{9}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 22.2233, size = 933, normalized size = 5.76 \begin{align*} -\frac{10 \, \sqrt{6} 2^{\frac{1}{6}} \left (-1\right )^{\frac{1}{3}}{\left (x^{11} - x^{8}\right )} \arctan \left (\frac{2^{\frac{1}{6}}{\left (6 \, \sqrt{6} 2^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (5 \, x^{7} + 4 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 12 \, \sqrt{6} \left (-1\right )^{\frac{1}{3}}{\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{6} 2^{\frac{1}{3}}{\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \,{\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{11} - x^{8}\right )} \log \left (\frac{6 \cdot 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{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 (-1\right )^{\frac{1}{3}}{\left (x^{11} - x^{8}\right )} \log \left (-\frac{3 \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{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 ) + 9 \,{\left (49 \, x^{9} - 23 \, x^{6} - x^{3} - 5\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{360 \,{\left (x^{11} - x^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/360*(10*sqrt(6)*2^(1/6)*(-1)^(1/3)*(x^11 - x^8)*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/3)*(-1)^(2/3)*(5*x^7 + 4
*x^4 - x)*(-x^3 + 1)^(2/3) - 12*sqrt(6)*(-1)^(1/3)*(19*x^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3) - sqrt(6)*2^(1/3)*
(71*x^9 - 111*x^6 + 33*x^3 - 1))/(109*x^9 - 105*x^6 + 3*x^3 + 1)) - 10*2^(2/3)*(-1)^(1/3)*(x^11 - x^8)*log((6*
2^(1/3)*(-1)^(2/3)*(-x^3 + 1)^(1/3)*x^2 - 2^(2/3)*(-1)^(1/3)*(x^3 + 1) + 6*(-x^3 + 1)^(2/3)*x)/(x^3 + 1)) + 5*
2^(2/3)*(-1)^(1/3)*(x^11 - x^8)*log(-(3*2^(2/3)*(-1)^(1/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) - 2^(1/3)*(-1)^(2/3)*(
19*x^6 - 16*x^3 + 1) + 12*(2*x^5 - x^2)*(-x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) + 9*(49*x^9 - 23*x^6 - x^3 - 5)*(
-x^3 + 1)^(2/3))/(x^11 - x^8)

________________________________________________________________________________________

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

[Out]

Integral(1/(x**9*(-(x - 1)*(x**2 + x + 1))**(4/3)*(x + 1)*(x**2 - x + 1)), x)

________________________________________________________________________________________

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^{9}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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