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

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

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Rubi [C]  time = 10.59, 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} \begin {gather*} -\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}} \end {gather*}

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*}

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Mathematica [A]  time = 5.21, size = 136, normalized size = 0.84 \begin {gather*} \frac {1}{120} \left (5\ 2^{2/3} \left (-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 )+\log \left (-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )\right )-\frac {3 \left (-49 x^9+23 x^6+x^3+5\right )}{x^8 \sqrt [3]{1-x^3}}\right ) \end {gather*}

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

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IntegrateAlgebraic [A]  time = 0.39, size = 169, normalized size = 1.04 \begin {gather*} -\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}+2 x\right )}{6 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{1-x^3}-x}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3} x-\sqrt [3]{2} \left (1-x^3\right )^{2/3}-2 x^2\right )}{12 \sqrt [3]{2}}+\frac {\left (1-x^3\right )^{2/3} \left (-49 x^9+23 x^6+x^3+5\right )}{40 x^8 \left (x^3-1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - x^3)^(2/3)*(5 + x^3 + 23*x^6 - 49*x^9))/(40*x^8*(-1 + x^3)) - ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 - x^3)
^(1/3))]/(2*2^(1/3)*Sqrt[3]) - Log[2*x + 2^(2/3)*(1 - x^3)^(1/3)]/(6*2^(1/3)) + Log[-2*x^2 + 2^(2/3)*x*(1 - x^
3)^(1/3) - 2^(1/3)*(1 - x^3)^(2/3)]/(12*2^(1/3))

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fricas [B]  time = 2.59, size = 351, normalized size = 2.17 \begin {gather*} -\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 {gather*}

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}} x^{9}}\,{d x} \end {gather*}

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)

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maple [C]  time = 2.68, size = 649, normalized size = 4.01

result too large to display

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]

1/40*(49*x^9-23*x^6-x^3-5)/x^8/(-x^3+1)^(1/3)-1/2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*ln(-(-9
*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^3*x^3-36*RootOf(RootOf(_Z^3-4)^2+6*_Z*Roo
tOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^3+12*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_
Z^2)*RootOf(_Z^3-4)^2*x+4*RootOf(_Z^3-4)^2*(-x^3+1)^(1/3)*x^2+30*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*R
ootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^2+3*RootOf(_Z^3-4)*x^3+12*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+
36*_Z^2)*x^3-2*(-x^3+1)^(2/3)*x-3*RootOf(_Z^3-4)-12*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2))/(x+1
)/(x^2-x+1))-1/12*RootOf(_Z^3-4)*ln(-(3*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^3*
x^3+27*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^3-6*(-x^3+1)^(2/3)*RootOf(Roo
tOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x-2*RootOf(_Z^3-4)^2*(-x^3+1)^(1/3)*x^2+3*(-x^3+1)
^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^2+3*RootOf(_Z^3-4)*x^3+27*RootOf(
RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^3-5*(-x^3+1)^(2/3)*x-RootOf(_Z^3-4)-9*RootOf(RootOf(_Z^3-4)^2+
6*_Z*RootOf(_Z^3-4)+36*_Z^2))/(x+1)/(x^2-x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}} x^{9}}\,{d x} \end {gather*}

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^9\,{\left (1-x^3\right )}^{4/3}\,\left (x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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)

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