3.16.3 \(\int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx\)

Optimal. Leaf size=105 \[ -\frac {2}{27} \log \left (\sqrt [3]{x^3+1}-1\right )+\frac {1}{27} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3} \left (27 x^4-20 x^3-18 x+15\right )}{90 x^6} \]

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Rubi [A]  time = 0.10, antiderivative size = 118, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1844, 266, 51, 55, 618, 204, 31, 271, 264} \begin {gather*} -\frac {2 \left (x^3+1\right )^{2/3}}{9 x^3}-\frac {1}{9} \log \left (1-\sqrt [3]{x^3+1}\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3}}{6 x^6}-\frac {\left (x^3+1\right )^{2/3}}{5 x^5}+\frac {3 \left (x^3+1\right )^{2/3}}{10 x^2}+\frac {\log (x)}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x)/(x^7*(1 + x^3)^(1/3)),x]

[Out]

(1 + x^3)^(2/3)/(6*x^6) - (1 + x^3)^(2/3)/(5*x^5) - (2*(1 + x^3)^(2/3))/(9*x^3) + (3*(1 + x^3)^(2/3))/(10*x^2)
- (2*ArcTan[(1 + 2*(1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) + Log[x]/9 - Log[1 - (1 + x^3)^(1/3)]/9

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 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[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] && PosQ
[(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 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

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 1844

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {-1+x}{x^7 \sqrt [3]{1+x^3}} \, dx &=\int \left (-\frac {1}{x^7 \sqrt [3]{1+x^3}}+\frac {1}{x^6 \sqrt [3]{1+x^3}}\right ) \, dx\\ &=-\int \frac {1}{x^7 \sqrt [3]{1+x^3}} \, dx+\int \frac {1}{x^6 \sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt [3]{1+x}} \, dx,x,x^3\right )-\frac {3}{5} \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}+\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [3]{1+x}} \, dx,x,x^3\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {2}{27} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^3\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}+\frac {\log (x)}{9}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}+\frac {\log (x)}{9}-\frac {1}{9} \log \left (1-\sqrt [3]{1+x^3}\right )+\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{6 x^6}-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{9 x^3}+\frac {3 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {2 \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\log (x)}{9}-\frac {1}{9} \log \left (1-\sqrt [3]{1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 41, normalized size = 0.39 \begin {gather*} \frac {\left (x^3+1\right )^{2/3} \left (5 x^5 \, _2F_1\left (\frac {2}{3},3;\frac {5}{3};x^3+1\right )+3 x^3-2\right )}{10 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x)/(x^7*(1 + x^3)^(1/3)),x]

[Out]

((1 + x^3)^(2/3)*(-2 + 3*x^3 + 5*x^5*Hypergeometric2F1[2/3, 3, 5/3, 1 + x^3]))/(10*x^5)

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IntegrateAlgebraic [A]  time = 15.71, size = 105, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (15-18 x-20 x^3+27 x^4\right )}{90 x^6}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2}{27} \log \left (-1+\sqrt [3]{1+x^3}\right )+\frac {1}{27} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x)/(x^7*(1 + x^3)^(1/3)),x]

[Out]

((1 + x^3)^(2/3)*(15 - 18*x - 20*x^3 + 27*x^4))/(90*x^6) - (2*ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]])
/(9*Sqrt[3]) - (2*Log[-1 + (1 + x^3)^(1/3)])/27 + Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/27

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fricas [A]  time = 0.83, size = 114, normalized size = 1.09 \begin {gather*} \frac {20 \, \sqrt {3} x^{6} \arctan \left (-\frac {\sqrt {3} {\left (x^{3} + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3} + 9}\right ) - 10 \, x^{6} \log \left (\frac {x^{3} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3}}\right ) + 3 \, {\left (27 \, x^{4} - 20 \, x^{3} - 18 \, x + 15\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{270 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/x^7/(x^3+1)^(1/3),x, algorithm="fricas")

[Out]

1/270*(20*sqrt(3)*x^6*arctan(-(sqrt(3)*(x^3 + 1) - 2*sqrt(3)*(x^3 + 1)^(2/3) + 4*sqrt(3)*(x^3 + 1)^(1/3))/(x^3
+ 9)) - 10*x^6*log((x^3 - 3*(x^3 + 1)^(2/3) + 3*(x^3 + 1)^(1/3))/x^3) + 3*(27*x^4 - 20*x^3 - 18*x + 15)*(x^3
+ 1)^(2/3))/x^6

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/x^7/(x^3+1)^(1/3),x, algorithm="giac")

[Out]

integrate((x - 1)/((x^3 + 1)^(1/3)*x^7), x)

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maple [C]  time = 2.76, size = 101, normalized size = 0.96

method	result	size
risch	\(\frac {27 x^{7}-20 x^{6}+9 x^{4}-5 x^{3}-18 x +15}{90 x^{6} \left (x^{3}+1\right )^{\frac {1}{3}}}-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{27 \pi }\)	\(101\)
meijerg	\(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {28 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {10}{3}\right ], \left [2, 4\right ], -x^{3}\right )}{243 \Gamma \left (\frac {2}{3}\right )}+\frac {4 \left (\frac {9}{4}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{6}}+\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{3}}\right )}{6 \pi }-\frac {\left (1-\frac {3 x^{3}}{2}\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{5 x^{5}}\)	\(110\)
trager	\(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (27 x^{4}-20 x^{3}-18 x +15\right )}{90 x^{6}}+\frac {4 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{3}+18 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{3}+30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}+30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+9 \left (x^{3}+1\right )^{\frac {2}{3}}+38 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right )}{27}+\frac {2 \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{3}-34 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{3}-30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}-30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}-22 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right )}{27}-\frac {4 \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{3}-34 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{3}-30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}-30 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}-22 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right ) \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{27}\)	\(456\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+x)/x^7/(x^3+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/90*(27*x^7-20*x^6+9*x^4-5*x^3-18*x+15)/x^6/(x^3+1)^(1/3)-1/27/Pi*3^(1/2)*GAMMA(2/3)*(-2/9*Pi*3^(1/2)/GAMMA(2
/3)*x^3*hypergeom([1,1,4/3],[2,2],-x^3)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x))*Pi*3^(1/2)/GAMMA(2/3))

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maxima [A]  time = 0.43, size = 117, normalized size = 1.11 \begin {gather*} -\frac {2}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {4 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} - 7 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{18 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} - \frac {{\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} + \frac {1}{27} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{27} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/x^7/(x^3+1)^(1/3),x, algorithm="maxima")

[Out]

-2/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) + 1/18*(4*(x^3 + 1)^(5/3) - 7*(x^3 + 1)^(2/3))/(2*x^
3 - (x^3 + 1)^2 + 1) + 1/2*(x^3 + 1)^(2/3)/x^2 - 1/5*(x^3 + 1)^(5/3)/x^5 + 1/27*log((x^3 + 1)^(2/3) + (x^3 + 1
)^(1/3) + 1) - 2/27*log((x^3 + 1)^(1/3) - 1)

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mupad [B]  time = 1.15, size = 146, normalized size = 1.39 \begin {gather*} -\frac {2\,\ln \left (\frac {4\,{\left (x^3+1\right )}^{1/3}}{81}-\frac {4}{81}\right )}{27}-\ln \left (\frac {4\,{\left (x^3+1\right )}^{1/3}}{81}-9\,{\left (-\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )}^2\right )\,\left (-\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )+\ln \left (\frac {4\,{\left (x^3+1\right )}^{1/3}}{81}-9\,{\left (\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )}^2\right )\,\left (\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )-\frac {2\,{\left (x^3+1\right )}^{2/3}-3\,x^3\,{\left (x^3+1\right )}^{2/3}}{10\,x^5}-\frac {\frac {7\,{\left (x^3+1\right )}^{2/3}}{18}-\frac {2\,{\left (x^3+1\right )}^{5/3}}{9}}{2\,x^3-{\left (x^3+1\right )}^2+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/(x^7*(x^3 + 1)^(1/3)),x)

[Out]

log((4*(x^3 + 1)^(1/3))/81 - 9*((3^(1/2)*1i)/27 + 1/27)^2)*((3^(1/2)*1i)/27 + 1/27) - log((4*(x^3 + 1)^(1/3))/
81 - 9*((3^(1/2)*1i)/27 - 1/27)^2)*((3^(1/2)*1i)/27 - 1/27) - (2*log((4*(x^3 + 1)^(1/3))/81 - 4/81))/27 - (2*(
x^3 + 1)^(2/3) - 3*x^3*(x^3 + 1)^(2/3))/(10*x^5) - ((7*(x^3 + 1)^(2/3))/18 - (2*(x^3 + 1)^(5/3))/9)/(2*x^3 - (
x^3 + 1)^2 + 1)

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sympy [C]  time = 2.16, size = 82, normalized size = 0.78 \begin {gather*} \frac {\left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} - \frac {2 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 x^{5} \Gamma \left (\frac {1}{3}\right )} + \frac {\Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{7} \Gamma \left (\frac {10}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/x**7/(x**3+1)**(1/3),x)

[Out]

(x**3 + 1)**(2/3)*gamma(-5/3)/(3*x**2*gamma(1/3)) - 2*(x**3 + 1)**(2/3)*gamma(-5/3)/(9*x**5*gamma(1/3)) + gamm
a(7/3)*hyper((1/3, 7/3), (10/3,), exp_polar(I*pi)/x**3)/(3*x**7*gamma(10/3))

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