3.13.73 \(\int \frac {(-1+x^5)^{2/3}}{x^6} \, dx\)

Optimal. Leaf size=92 \[ -\frac {\left (x^5-1\right )^{2/3}}{5 x^5}-\frac {2}{15} \log \left (\sqrt [3]{x^5-1}+1\right )+\frac {1}{15} \log \left (\left (x^5-1\right )^{2/3}-\sqrt [3]{x^5-1}+1\right )-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^5-1}}{\sqrt {3}}\right )}{5 \sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 68, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 47, 56, 618, 204, 31} \begin {gather*} -\frac {\left (x^5-1\right )^{2/3}}{5 x^5}-\frac {1}{5} \log \left (\sqrt [3]{x^5-1}+1\right )-\frac {2 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^5-1}}{\sqrt {3}}\right )}{5 \sqrt {3}}+\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^5)^(2/3)/x^6,x]

[Out]

-1/5*(-1 + x^5)^(2/3)/x^5 - (2*ArcTan[(1 - 2*(-1 + x^5)^(1/3))/Sqrt[3]])/(5*Sqrt[3]) + Log[x]/3 - Log[1 + (-1
+ x^5)^(1/3)]/5

Rule 31

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

Rule 47

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

Rule 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(
1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && Ne
gQ[(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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 {\left (-1+x^5\right )^{2/3}}{x^6} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {(-1+x)^{2/3}}{x^2} \, dx,x,x^5\right )\\ &=-\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {2}{15} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^5\right )\\ &=-\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {\log (x)}{3}-\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^5}\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^5}\right )\\ &=-\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {\log (x)}{3}-\frac {1}{5} \log \left (1+\sqrt [3]{-1+x^5}\right )-\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^5}\right )\\ &=-\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}-\frac {2 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^5}}{\sqrt {3}}\right )}{5 \sqrt {3}}+\frac {\log (x)}{3}-\frac {1}{5} \log \left (1+\sqrt [3]{-1+x^5}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 28, normalized size = 0.30 \begin {gather*} \frac {3}{25} \left (x^5-1\right )^{5/3} \, _2F_1\left (\frac {5}{3},2;\frac {8}{3};1-x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^5)^(2/3)/x^6,x]

[Out]

(3*(-1 + x^5)^(5/3)*Hypergeometric2F1[5/3, 2, 8/3, 1 - x^5])/25

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IntegrateAlgebraic [A]  time = 0.06, size = 92, normalized size = 1.00 \begin {gather*} -\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^5}}{\sqrt {3}}\right )}{5 \sqrt {3}}-\frac {2}{15} \log \left (1+\sqrt [3]{-1+x^5}\right )+\frac {1}{15} \log \left (1-\sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^5)^(2/3)/x^6,x]

[Out]

-1/5*(-1 + x^5)^(2/3)/x^5 - (2*ArcTan[1/Sqrt[3] - (2*(-1 + x^5)^(1/3))/Sqrt[3]])/(5*Sqrt[3]) - (2*Log[1 + (-1
+ x^5)^(1/3)])/15 + Log[1 - (-1 + x^5)^(1/3) + (-1 + x^5)^(2/3)]/15

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fricas [A]  time = 0.46, size = 80, normalized size = 0.87 \begin {gather*} \frac {2 \, \sqrt {3} x^{5} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{5} \log \left ({\left (x^{5} - 1\right )}^{\frac {2}{3}} - {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{5} \log \left ({\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-1)^(2/3)/x^6,x, algorithm="fricas")

[Out]

1/15*(2*sqrt(3)*x^5*arctan(2/3*sqrt(3)*(x^5 - 1)^(1/3) - 1/3*sqrt(3)) + x^5*log((x^5 - 1)^(2/3) - (x^5 - 1)^(1
/3) + 1) - 2*x^5*log((x^5 - 1)^(1/3) + 1) - 3*(x^5 - 1)^(2/3))/x^5

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giac [A]  time = 0.30, size = 69, normalized size = 0.75 \begin {gather*} \frac {2}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{5} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{5} - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} + \frac {1}{15} \, \log \left ({\left (x^{5} - 1\right )}^{\frac {2}{3}} - {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{15} \, \log \left ({\left | {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-1)^(2/3)/x^6,x, algorithm="giac")

[Out]

2/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^5 - 1)^(1/3) - 1)) - 1/5*(x^5 - 1)^(2/3)/x^5 + 1/15*log((x^5 - 1)^(2/3)
- (x^5 - 1)^(1/3) + 1) - 2/15*log(abs((x^5 - 1)^(1/3) + 1))

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maple [C]  time = 6.72, size = 96, normalized size = 1.04

method	result	size
risch	\(-\frac {\left (x^{5}-1\right )^{\frac {2}{3}}}{5 x^{5}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{5} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+5 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{15 \pi \mathrm {signum}\left (x^{5}-1\right )^{\frac {1}{3}}}\)	\(96\)
meijerg	\(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{5}-1\right )^{\frac {2}{3}} \left (-\frac {\pi \sqrt {3}\, x^{5} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+5 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{15 \pi \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {2}{3}}}\)	\(97\)
trager	\(-\frac {\left (x^{5}-1\right )^{\frac {2}{3}}}{5 x^{5}}-\frac {2 \ln \left (\frac {30885108600167424 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}+9590347318616160 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-19140349667550 x^{5}+12071260649204448 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}}[phone] \left (x^{5}-1\right )^{\frac {2}{3}}-5593680301194816 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}}-988323475205357568 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}[phone] \left (x^{5}-1\right )^{\frac {1}{3}}-7369904750343456 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )[phone]}{x^{5}}\right )}{15}+\frac {2 \ln \left (-\frac {25005193182296064 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}-8013344512846944 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-19778361323135 x^{5}+12071260649204448 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}}-184009801566659 \left (x^{5}-1\right )^{\frac {2}{3}}+17664940950399264 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}}-800166181833474048 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-58267503137446 \left (x^{5}-1\right )^{\frac {1}{3}}+13928744695293504 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )[phone]}{x^{5}}\right )}{15}-\frac {64 \ln \left (-\frac {25005193182296064 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}-8013344512846944 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-19778361323135 x^{5}+12071260649204448 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}}-184009801566659 \left (x^{5}-1\right )^{\frac {2}{3}}+17664940950399264 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}}-800166181833474048 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-58267503137446 \left (x^{5}-1\right )^{\frac {1}{3}}+13928744695293504 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )[phone]}{x^{5}}\right ) \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )}{5}\)	\(429\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5-1)^(2/3)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*(x^5-1)^(2/3)/x^5+1/15/Pi*3^(1/2)*GAMMA(2/3)/signum(x^5-1)^(1/3)*(-signum(x^5-1))^(1/3)*(2/9*Pi*3^(1/2)/G
AMMA(2/3)*x^5*hypergeom([1,1,4/3],[2,2],x^5)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+5*ln(x)+I*Pi)*Pi*3^(1/2)/GAMMA(2/3
))

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maxima [A]  time = 0.47, size = 68, normalized size = 0.74 \begin {gather*} \frac {2}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{5} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{5} - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} + \frac {1}{15} \, \log \left ({\left (x^{5} - 1\right )}^{\frac {2}{3}} - {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{15} \, \log \left ({\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-1)^(2/3)/x^6,x, algorithm="maxima")

[Out]

2/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^5 - 1)^(1/3) - 1)) - 1/5*(x^5 - 1)^(2/3)/x^5 + 1/15*log((x^5 - 1)^(2/3)
- (x^5 - 1)^(1/3) + 1) - 2/15*log((x^5 - 1)^(1/3) + 1)

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mupad [B]  time = 0.94, size = 92, normalized size = 1.00 \begin {gather*} -\frac {2\,\ln \left (\frac {4\,{\left (x^5-1\right )}^{1/3}}{25}+\frac {4}{25}\right )}{15}-\ln \left (9\,{\left (-\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )}^2+\frac {4\,{\left (x^5-1\right )}^{1/3}}{25}\right )\,\left (-\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )+\ln \left (9\,{\left (\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )}^2+\frac {4\,{\left (x^5-1\right )}^{1/3}}{25}\right )\,\left (\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )-\frac {{\left (x^5-1\right )}^{2/3}}{5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5 - 1)^(2/3)/x^6,x)

[Out]

log(9*((3^(1/2)*1i)/15 + 1/15)^2 + (4*(x^5 - 1)^(1/3))/25)*((3^(1/2)*1i)/15 + 1/15) - log(9*((3^(1/2)*1i)/15 -
1/15)^2 + (4*(x^5 - 1)^(1/3))/25)*((3^(1/2)*1i)/15 - 1/15) - (2*log((4*(x^5 - 1)^(1/3))/25 + 4/25))/15 - (x^5
- 1)^(2/3)/(5*x^5)

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sympy [C]  time = 1.09, size = 36, normalized size = 0.39 \begin {gather*} - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {5}{3}} \Gamma \left (\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5-1)**(2/3)/x**6,x)

[Out]

-gamma(1/3)*hyper((-2/3, 1/3), (4/3,), exp_polar(2*I*pi)/x**5)/(5*x**(5/3)*gamma(4/3))

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