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

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

[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

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Rubi [A]  time = 0.0400664, antiderivative size = 104, normalized size of antiderivative = 1., 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 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 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]

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 31

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

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*}

Mathematica [C]  time = 0.0048085, 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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Maple [B]  time = 0.015, size = 158, normalized size = 1.5 \begin{align*} -{\frac{1}{324} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-4}}-{\frac{5}{243} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-3}}-{\frac{20}{243} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-2}}-{\frac{25}{81} \left ( 1+\sqrt [3]{-1+x} \right ) ^{-1}}+{\frac{110}{243}\ln \left ( 1+\sqrt [3]{-1+x} \right ) }-{\frac{1}{243} \left ( -75\, \left ( -1+x \right ) ^{7/3}+190\, \left ( -1+x \right ) ^{2}-350\, \left ( -1+x \right ) ^{5/3}+{\frac{1157}{4} \left ( -1+x \right ) ^{{\frac{4}{3}}}}+{\frac{149}{4}}-138\,x-116\, \left ( -1+x \right ) ^{2/3}+137\,\sqrt [3]{-1+x} \right ) \left ( \left ( -1+x \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-1+x}+1 \right ) ^{-4}}-{\frac{55}{243}\ln \left ( \left ( -1+x \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-1+x}+1 \right ) }+{\frac{110\,\sqrt{3}}{243}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,\sqrt [3]{-1+x}-1 \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/324/(1+(-1+x)^(1/3))^4-5/243/(1+(-1+x)^(1/3))^3-20/243/(1+(-1+x)^(1/3))^2-25/81/(1+(-1+x)^(1/3))+110/243*ln
(1+(-1+x)^(1/3))-1/243*(-75*(-1+x)^(7/3)+190*(-1+x)^2-350*(-1+x)^(5/3)+1157/4*(-1+x)^(4/3)+149/4-138*x-116*(-1
+x)^(2/3)+137*(-1+x)^(1/3))/((-1+x)^(2/3)-(-1+x)^(1/3)+1)^4-55/243*ln((-1+x)^(2/3)-(-1+x)^(1/3)+1)+110/243*3^(
1/2)*arctan(1/3*(2*(-1+x)^(1/3)-1)*3^(1/2))

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Maxima [A]  time = 1.41639, size = 142, normalized size = 1.37 \begin{align*} \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 ) \end{align*}

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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Fricas [A]  time = 1.69599, size = 282, normalized size = 2.71 \begin{align*} \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}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 1.07167, size = 111, normalized size = 1.07 \begin{align*} \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 ) \end{align*}

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)