3.18.56 \(\int \frac {(-1+x^3)^{2/3} (1-5 x^3+4 x^6)}{x^6 (-1+2 x^3)^2} \, dx\)

Optimal. Leaf size=118 \[ \frac {7}{9} \log \left (\sqrt [3]{x^3-1}+x\right )+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}-x}\right )}{3 \sqrt {3}}-\frac {7}{18} \log \left (-\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )+\frac {\left (x^3-1\right )^{2/3} \left (62 x^6-33 x^3+6\right )}{30 x^5 \left (2 x^3-1\right )} \]

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Rubi [C]  time = 0.48, antiderivative size = 224, normalized size of antiderivative = 1.90, number of steps used = 15, number of rules used = 14, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6742, 264, 277, 239, 378, 377, 200, 31, 634, 618, 204, 628, 430, 429} \begin {gather*} -\frac {2 x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {2 x \left (x^3-1\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {4}{9} \log \left (\frac {x}{\sqrt [3]{x^3-1}}+1\right )+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {4 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}-\frac {2}{9} \log \left (-\frac {x}{\sqrt [3]{x^3-1}}+\frac {x^2}{\left (x^3-1\right )^{2/3}}+1\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((-1 + x^3)^(2/3)*(1 - 5*x^3 + 4*x^6))/(x^6*(-1 + 2*x^3)^2),x]

[Out]

(-1 + x^3)^(2/3)/(2*x^2) - (2*x*(-1 + x^3)^(2/3))/(3*(1 - 2*x^3)) + (-1 + x^3)^(5/3)/(5*x^5) - (2*x*(-1 + x^3)
^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, x^3, 2*x^3])/(1 - x^3)^(2/3) - (4*ArcTan[(1 - (2*x)/(-1 + x^3)^(1/3))/Sqrt[
3]])/(3*Sqrt[3]) - ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - (2*Log[1 + x^2/(-1 + x^3)^(2/3) - x/
(-1 + x^3)^(1/3)])/9 + (4*Log[1 + x/(-1 + x^3)^(1/3)])/9 + Log[-x + (-1 + x^3)^(1/3)]/2

Rule 31

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

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

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 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

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 277

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

Rule 377

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

Rule 378

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

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 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]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx &=\int \left (\frac {\left (-1+x^3\right )^{2/3}}{x^6}-\frac {\left (-1+x^3\right )^{2/3}}{x^3}-\frac {2 \left (-1+x^3\right )^{2/3}}{\left (-1+2 x^3\right )^2}+\frac {2 \left (-1+x^3\right )^{2/3}}{-1+2 x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-1+x^3\right )^{2/3}}{\left (-1+2 x^3\right )^2} \, dx\right )+2 \int \frac {\left (-1+x^3\right )^{2/3}}{-1+2 x^3} \, dx+\int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx-\int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {4}{3} \int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx+\frac {\left (2 \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-1+2 x^3} \, dx}{\left (1-x^3\right )^{2/3}}-\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-1-x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {4}{9} \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {4}{9} \operatorname {Subst}\left (\int \frac {-2+x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {4}{9} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {2}{9} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{9} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {4}{9} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x}{\sqrt [3]{-1+x^3}}\right )\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {4 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{9} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {4}{9} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 125, normalized size = 1.06 \begin {gather*} \frac {7}{18} \left (2 \log \left (\frac {x}{\sqrt [3]{1-x^3}}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{1-x^3}}-1}{\sqrt {3}}\right )-\log \left (-\frac {x}{\sqrt [3]{1-x^3}}+\frac {x^2}{\left (1-x^3\right )^{2/3}}+1\right )\right )+\frac {\left (x^3-1\right )^{2/3} \left (62 x^6-33 x^3+6\right )}{30 x^5 \left (2 x^3-1\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 + x^3)^(2/3)*(1 - 5*x^3 + 4*x^6))/(x^6*(-1 + 2*x^3)^2),x]

[Out]

((-1 + x^3)^(2/3)*(6 - 33*x^3 + 62*x^6))/(30*x^5*(-1 + 2*x^3)) + (7*(2*Sqrt[3]*ArcTan[(-1 + (2*x)/(1 - x^3)^(1
/3))/Sqrt[3]] - Log[1 + x^2/(1 - x^3)^(2/3) - x/(1 - x^3)^(1/3)] + 2*Log[1 + x/(1 - x^3)^(1/3)]))/18

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IntegrateAlgebraic [A]  time = 0.26, size = 118, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (6-33 x^3+62 x^6\right )}{30 x^5 \left (-1+2 x^3\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}+\frac {7}{9} \log \left (x+\sqrt [3]{-1+x^3}\right )-\frac {7}{18} \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(2/3)*(1 - 5*x^3 + 4*x^6))/(x^6*(-1 + 2*x^3)^2),x]

[Out]

((-1 + x^3)^(2/3)*(6 - 33*x^3 + 62*x^6))/(30*x^5*(-1 + 2*x^3)) + (7*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^3)^(1/3
))])/(3*Sqrt[3]) + (7*Log[x + (-1 + x^3)^(1/3)])/9 - (7*Log[x^2 - x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/18

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fricas [A]  time = 1.02, size = 155, normalized size = 1.31 \begin {gather*} -\frac {70 \, \sqrt {3} {\left (2 \, x^{8} - x^{5}\right )} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} - 1\right )}}{7 \, x^{3} + 1}\right ) - 35 \, {\left (2 \, x^{8} - x^{5}\right )} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{2 \, x^{3} - 1}\right ) - 3 \, {\left (62 \, x^{6} - 33 \, x^{3} + 6\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, {\left (2 \, x^{8} - x^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(4*x^6-5*x^3+1)/x^6/(2*x^3-1)^2,x, algorithm="fricas")

[Out]

-1/90*(70*sqrt(3)*(2*x^8 - x^5)*arctan((4*sqrt(3)*(x^3 - 1)^(1/3)*x^2 + 2*sqrt(3)*(x^3 - 1)^(2/3)*x + sqrt(3)*
(x^3 - 1))/(7*x^3 + 1)) - 35*(2*x^8 - x^5)*log((2*x^3 + 3*(x^3 - 1)^(1/3)*x^2 + 3*(x^3 - 1)^(2/3)*x - 1)/(2*x^
3 - 1)) - 3*(62*x^6 - 33*x^3 + 6)*(x^3 - 1)^(2/3))/(2*x^8 - x^5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{6} - 5 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )}^{2} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(4*x^6-5*x^3+1)/x^6/(2*x^3-1)^2,x, algorithm="giac")

[Out]

integrate((4*x^6 - 5*x^3 + 1)*(x^3 - 1)^(2/3)/((2*x^3 - 1)^2*x^6), x)

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maple [C]  time = 4.25, size = 274, normalized size = 2.32

method	result	size
risch	\(\frac {62 x^{9}-95 x^{6}+39 x^{3}-6}{30 \left (2 x^{3}-1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{5}}+\frac {7 \ln \left (\frac {3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-x \left (x^{3}-1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+1}{2 x^{3}-1}\right )}{9}+\frac {7 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 x \left (x^{3}-1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2 x^{3}-1}\right )}{3}\)	\(274\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (62 x^{6}-33 x^{3}+6\right )}{30 x^{5} \left (2 x^{3}-1\right )}+\frac {7 \ln \left (-\frac {65382681600 \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x [phone] \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{2}[phone] \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{3}-421089 x \left (x^{3}-1\right )^{\frac {2}{3}}+106800 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-73232 x^{3}-523061452800 \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2}-212378400 \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )+64078}{2 x^{3}-1}\right )}{9}+1120 \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) \ln \left (\frac {-18981734400 \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -606368160 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{2}[phone] \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{3}+527889 x \left (x^{3}-1\right )^{\frac {2}{3}}+106800 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-220717 x^{3}[phone] \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2}-350055360 \RootOf \left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )-31531}{2 x^{3}-1}\right )\)	\(343\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(2/3)*(4*x^6-5*x^3+1)/x^6/(2*x^3-1)^2,x,method=_RETURNVERBOSE)

[Out]

1/30*(62*x^9-95*x^6+39*x^3-6)/(2*x^3-1)/(x^3-1)^(1/3)/x^5+7/9*ln((3*RootOf(9*_Z^2+3*_Z+1)*(x^3-1)^(2/3)*x+6*Ro
otOf(9*_Z^2+3*_Z+1)*(x^3-1)^(1/3)*x^2+3*RootOf(9*_Z^2+3*_Z+1)*x^3-x*(x^3-1)^(2/3)+x^2*(x^3-1)^(1/3)+1)/(2*x^3-
1))+7/3*RootOf(9*_Z^2+3*_Z+1)*ln(-(9*RootOf(9*_Z^2+3*_Z+1)^2*x^3+3*RootOf(9*_Z^2+3*_Z+1)*(x^3-1)^(2/3)*x-3*Roo
tOf(9*_Z^2+3*_Z+1)*(x^3-1)^(1/3)*x^2+3*RootOf(9*_Z^2+3*_Z+1)*x^3+2*x*(x^3-1)^(2/3)+x^2*(x^3-1)^(1/3)-3*RootOf(
9*_Z^2+3*_Z+1))/(2*x^3-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{6} - 5 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )}^{2} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(4*x^6-5*x^3+1)/x^6/(2*x^3-1)^2,x, algorithm="maxima")

[Out]

integrate((4*x^6 - 5*x^3 + 1)*(x^3 - 1)^(2/3)/((2*x^3 - 1)^2*x^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-1\right )}^{2/3}\,\left (4\,x^6-5\,x^3+1\right )}{x^6\,{\left (2\,x^3-1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(2/3)*(4*x^6 - 5*x^3 + 1))/(x^6*(2*x^3 - 1)^2),x)

[Out]

int(((x^3 - 1)^(2/3)*(4*x^6 - 5*x^3 + 1))/(x^6*(2*x^3 - 1)^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)**(2/3)*(4*x**6-5*x**3+1)/x**6/(2*x**3-1)**2,x)

[Out]

Timed out

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