3.23.97 \(\int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx\)

Optimal. Leaf size=175 \[ \frac {(x-1)^{2/3} \left (x^2+x-1\right ) \left (\text {RootSum}\left [\text {$\#$1}^6+3 \text {$\#$1}^3+1\& ,\frac {\text {$\#$1}^3 \log \left (\sqrt [3]{x-1}-\text {$\#$1}\right )+3 \log \left (\sqrt [3]{x-1}-\text {$\#$1}\right )}{2 \text {$\#$1}^5+3 \text {$\#$1}^2}\& \right ]-\frac {\sqrt [3]{x-1}}{x}-\frac {5}{3} \log \left (\sqrt [3]{x-1}+1\right )+\frac {5}{6} \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )+\frac {5 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{\sqrt {3}}\right )}{\sqrt [3]{(x-1)^2 \left (x^2+x-1\right )^3}} \]

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Rubi [B]  time = 0.93, antiderivative size = 946, normalized size of antiderivative = 5.41, number of steps used = 25, number of rules used = 13, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6688, 6719, 897, 1424, 638, 618, 204, 634, 628, 1422, 200, 31, 617} \begin {gather*} -\frac {5 (x-1)^{2/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x-1}}{\sqrt {3}}\right ) \left (-x^2-x+1\right )}{\sqrt {3} \sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}-\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} (x-1)^{2/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\frac {2}{3+\sqrt {5}}} \sqrt [3]{x-1}}{\sqrt {3}}\right ) \left (-x^2-x+1\right )}{\sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}+\frac {\sqrt {\frac {3}{5}} \left (3+\sqrt {5}\right )^{5/3} (x-1)^{2/3} \tan ^{-1}\left (\frac {1-2^{2/3} \sqrt [3]{3+\sqrt {5}} \sqrt [3]{x-1}}{\sqrt {3}}\right ) \left (-x^2-x+1\right )}{2\ 2^{2/3} \sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}+\frac {5 (x-1)^{2/3} \log \left (\sqrt [3]{x-1}+1\right ) \left (-x^2-x+1\right )}{3 \sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}-\frac {\left (3+\sqrt {5}\right )^{5/3} (x-1)^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{x-1}+\sqrt [3]{3-\sqrt {5}}\right ) \left (-x^2-x+1\right )}{2\ 2^{2/3} \sqrt {5} \sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} (x-1)^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{x-1}+\sqrt [3]{3+\sqrt {5}}\right ) \left (-x^2-x+1\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}-\frac {5 (x-1)^{2/3} \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right ) \left (-x^2-x+1\right )}{6 \sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}+\frac {\left (3+\sqrt {5}\right )^{5/3} (x-1)^{2/3} \log \left (2^{2/3} (x-1)^{2/3}-\sqrt [3]{2 \left (3-\sqrt {5}\right )} \sqrt [3]{x-1}+\left (3-\sqrt {5}\right )^{2/3}\right ) \left (-x^2-x+1\right )}{4\ 2^{2/3} \sqrt {5} \sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} (x-1)^{2/3} \log \left (2^{2/3} (x-1)^{2/3}-\sqrt [3]{2 \left (3+\sqrt {5}\right )} \sqrt [3]{x-1}+\left (3+\sqrt {5}\right )^{2/3}\right ) \left (-x^2-x+1\right )}{2 \sqrt {5} \sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}-\frac {(x-1)^{2/3} \left (-x^2-x+1\right )}{3 \left (\sqrt [3]{x-1}+1\right ) \sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}}+\frac {\left (\sqrt [3]{x-1}+1\right ) (x-1)^{2/3} \left (-x^2-x+1\right )}{3 \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right ) \sqrt [3]{-(1-x)^2 \left (-x^2-x+1\right )^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[1/(x^2*(-1 + 5*x - 7*x^2 - 2*x^3 + 10*x^4 - 2*x^5 - 5*x^6 + x^7 + x^8)^(1/3)),x]

[Out]

-1/3*((-1 + x)^(2/3)*(1 - x - x^2))/((1 + (-1 + x)^(1/3))*(-((1 - x)^2*(1 - x - x^2)^3))^(1/3)) + ((1 + (-1 +
x)^(1/3))*(-1 + x)^(2/3)*(1 - x - x^2))/(3*(1 - (-1 + x)^(1/3) + (-1 + x)^(2/3))*(-((1 - x)^2*(1 - x - x^2)^3)
)^(1/3)) - (5*(-1 + x)^(2/3)*(1 - x - x^2)*ArcTan[(1 - 2*(-1 + x)^(1/3))/Sqrt[3]])/(Sqrt[3]*(-((1 - x)^2*(1 -
x - x^2)^3))^(1/3)) - (Sqrt[3/5]*((123 - 55*Sqrt[5])/2)^(1/3)*(-1 + x)^(2/3)*(1 - x - x^2)*ArcTan[(1 - 2*(2/(3
+ Sqrt[5]))^(1/3)*(-1 + x)^(1/3))/Sqrt[3]])/(-((1 - x)^2*(1 - x - x^2)^3))^(1/3) + (Sqrt[3/5]*(3 + Sqrt[5])^(
5/3)*(-1 + x)^(2/3)*(1 - x - x^2)*ArcTan[(1 - 2^(2/3)*(3 + Sqrt[5])^(1/3)*(-1 + x)^(1/3))/Sqrt[3]])/(2*2^(2/3)
*(-((1 - x)^2*(1 - x - x^2)^3))^(1/3)) + (5*(-1 + x)^(2/3)*(1 - x - x^2)*Log[1 + (-1 + x)^(1/3)])/(3*(-((1 - x
)^2*(1 - x - x^2)^3))^(1/3)) - ((3 + Sqrt[5])^(5/3)*(-1 + x)^(2/3)*(1 - x - x^2)*Log[(3 - Sqrt[5])^(1/3) + 2^(
1/3)*(-1 + x)^(1/3)])/(2*2^(2/3)*Sqrt[5]*(-((1 - x)^2*(1 - x - x^2)^3))^(1/3)) + (((123 - 55*Sqrt[5])/2)^(1/3)
*(-1 + x)^(2/3)*(1 - x - x^2)*Log[(3 + Sqrt[5])^(1/3) + 2^(1/3)*(-1 + x)^(1/3)])/(Sqrt[5]*(-((1 - x)^2*(1 - x
- x^2)^3))^(1/3)) - (5*(-1 + x)^(2/3)*(1 - x - x^2)*Log[1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)])/(6*(-((1 - x)^2*
(1 - x - x^2)^3))^(1/3)) + ((3 + Sqrt[5])^(5/3)*(-1 + x)^(2/3)*(1 - x - x^2)*Log[(3 - Sqrt[5])^(2/3) - (2*(3 -
Sqrt[5]))^(1/3)*(-1 + x)^(1/3) + 2^(2/3)*(-1 + x)^(2/3)])/(4*2^(2/3)*Sqrt[5]*(-((1 - x)^2*(1 - x - x^2)^3))^(
1/3)) - (((123 - 55*Sqrt[5])/2)^(1/3)*(-1 + x)^(2/3)*(1 - x - x^2)*Log[(3 + Sqrt[5])^(2/3) - (2*(3 + Sqrt[5]))
^(1/3)*(-1 + x)^(1/3) + 2^(2/3)*(-1 + x)^(2/3)])/(2*Sqrt[5]*(-((1 - x)^2*(1 - x - x^2)^3))^(1/3))

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1422

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 1424

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Int[ExpandIntegran
d[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4
*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]

Rule 6688

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

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx &=\int \frac {1}{x^2 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}} \, dx\\ &=\frac {\left ((-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \int \frac {1}{(-1+x)^{2/3} x^2 \left (-1+x+x^2\right )} \, dx}{\sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}\\ &=\frac {\left (3 (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^3\right )^2 \left (1+3 x^3+x^6\right )} \, dx,x,\sqrt [3]{-1+x}\right )}{\sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}\\ &=\frac {\left (3 (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{9 (1+x)^2}-\frac {5}{9 (1+x)}+\frac {-1+x}{3 \left (1-x+x^2\right )^2}+\frac {-9+5 x}{9 \left (1-x+x^2\right )}+\frac {3+x^3}{1+3 x^3+x^6}\right ) \, dx,x,\sqrt [3]{-1+x}\right )}{\sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}\\ &=-\frac {(-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\left ((-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {-9+5 x}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left ((-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {-1+x}{\left (1-x+x^2\right )^2} \, dx,x,\sqrt [3]{-1+x}\right )}{\sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (3 (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {3+x^3}{1+3 x^3+x^6} \, dx,x,\sqrt [3]{-1+x}\right )}{\sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}\\ &=-\frac {(-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\left (1+\sqrt [3]{-1+x}\right ) (-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {\left ((-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (5 (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{6 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}-\frac {\left (13 (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{6 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (3 \left (5-3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^3} \, dx,x,\sqrt [3]{-1+x}\right )}{10 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (3 \left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^3} \, dx,x,\sqrt [3]{-1+x}\right )}{10 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}\\ &=-\frac {(-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\left (1+\sqrt [3]{-1+x}\right ) (-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )}{6 \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\left (2 (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x}\right )}{3 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (13 (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x}\right )}{3 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (\left (5-3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )}+x} \, dx,x,\sqrt [3]{-1+x}\right )}{5 \sqrt [3]{2} \left (3+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (\left (5-3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{3+\sqrt {5}}-x}{\left (\frac {1}{2} \left (3+\sqrt {5}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{5 \sqrt [3]{2} \left (3+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (\left (\frac {1}{2} \left (3+\sqrt {5}\right )\right )^{2/3} \left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )}+x} \, dx,x,\sqrt [3]{-1+x}\right )}{10 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (\left (\frac {1}{2} \left (3+\sqrt {5}\right )\right )^{2/3} \left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{3-\sqrt {5}}-x}{\left (\frac {1}{2} \left (3-\sqrt {5}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{10 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}\\ &=-\frac {(-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\left (1+\sqrt [3]{-1+x}\right ) (-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {\left (3+\sqrt {5}\right )^{5/3} (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (\sqrt [3]{3-\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{2\ 2^{2/3} \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (\sqrt [3]{3+\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )}{6 \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {\left (\left (5-3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )}+2 x}{\left (\frac {1}{2} \left (3+\sqrt {5}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{10 \sqrt [3]{2} \left (3+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (3 \left (5-3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (3+\sqrt {5}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{10\ 2^{2/3} \sqrt [3]{3+\sqrt {5}} \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}-\frac {\left (\left (\frac {1}{2} \left (3+\sqrt {5}\right )\right )^{2/3} \left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )}+2 x}{\left (\frac {1}{2} \left (3-\sqrt {5}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{20 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (3 \sqrt [3]{3-\sqrt {5}} \left (3+\sqrt {5}\right )^{2/3} \left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (3-\sqrt {5}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{40 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}\\ &=-\frac {(-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\left (1+\sqrt [3]{-1+x}\right ) (-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {\left (3+\sqrt {5}\right )^{5/3} (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (\sqrt [3]{3-\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{2\ 2^{2/3} \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (\sqrt [3]{3+\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )}{6 \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\left (3+\sqrt {5}\right )^{5/3} (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (\left (3-\sqrt {5}\right )^{2/3}-\sqrt [3]{2 \left (3-\sqrt {5}\right )} \sqrt [3]{-1+x}+2^{2/3} (-1+x)^{2/3}\right )}{4\ 2^{2/3} \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (\left (3+\sqrt {5}\right )^{2/3}-\sqrt [3]{2 \left (3+\sqrt {5}\right )} \sqrt [3]{-1+x}+2^{2/3} (-1+x)^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\left (3 \left (5-3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{\frac {2}{3+\sqrt {5}}} \sqrt [3]{-1+x}\right )}{5 \sqrt [3]{2} \left (3+\sqrt {5}\right )^{2/3} \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}+\frac {\left (3 \sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} \left (3+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right ) (-1+x)^{2/3} \left (-1+x+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{\frac {2}{3-\sqrt {5}}} \sqrt [3]{-1+x}\right )}{20 \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}}\\ &=-\frac {(-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1+\sqrt [3]{-1+x}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\left (1+\sqrt [3]{-1+x}\right ) (-1+x)^{2/3} \left (1-x-x^2\right )}{3 \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right ) \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} (-1+x)^{2/3} \left (1-x-x^2\right ) \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\frac {2}{3+\sqrt {5}}} \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} (-1+x)^{2/3} \left (1-x-x^2\right ) \tan ^{-1}\left (\frac {1-2^{2/3} \sqrt [3]{3+\sqrt {5}} \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (1+\sqrt [3]{-1+x}\right )}{3 \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {\left (3+\sqrt {5}\right )^{5/3} (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (\sqrt [3]{3-\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{2\ 2^{2/3} \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (\sqrt [3]{3+\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{\sqrt {5} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {5 (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )}{6 \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}+\frac {\left (3+\sqrt {5}\right )^{5/3} (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (\left (3-\sqrt {5}\right )^{2/3}-\sqrt [3]{2 \left (3-\sqrt {5}\right )} \sqrt [3]{-1+x}+2^{2/3} (-1+x)^{2/3}\right )}{4\ 2^{2/3} \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} (-1+x)^{2/3} \left (1-x-x^2\right ) \log \left (\left (3+\sqrt {5}\right )^{2/3}-\sqrt [3]{2 \left (3+\sqrt {5}\right )} \sqrt [3]{-1+x}+2^{2/3} (-1+x)^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{-(1-x)^2 \left (1-x-x^2\right )^3}}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 193, normalized size = 1.10 \begin {gather*} -\frac {\left (x^2+x-1\right ) \left (-6 (x-1)^{2/3} x \text {RootSum}\left [\text {$\#$1}^6+3 \text {$\#$1}^3+1\&,\frac {\text {$\#$1}^3 \log \left (\sqrt [3]{x-1}-\text {$\#$1}\right )+3 \log \left (\sqrt [3]{x-1}-\text {$\#$1}\right )}{2 \text {$\#$1}^5+3 \text {$\#$1}^2}\&\right ]+6 x+10 (x-1)^{2/3} x \log \left (\sqrt [3]{x-1}+1\right )-5 (x-1)^{2/3} x \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )+10 \sqrt {3} (x-1)^{2/3} x \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}-1}{\sqrt {3}}\right )-6\right )}{6 x \sqrt [3]{(x-1)^2 \left (x^2+x-1\right )^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(-1 + 5*x - 7*x^2 - 2*x^3 + 10*x^4 - 2*x^5 - 5*x^6 + x^7 + x^8)^(1/3)),x]

[Out]

-1/6*((-1 + x + x^2)*(-6 + 6*x + 10*Sqrt[3]*(-1 + x)^(2/3)*x*ArcTan[(-1 + 2*(-1 + x)^(1/3))/Sqrt[3]] + 10*(-1
+ x)^(2/3)*x*Log[1 + (-1 + x)^(1/3)] - 5*(-1 + x)^(2/3)*x*Log[1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)] - 6*(-1 + x
)^(2/3)*x*RootSum[1 + 3*#1^3 + #1^6 & , (3*Log[(-1 + x)^(1/3) - #1] + Log[(-1 + x)^(1/3) - #1]*#1^3)/(3*#1^2 +
2*#1^5) & ]))/(x*((-1 + x)^2*(-1 + x + x^2)^3)^(1/3))

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IntegrateAlgebraic [A]  time = 27.06, size = 175, normalized size = 1.00 \begin {gather*} \frac {(-1+x)^{2/3} \left (-1+x+x^2\right ) \left (-\frac {\sqrt [3]{-1+x}}{x}+\frac {5 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5}{3} \log \left (1+\sqrt [3]{-1+x}\right )+\frac {5}{6} \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )+\text {RootSum}\left [1+3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {3 \log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right )}{\sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^2*(-1 + 5*x - 7*x^2 - 2*x^3 + 10*x^4 - 2*x^5 - 5*x^6 + x^7 + x^8)^(1/3)),x]

[Out]

((-1 + x)^(2/3)*(-1 + x + x^2)*(-((-1 + x)^(1/3)/x) + (5*ArcTan[1/Sqrt[3] - (2*(-1 + x)^(1/3))/Sqrt[3]])/Sqrt[
3] - (5*Log[1 + (-1 + x)^(1/3)])/3 + (5*Log[1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)])/6 + RootSum[1 + 3*#1^3 + #1^
6 & , (3*Log[(-1 + x)^(1/3) - #1] + Log[(-1 + x)^(1/3) - #1]*#1^3)/(3*#1^2 + 2*#1^5) & ]))/((-1 + x)^2*(-1 + x
+ x^2)^3)^(1/3)

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fricas [B]  time = 0.75, size = 1949, normalized size = 11.14

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8+x^7-5*x^6-2*x^5+10*x^4-2*x^3-7*x^2+5*x-1)^(1/3),x, algorithm="fricas")

[Out]

1/300*(12*50^(2/3)*sqrt(3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x)*(123*sqrt(5) + 275)^(1/3)*arctan(1/1500*(50^(
2/3)*(7*sqrt(5)*sqrt(3)*(x^3 - 2*x + 1) - 15*sqrt(3)*(x^3 - 2*x + 1))*(123*sqrt(5) + 275)^(1/3)*sqrt(-(10*50^(
1/3)*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3)*(9*x^3 - 4*sqrt(5)*(x^3 - 2*x + 1) -
18*x + 9)*(123*sqrt(5) + 275)^(2/3) + 50^(2/3)*(5*x^6 - 20*x^4 + 10*x^3 + 20*x^2 - 3*sqrt(5)*(x^6 - 4*x^4 + 2
*x^3 + 4*x^2 - 4*x + 1) - 20*x + 5)*(123*sqrt(5) + 275)^(1/3) - 100*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^
3 - 7*x^2 + 5*x - 1)^(2/3))/(x^6 - 4*x^4 + 2*x^3 + 4*x^2 - 4*x + 1)) - 10*50^(2/3)*(x^8 + x^7 - 5*x^6 - 2*x^5
+ 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3)*(7*sqrt(5)*sqrt(3) - 15*sqrt(3))*(123*sqrt(5) + 275)^(1/3) + 500*sqr
t(3)*(x^3 - 2*x + 1))/(x^3 - 2*x + 1)) - 12*50^(2/3)*sqrt(3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x)*(-123*sqrt(
5) + 275)^(1/3)*arctan(1/1500*(50^(2/3)*(7*sqrt(5)*sqrt(3)*(x^3 - 2*x + 1) + 15*sqrt(3)*(x^3 - 2*x + 1))*(-123
*sqrt(5) + 275)^(1/3)*sqrt(-(10*50^(1/3)*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3)*
(9*x^3 + 4*sqrt(5)*(x^3 - 2*x + 1) - 18*x + 9)*(-123*sqrt(5) + 275)^(2/3) + 50^(2/3)*(5*x^6 - 20*x^4 + 10*x^3
+ 20*x^2 + 3*sqrt(5)*(x^6 - 4*x^4 + 2*x^3 + 4*x^2 - 4*x + 1) - 20*x + 5)*(-123*sqrt(5) + 275)^(1/3) - 100*(x^8
+ x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(2/3))/(x^6 - 4*x^4 + 2*x^3 + 4*x^2 - 4*x + 1)) - 1
0*50^(2/3)*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3)*(7*sqrt(5)*sqrt(3) + 15*sqrt(3
))*(-123*sqrt(5) + 275)^(1/3) - 500*sqrt(3)*(x^3 - 2*x + 1))/(x^3 - 2*x + 1)) - 3*50^(2/3)*(x^6 + x^5 - 3*x^4
- x^3 + 3*x^2 - x)*(123*sqrt(5) + 275)^(1/3)*log(-4*(10*50^(1/3)*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 -
7*x^2 + 5*x - 1)^(1/3)*(9*x^3 - 4*sqrt(5)*(x^3 - 2*x + 1) - 18*x + 9)*(123*sqrt(5) + 275)^(2/3) + 50^(2/3)*(5
*x^6 - 20*x^4 + 10*x^3 + 20*x^2 - 3*sqrt(5)*(x^6 - 4*x^4 + 2*x^3 + 4*x^2 - 4*x + 1) - 20*x + 5)*(123*sqrt(5) +
275)^(1/3) - 100*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(2/3))/(x^6 - 4*x^4 + 2*x^3 +
4*x^2 - 4*x + 1)) - 3*50^(2/3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x)*(-123*sqrt(5) + 275)^(1/3)*log(-4*(10*50
^(1/3)*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3)*(9*x^3 + 4*sqrt(5)*(x^3 - 2*x + 1)
- 18*x + 9)*(-123*sqrt(5) + 275)^(2/3) + 50^(2/3)*(5*x^6 - 20*x^4 + 10*x^3 + 20*x^2 + 3*sqrt(5)*(x^6 - 4*x^4
+ 2*x^3 + 4*x^2 - 4*x + 1) - 20*x + 5)*(-123*sqrt(5) + 275)^(1/3) - 100*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 -
2*x^3 - 7*x^2 + 5*x - 1)^(2/3))/(x^6 - 4*x^4 + 2*x^3 + 4*x^2 - 4*x + 1)) + 6*50^(2/3)*(x^6 + x^5 - 3*x^4 - x^3
+ 3*x^2 - x)*(123*sqrt(5) + 275)^(1/3)*log((50^(1/3)*(9*x^3 - 4*sqrt(5)*(x^3 - 2*x + 1) - 18*x + 9)*(123*sqrt
(5) + 275)^(2/3) + 10*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3))/(x^3 - 2*x + 1)) +
6*50^(2/3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x)*(-123*sqrt(5) + 275)^(1/3)*log((50^(1/3)*(9*x^3 + 4*sqrt(5)*
(x^3 - 2*x + 1) - 18*x + 9)*(-123*sqrt(5) + 275)^(2/3) + 10*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^
2 + 5*x - 1)^(1/3))/(x^3 - 2*x + 1)) + 500*sqrt(3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x)*arctan(-1/3*(sqrt(3)*
(x^3 - 2*x + 1) - 2*sqrt(3)*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3))/(x^3 - 2*x +
1)) + 250*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x)*log((x^6 - 4*x^4 + 2*x^3 + 4*x^2 - (x^8 + x^7 - 5*x^6 - 2*x^5
+ 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3)*(x^3 - 2*x + 1) - 4*x + (x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3
- 7*x^2 + 5*x - 1)^(2/3) + 1)/(x^6 - 4*x^4 + 2*x^3 + 4*x^2 - 4*x + 1)) - 500*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2
- x)*log((x^3 - 2*x + (x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3) + 1)/(x^3 - 2*x +
1)) - 300*(x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(2/3))/(x^6 + x^5 - 3*x^4 - x^3 + 3*x
^2 - x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{8} + x^{7} - 5 \, x^{6} - 2 \, x^{5} + 10 \, x^{4} - 2 \, x^{3} - 7 \, x^{2} + 5 \, x - 1\right )}^{\frac {1}{3}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8+x^7-5*x^6-2*x^5+10*x^4-2*x^3-7*x^2+5*x-1)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3)*x^2), x)

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maple [B]  time = 43.44, size = 11794, normalized size = 67.39

method	result	size
risch	\(\text {Expression too large to display}\)	\(11794\)
trager	\(\text {Expression too large to display}\)	\(106745\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(x^8+x^7-5*x^6-2*x^5+10*x^4-2*x^3-7*x^2+5*x-1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{8} + x^{7} - 5 \, x^{6} - 2 \, x^{5} + 10 \, x^{4} - 2 \, x^{3} - 7 \, x^{2} + 5 \, x - 1\right )}^{\frac {1}{3}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8+x^7-5*x^6-2*x^5+10*x^4-2*x^3-7*x^2+5*x-1)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((x^8 + x^7 - 5*x^6 - 2*x^5 + 10*x^4 - 2*x^3 - 7*x^2 + 5*x - 1)^(1/3)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^2\,{\left (x^8+x^7-5\,x^6-2\,x^5+10\,x^4-2\,x^3-7\,x^2+5\,x-1\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(5*x - 7*x^2 - 2*x^3 + 10*x^4 - 2*x^5 - 5*x^6 + x^7 + x^8 - 1)^(1/3)),x)

[Out]

int(1/(x^2*(5*x - 7*x^2 - 2*x^3 + 10*x^4 - 2*x^5 - 5*x^6 + x^7 + x^8 - 1)^(1/3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \sqrt [3]{\left (x - 1\right )^{2} \left (x^{2} + x - 1\right )^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(x**8+x**7-5*x**6-2*x**5+10*x**4-2*x**3-7*x**2+5*x-1)**(1/3),x)

[Out]

Integral(1/(x**2*((x - 1)**2*(x**2 + x - 1)**3)**(1/3)), x)

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