\(\int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx\) [2297]

Optimal result

Integrand size = 44, antiderivative size = 175 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\frac {(-1+x)^{2/3} \left (-1+x+x^2\right ) \left (-\frac {\sqrt [3]{-1+x}}{x}+\frac {5 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5}{3} \log \left (1+\sqrt [3]{-1+x}\right )+\frac {5}{6} \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )+\text {RootSum}\left [1+3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {3 \log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right )}{\sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}} \]

[Out]

Unintegrable

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(946\) vs. \(2(175)=350\).

Time = 0.64 (sec) , antiderivative size = 946, normalized size of antiderivative = 5.41, number of steps used = 25, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.295, Rules used = {6820, 6851, 911, 1438, 652, 632, 210, 648, 642, 1436, 206, 31, 631} \[ \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=-\frac {5 (x-1)^{2/3} \arctan \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} \arctan \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} \arctan \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}} \]

[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 206

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

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 632

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 642

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 648

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 652

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)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), 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 911

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 1436

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 1438

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 6820

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

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[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*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \arctan \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \arctan \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 ) \text {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 ) \text {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 ) \arctan \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 ) \arctan \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 ) \arctan \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*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\frac {\left (-1+x+x^2\right ) \left (6-6 x+10 \sqrt {3} (-1+x)^{2/3} x \arctan \left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )-10 (-1+x)^{2/3} x \log \left (1+\sqrt [3]{-1+x}\right )+5 (-1+x)^{2/3} x \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )+6 (-1+x)^{2/3} x \text {RootSum}\left [1+3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {3 \log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right )}{6 x \sqrt [3]{(-1+x)^2 \left (-1+x+x^2\right )^3}} \]

[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 + 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
) & ]))/(6*x*((-1 + x)^2*(-1 + x + x^2)^3)^(1/3))

Maple [N/A]

Time = 54.56 (sec) , antiderivative size = 13654, normalized size of antiderivative = 78.02

method	result	size
risch	\(\text {Expression too large to display}\)	\(13654\)
trager	\(\text {Expression too large to display}\)	\(65982\)

[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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.29 (sec) , antiderivative size = 1409, normalized size of antiderivative = 8.05 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\text {Too large to display} \]

[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*(3*50^(2/3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - sqrt(-3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x) - x)*(123
*sqrt(5) + 275)^(1/3)*log(-(50^(1/3)*(9*x^3 - 4*sqrt(5)*(x^3 + sqrt(-3)*(x^3 - 2*x + 1) - 2*x + 1) + 9*sqrt(-3
)*(x^3 - 2*x + 1) - 18*x + 9)*(123*sqrt(5) + 275)^(2/3) - 20*(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)) + 3*50^(2/3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 + sqrt(-3)*(x^6 + x^5 - 3*
x^4 - x^3 + 3*x^2 - x) - x)*(123*sqrt(5) + 275)^(1/3)*log(-(50^(1/3)*(9*x^3 - 4*sqrt(5)*(x^3 - sqrt(-3)*(x^3 -
2*x + 1) - 2*x + 1) - 9*sqrt(-3)*(x^3 - 2*x + 1) - 18*x + 9)*(123*sqrt(5) + 275)^(2/3) - 20*(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)) + 3*
50^(2/3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - sqrt(-3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 - x) - x)*(-123*sqrt(5)
+ 275)^(1/3)*log(-(50^(1/3)*(9*x^3 + 4*sqrt(5)*(x^3 + sqrt(-3)*(x^3 - 2*x + 1) - 2*x + 1) + 9*sqrt(-3)*(x^3 -
2*x + 1) - 18*x + 9)*(-123*sqrt(5) + 275)^(2/3) - 20*(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)) + 3*50^(2/3)*(x^6 + x^5 - 3*x^4 - x^3 + 3*x^2 + sqrt(-3)*(x^6 + x^5 - 3*x^4 - x^
3 + 3*x^2 - x) - x)*(-123*sqrt(5) + 275)^(1/3)*log(-(50^(1/3)*(9*x^3 + 4*sqrt(5)*(x^3 - sqrt(-3)*(x^3 - 2*x +
1) - 2*x + 1) - 9*sqrt(-3)*(x^3 - 2*x + 1) - 18*x + 9)*(-123*sqrt(5) + 275)^(2/3) - 20*(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) + 2
75)^(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*sq
rt(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)

Sympy [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\int \frac {1}{x^{2} \sqrt [3]{\left (x - 1\right )^{2} \left (x^{2} + x - 1\right )^{3}}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} + x^{7} - 5 \, x^{6} - 2 \, x^{5} + 10 \, x^{4} - 2 \, x^{3} - 7 \, x^{2} + 5 \, x - 1\right )}^{\frac {1}{3}} x^{2}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} + x^{7} - 5 \, x^{6} - 2 \, x^{5} + 10 \, x^{4} - 2 \, x^{3} - 7 \, x^{2} + 5 \, x - 1\right )}^{\frac {1}{3}} x^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 6.46 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \sqrt [3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx=\int \frac {1}{x^2\,{\left (x^8+x^7-5\,x^6-2\,x^5+10\,x^4-2\,x^3-7\,x^2+5\,x-1\right )}^{1/3}} \,d x \]

[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)