3.217 \(\int \frac {1}{(-1+x)^{2/3} x^5} \, dx\)

Optimal. Leaf size=104 \[ \frac {\sqrt [3]{x-1}}{4 x^4}+\frac {11 \sqrt [3]{x-1}}{36 x^3}+\frac {11 \sqrt [3]{x-1}}{27 x^2}+\frac {55 \sqrt [3]{x-1}}{81 x}+\frac {55}{81} \log \left (\sqrt [3]{x-1}+1\right )-\frac {55 \log (x)}{243}-\frac {110 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{81 \sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {51, 58, 618, 204, 31} \[ \frac {11 \sqrt [3]{x-1}}{27 x^2}+\frac {11 \sqrt [3]{x-1}}{36 x^3}+\frac {\sqrt [3]{x-1}}{4 x^4}+\frac {55 \sqrt [3]{x-1}}{81 x}+\frac {55}{81} \log \left (\sqrt [3]{x-1}+1\right )-\frac {55 \log (x)}{243}-\frac {110 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{81 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((-1 + x)^(2/3)*x^5),x]

[Out]

(-1 + x)^(1/3)/(4*x^4) + (11*(-1 + x)^(1/3))/(36*x^3) + (11*(-1 + x)^(1/3))/(27*x^2) + (55*(-1 + x)^(1/3))/(81
*x) - (110*ArcTan[(1 - 2*(-1 + x)^(1/3))/Sqrt[3]])/(81*Sqrt[3]) + (55*Log[1 + (-1 + x)^(1/3)])/81 - (55*Log[x]
)/243

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d
*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x
] && NegQ[(b*c - a*d)/b]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {1}{(-1+x)^{2/3} x^5} \, dx &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11}{12} \int \frac {1}{(-1+x)^{2/3} x^4} \, dx\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {22}{27} \int \frac {1}{(-1+x)^{2/3} x^3} \, dx\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55}{81} \int \frac {1}{(-1+x)^{2/3} x^2} \, dx\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55 \sqrt [3]{-1+x}}{81 x}+\frac {110}{243} \int \frac {1}{(-1+x)^{2/3} x} \, dx\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55 \sqrt [3]{-1+x}}{81 x}-\frac {55 \log (x)}{243}+\frac {55}{81} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x}\right )+\frac {55}{81} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55 \sqrt [3]{-1+x}}{81 x}+\frac {55}{81} \log \left (1+\sqrt [3]{-1+x}\right )-\frac {55 \log (x)}{243}-\frac {110}{81} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x}\right )\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55 \sqrt [3]{-1+x}}{81 x}-\frac {110 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {55}{81} \log \left (1+\sqrt [3]{-1+x}\right )-\frac {55 \log (x)}{243}\\ \end {align*}

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Mathematica [C]  time = 0.00, size = 22, normalized size = 0.21 \[ 3 \sqrt [3]{x-1} \, _2F_1\left (\frac {1}{3},5;\frac {4}{3};1-x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((-1 + x)^(2/3)*x^5),x]

[Out]

3*(-1 + x)^(1/3)*Hypergeometric2F1[1/3, 5, 4/3, 1 - x]

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IntegrateAlgebraic [A]  time = 0.07, size = 97, normalized size = 0.93 \[ \frac {\sqrt [3]{x-1} \left (220 x^3+132 x^2+99 x+81\right )}{324 x^4}+\frac {110}{243} \log \left (\sqrt [3]{x-1}+1\right )-\frac {55}{243} \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )-\frac {110 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{81 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x)^(1/3)*(81 + 99*x + 132*x^2 + 220*x^3))/(324*x^4) - (110*ArcTan[1/Sqrt[3] - (2*(-1 + x)^(1/3))/Sqrt[3
]])/(81*Sqrt[3]) + (110*Log[1 + (-1 + x)^(1/3)])/243 - (55*Log[1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)])/243

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fricas [A]  time = 1.19, size = 86, normalized size = 0.83 \[ \frac {440 \, \sqrt {3} x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 220 \, x^{4} \log \left ({\left (x - 1\right )}^{\frac {2}{3}} - {\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + 440 \, x^{4} \log \left ({\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (220 \, x^{3} + 132 \, x^{2} + 99 \, x + 81\right )} {\left (x - 1\right )}^{\frac {1}{3}}}{972 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(2/3)/x^5,x, algorithm="fricas")

[Out]

1/972*(440*sqrt(3)*x^4*arctan(2/3*sqrt(3)*(x - 1)^(1/3) - 1/3*sqrt(3)) - 220*x^4*log((x - 1)^(2/3) - (x - 1)^(
1/3) + 1) + 440*x^4*log((x - 1)^(1/3) + 1) + 3*(220*x^3 + 132*x^2 + 99*x + 81)*(x - 1)^(1/3))/x^4

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giac [A]  time = 1.14, size = 82, normalized size = 0.79 \[ \frac {110}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {220 \, {\left (x - 1\right )}^{\frac {10}{3}} + 792 \, {\left (x - 1\right )}^{\frac {7}{3}} + 1023 \, {\left (x - 1\right )}^{\frac {4}{3}} + 532 \, {\left (x - 1\right )}^{\frac {1}{3}}}{324 \, x^{4}} - \frac {55}{243} \, \log \left ({\left (x - 1\right )}^{\frac {2}{3}} - {\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {110}{243} \, \log \left ({\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(2/3)/x^5,x, algorithm="giac")

[Out]

110/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x - 1)^(1/3) - 1)) + 1/324*(220*(x - 1)^(10/3) + 792*(x - 1)^(7/3) + 10
23*(x - 1)^(4/3) + 532*(x - 1)^(1/3))/x^4 - 55/243*log((x - 1)^(2/3) - (x - 1)^(1/3) + 1) + 110/243*log((x - 1
)^(1/3) + 1)

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maple [C]  time = 0.54, size = 85, normalized size = 0.82

method	result	size
meijerg	\(\frac {\left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {2}{3}} \left (\frac {308 \Gamma \left (\frac {2}{3}\right ) x \hypergeom \left (\left [1, 1, \frac {17}{3}\right ], \left [2, 6\right ], x\right )}{729}+\frac {110 \left (\frac {877}{1320}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+\ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{243}-\frac {\Gamma \left (\frac {2}{3}\right )}{4 x^{4}}-\frac {2 \Gamma \left (\frac {2}{3}\right )}{9 x^{3}}-\frac {5 \Gamma \left (\frac {2}{3}\right )}{18 x^{2}}-\frac {40 \Gamma \left (\frac {2}{3}\right )}{81 x}\right )}{\Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (-1+x \right )^{\frac {2}{3}}}\)	\(85\)
risch	\(\frac {220 x^{4}-88 x^{3}-33 x^{2}-18 x -81}{324 x^{4} \left (-1+x \right )^{\frac {2}{3}}}+\frac {110 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+\ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{243 \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (-1+x \right )^{\frac {2}{3}}}\)	\(87\)
derivativedivides	\(-\frac {-75 \left (-1+x \right )^{\frac {7}{3}}+190 \left (-1+x \right )^{2}-350 \left (-1+x \right )^{\frac {5}{3}}+\frac {1157 \left (-1+x \right )^{\frac {4}{3}}}{4}+\frac {149}{4}-138 x -116 \left (-1+x \right )^{\frac {2}{3}}+137 \left (-1+x \right )^{\frac {1}{3}}}{243 \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )^{4}}-\frac {55 \ln \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )}{243}+\frac {110 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{243}-\frac {1}{324 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{4}}-\frac {5}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{3}}-\frac {20}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{2}}-\frac {25}{81 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}+\frac {110 \ln \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}{243}\)	\(158\)
default	\(-\frac {-75 \left (-1+x \right )^{\frac {7}{3}}+190 \left (-1+x \right )^{2}-350 \left (-1+x \right )^{\frac {5}{3}}+\frac {1157 \left (-1+x \right )^{\frac {4}{3}}}{4}+\frac {149}{4}-138 x -116 \left (-1+x \right )^{\frac {2}{3}}+137 \left (-1+x \right )^{\frac {1}{3}}}{243 \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )^{4}}-\frac {55 \ln \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )}{243}+\frac {110 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{243}-\frac {1}{324 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{4}}-\frac {5}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{3}}-\frac {20}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{2}}-\frac {25}{81 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}+\frac {110 \ln \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}{243}\)	\(158\)
trager	\(\frac {\left (220 x^{3}+132 x^{2}+99 x +81\right ) \left (-1+x \right )^{\frac {1}{3}}}{324 x^{4}}+\frac {110 \ln \left (\frac {144 \left (-1+x \right )^{\frac {2}{3}} \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )+4608 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2} x +144 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \left (-1+x \right )^{\frac {1}{3}}-9216 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2}+144 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) x +3 \left (-1+x \right )^{\frac {1}{3}}-192 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )+x -1}{x}\right )}{243}-\frac {110 \ln \left (-\frac {144 \left (-1+x \right )^{\frac {2}{3}} \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )+2304 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2} x +3 \left (-1+x \right )^{\frac {2}{3}}-4608 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2}+96 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) x +3 \left (-1+x \right )^{\frac {1}{3}}-240 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )-2}{x}\right )}{243}-\frac {1760 \ln \left (-\frac {144 \left (-1+x \right )^{\frac {2}{3}} \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )+2304 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2} x +3 \left (-1+x \right )^{\frac {2}{3}}-4608 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2}+96 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) x +3 \left (-1+x \right )^{\frac {1}{3}}-240 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )-2}{x}\right ) \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )}{81}\)	\(349\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-1+x)^(2/3)/x^5,x,method=_RETURNVERBOSE)

[Out]

1/GAMMA(2/3)/signum(-1+x)^(2/3)*(-signum(-1+x))^(2/3)*(308/729*GAMMA(2/3)*x*hypergeom([1,1,17/3],[2,6],x)+110/
243*(877/1320+1/6*Pi*3^(1/2)-3/2*ln(3)+ln(x)+I*Pi)*GAMMA(2/3)-1/4*GAMMA(2/3)/x^4-2/9*GAMMA(2/3)/x^3-5/18*GAMMA
(2/3)/x^2-40/81*GAMMA(2/3)/x)

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maxima [A]  time = 1.39, size = 105, normalized size = 1.01 \[ \frac {110}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {220 \, {\left (x - 1\right )}^{\frac {10}{3}} + 792 \, {\left (x - 1\right )}^{\frac {7}{3}} + 1023 \, {\left (x - 1\right )}^{\frac {4}{3}} + 532 \, {\left (x - 1\right )}^{\frac {1}{3}}}{324 \, {\left ({\left (x - 1\right )}^{4} + 4 \, {\left (x - 1\right )}^{3} + 6 \, {\left (x - 1\right )}^{2} + 4 \, x - 3\right )}} - \frac {55}{243} \, \log \left ({\left (x - 1\right )}^{\frac {2}{3}} - {\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {110}{243} \, \log \left ({\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(2/3)/x^5,x, algorithm="maxima")

[Out]

110/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x - 1)^(1/3) - 1)) + 1/324*(220*(x - 1)^(10/3) + 792*(x - 1)^(7/3) + 10
23*(x - 1)^(4/3) + 532*(x - 1)^(1/3))/((x - 1)^4 + 4*(x - 1)^3 + 6*(x - 1)^2 + 4*x - 3) - 55/243*log((x - 1)^(
2/3) - (x - 1)^(1/3) + 1) + 110/243*log((x - 1)^(1/3) + 1)

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mupad [B]  time = 0.21, size = 120, normalized size = 1.15 \[ \frac {110\,\ln \left (\frac {12100\,{\left (x-1\right )}^{1/3}}{6561}+\frac {12100}{6561}\right )}{243}+\frac {\frac {133\,{\left (x-1\right )}^{1/3}}{81}+\frac {341\,{\left (x-1\right )}^{4/3}}{108}+\frac {22\,{\left (x-1\right )}^{7/3}}{9}+\frac {55\,{\left (x-1\right )}^{10/3}}{81}}{4\,x+6\,{\left (x-1\right )}^2+4\,{\left (x-1\right )}^3+{\left (x-1\right )}^4-3}-\ln \left (\frac {55}{27}-\frac {110\,{\left (x-1\right )}^{1/3}}{27}+\frac {\sqrt {3}\,55{}\mathrm {i}}{27}\right )\,\left (\frac {55}{243}+\frac {\sqrt {3}\,55{}\mathrm {i}}{243}\right )+\ln \left (\frac {110\,{\left (x-1\right )}^{1/3}}{27}-\frac {55}{27}+\frac {\sqrt {3}\,55{}\mathrm {i}}{27}\right )\,\left (-\frac {55}{243}+\frac {\sqrt {3}\,55{}\mathrm {i}}{243}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(x - 1)^(2/3)),x)

[Out]

(110*log((12100*(x - 1)^(1/3))/6561 + 12100/6561))/243 + ((133*(x - 1)^(1/3))/81 + (341*(x - 1)^(4/3))/108 + (
22*(x - 1)^(7/3))/9 + (55*(x - 1)^(10/3))/81)/(4*x + 6*(x - 1)^2 + 4*(x - 1)^3 + (x - 1)^4 - 3) - log((3^(1/2)
*55i)/27 - (110*(x - 1)^(1/3))/27 + 55/27)*((3^(1/2)*55i)/243 + 55/243) + log((110*(x - 1)^(1/3))/27 + (3^(1/2
)*55i)/27 - 55/27)*((3^(1/2)*55i)/243 - 55/243)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)**(2/3)/x**5,x)

[Out]

Timed out

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