\(\int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx\) [2422]

Optimal result

Integrand size = 46, antiderivative size = 196 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\frac {(-2+x)^{2/3} \left (-1-x+x^2\right ) \left (-\frac {\sqrt [3]{-2+x}}{2 x}-\frac {\sqrt [3]{2} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \sqrt [3]{-2+x}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{2} \log \left (2+2^{2/3} \sqrt [3]{-2+x}\right )-\frac {\log \left (-2+2^{2/3} \sqrt [3]{-2+x}-\sqrt [3]{2} (-2+x)^{2/3}\right )}{3\ 2^{2/3}}-\text {RootSum}\left [1+3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log \left (\sqrt [3]{-2+x}-\text {$\#$1}\right ) \text {$\#$1}}{3+2 \text {$\#$1}^3}\&\right ]\right )}{\sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \]

[Out]

Unintegrable

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(845\) vs. \(2(196)=392\).

Time = 0.58 (sec) , antiderivative size = 845, normalized size of antiderivative = 4.31, number of steps used = 31, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6820, 6851, 911, 1438, 205, 206, 31, 648, 631, 210, 642, 1388} \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\frac {\sqrt [3]{2} (x-2)^{2/3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{x-2}}{\sqrt {3}}\right ) \left (-x^2+x+1\right )}{\sqrt {3} \sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}-\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} (x-2)^{2/3} \arctan \left (\frac {1-2 \sqrt [3]{\frac {2}{3+\sqrt {5}}} \sqrt [3]{x-2}}{\sqrt {3}}\right ) \left (-x^2+x+1\right )}{\sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}+\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} (x-2)^{2/3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{3+\sqrt {5}} \sqrt [3]{x-2}}{\sqrt {3}}\right ) \left (-x^2+x+1\right )}{\sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}-\frac {\sqrt [3]{2} (x-2)^{2/3} \log \left (\sqrt [3]{x-2}+\sqrt [3]{2}\right ) \left (-x^2+x+1\right )}{3 \sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} (x-2)^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{x-2}+\sqrt [3]{3-\sqrt {5}}\right ) \left (-x^2+x+1\right )}{\sqrt {5} \sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} (x-2)^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{x-2}+\sqrt [3]{3+\sqrt {5}}\right ) \left (-x^2+x+1\right )}{\sqrt {5} \sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}+\frac {(x-2)^{2/3} \log \left ((x-2)^{2/3}-\sqrt [3]{2} \sqrt [3]{x-2}+2^{2/3}\right ) \left (-x^2+x+1\right )}{3\ 2^{2/3} \sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} (x-2)^{2/3} \log \left (2^{2/3} (x-2)^{2/3}-\sqrt [3]{2 \left (3-\sqrt {5}\right )} \sqrt [3]{x-2}+\left (3-\sqrt {5}\right )^{2/3}\right ) \left (-x^2+x+1\right )}{2 \sqrt {5} \sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} (x-2)^{2/3} \log \left (2^{2/3} (x-2)^{2/3}-\sqrt [3]{2 \left (3+\sqrt {5}\right )} \sqrt [3]{x-2}+\left (3+\sqrt {5}\right )^{2/3}\right ) \left (-x^2+x+1\right )}{2 \sqrt {5} \sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}}-\frac {(2-x) \left (-x^2+x+1\right )}{2 x \sqrt [3]{-(2-x)^2 \left (-x^2+x+1\right )^3}} \]

[In]

Int[1/(x^2*(-4 - 8*x + 11*x^2 + 17*x^3 - 20*x^4 - 7*x^5 + 16*x^6 - 7*x^7 + x^8)^(1/3)),x]

[Out]

-1/2*((2 - x)*(1 + x - x^2))/(x*(-((2 - x)^2*(1 + x - x^2)^3))^(1/3)) + (2^(1/3)*(-2 + x)^(2/3)*(1 + x - x^2)*
ArcTan[(1 - 2^(2/3)*(-2 + x)^(1/3))/Sqrt[3]])/(Sqrt[3]*(-((2 - x)^2*(1 + x - x^2)^3))^(1/3)) - (Sqrt[3/5]*((3
+ Sqrt[5])/2)^(1/3)*(-2 + x)^(2/3)*(1 + x - x^2)*ArcTan[(1 - 2*(2/(3 + Sqrt[5]))^(1/3)*(-2 + x)^(1/3))/Sqrt[3]
])/(-((2 - x)^2*(1 + x - x^2)^3))^(1/3) + (Sqrt[3/5]*((3 - Sqrt[5])/2)^(1/3)*(-2 + x)^(2/3)*(1 + x - x^2)*ArcT
an[(1 - 2^(2/3)*(3 + Sqrt[5])^(1/3)*(-2 + x)^(1/3))/Sqrt[3]])/(-((2 - x)^2*(1 + x - x^2)^3))^(1/3) - (2^(1/3)*
(-2 + x)^(2/3)*(1 + x - x^2)*Log[2^(1/3) + (-2 + x)^(1/3)])/(3*(-((2 - x)^2*(1 + x - x^2)^3))^(1/3)) - (((3 -
Sqrt[5])/2)^(1/3)*(-2 + x)^(2/3)*(1 + x - x^2)*Log[(3 - Sqrt[5])^(1/3) + 2^(1/3)*(-2 + x)^(1/3)])/(Sqrt[5]*(-(
(2 - x)^2*(1 + x - x^2)^3))^(1/3)) + (((3 + Sqrt[5])/2)^(1/3)*(-2 + x)^(2/3)*(1 + x - x^2)*Log[(3 + Sqrt[5])^(
1/3) + 2^(1/3)*(-2 + x)^(1/3)])/(Sqrt[5]*(-((2 - x)^2*(1 + x - x^2)^3))^(1/3)) + ((-2 + x)^(2/3)*(1 + x - x^2)
*Log[2^(2/3) - 2^(1/3)*(-2 + x)^(1/3) + (-2 + x)^(2/3)])/(3*2^(2/3)*(-((2 - x)^2*(1 + x - x^2)^3))^(1/3)) + ((
(3 - Sqrt[5])/2)^(1/3)*(-2 + x)^(2/3)*(1 + x - x^2)*Log[(3 - Sqrt[5])^(2/3) - (2*(3 - Sqrt[5]))^(1/3)*(-2 + x)
^(1/3) + 2^(2/3)*(-2 + x)^(2/3)])/(2*Sqrt[5]*(-((2 - x)^2*(1 + x - x^2)^3))^(1/3)) - (((3 + Sqrt[5])/2)^(1/3)*
(-2 + x)^(2/3)*(1 + x - x^2)*Log[(3 + Sqrt[5])^(2/3) - (2*(3 + Sqrt[5]))^(1/3)*(-2 + x)^(1/3) + 2^(2/3)*(-2 +
x)^(2/3)])/(2*Sqrt[5]*(-((2 - 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 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 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 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 1388

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

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]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \, dx \\ & = \frac {\left ((-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \int \frac {1}{(-2+x)^{2/3} x^2 \left (-1-x+x^2\right )} \, dx}{\sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \\ & = \frac {\left (3 (-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (2+x^3\right )^2 \left (1+3 x^3+x^6\right )} \, dx,x,\sqrt [3]{-2+x}\right )}{\sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \\ & = \frac {\left (3 (-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \left (-\frac {1}{\left (2+x^3\right )^2}+\frac {1}{2+x^3}-\frac {x^3}{1+3 x^3+x^6}\right ) \, dx,x,\sqrt [3]{-2+x}\right )}{\sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {\left (3 (-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (2+x^3\right )^2} \, dx,x,\sqrt [3]{-2+x}\right )}{\sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (3 (-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{2+x^3} \, dx,x,\sqrt [3]{-2+x}\right )}{\sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (3 (-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {x^3}{1+3 x^3+x^6} \, dx,x,\sqrt [3]{-2+x}\right )}{\sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(2-x) \left (1+x-x^2\right )}{2 x \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\left ((-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{2+x^3} \, dx,x,\sqrt [3]{-2+x}\right )}{\sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left ((-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2}+x} \, dx,x,\sqrt [3]{-2+x}\right )}{2^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left ((-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {2 \sqrt [3]{2}-x}{2^{2/3}-\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{-2+x}\right )}{2^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (3 \left (-5+3 \sqrt {5}\right ) (-2+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]{-2+x}\right )}{10 \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (3 \left (5+3 \sqrt {5}\right ) (-2+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]{-2+x}\right )}{10 \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(2-x) \left (1+x-x^2\right )}{2 x \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {(-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{2}+\sqrt [3]{-2+x}\right )}{2^{2/3} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\left ((-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2}+x} \, dx,x,\sqrt [3]{-2+x}\right )}{3\ 2^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left ((-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {2 \sqrt [3]{2}-x}{2^{2/3}-\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{-2+x}\right )}{3\ 2^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left ((-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{2}+2 x}{2^{2/3}-\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{-2+x}\right )}{2\ 2^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (3 (-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{-2+x}\right )}{2 \sqrt [3]{2} \sqrt [3]{(-2+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 ) (-2+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]{-2+x}\right )}{10 \sqrt [3]{(-2+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 ) (-2+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]{-2+x}\right )}{10 \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (\left (5+3 \sqrt {5}\right ) (-2+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]{-2+x}\right )}{5 \sqrt [3]{2} \left (3+\sqrt {5}\right )^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (\left (5+3 \sqrt {5}\right ) (-2+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]{-2+x}\right )}{5 \sqrt [3]{2} \left (3+\sqrt {5}\right )^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(2-x) \left (1+x-x^2\right )}{2 x \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{2} (-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{2}+\sqrt [3]{-2+x}\right )}{3 \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} (-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{3-\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-2+x}\right )}{\sqrt {5} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} (-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{3+\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-2+x}\right )}{\sqrt {5} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {(-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (2^{2/3}-\sqrt [3]{2} \sqrt [3]{-2+x}+(-2+x)^{2/3}\right )}{2\ 2^{2/3} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {\left ((-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{2}+2 x}{2^{2/3}-\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{-2+x}\right )}{6\ 2^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (3 (-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2^{2/3} \sqrt [3]{-2+x}\right )}{2^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left ((-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{-2+x}\right )}{2 \sqrt [3]{2} \sqrt [3]{(-2+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 ) (-2+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]{-2+x}\right )}{20 \sqrt [3]{(-2+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 ) (-2+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]{-2+x}\right )}{40 \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}+\frac {\left (\left (5+3 \sqrt {5}\right ) (-2+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]{-2+x}\right )}{10 \sqrt [3]{2} \left (3+\sqrt {5}\right )^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (3 \left (5+3 \sqrt {5}\right ) (-2+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]{-2+x}\right )}{10\ 2^{2/3} \sqrt [3]{3+\sqrt {5}} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(2-x) \left (1+x-x^2\right )}{2 x \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt {3} (-2+x)^{2/3} \left (1+x-x^2\right ) \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-2+x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{2} (-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{2}+\sqrt [3]{-2+x}\right )}{3 \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} (-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{3-\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-2+x}\right )}{\sqrt {5} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} (-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{3+\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-2+x}\right )}{\sqrt {5} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {(-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (2^{2/3}-\sqrt [3]{2} \sqrt [3]{-2+x}+(-2+x)^{2/3}\right )}{3\ 2^{2/3} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} (-2+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]{-2+x}+2^{2/3} (-2+x)^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} (-2+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]{-2+x}+2^{2/3} (-2+x)^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\left ((-2+x)^{2/3} \left (-1-x+x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2^{2/3} \sqrt [3]{-2+x}\right )}{2^{2/3} \sqrt [3]{(-2+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 ) (-2+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]{-2+x}\right )}{20 \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}}-\frac {\left (3 \left (5+3 \sqrt {5}\right ) (-2+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]{-2+x}\right )}{5 \sqrt [3]{2} \left (3+\sqrt {5}\right )^{2/3} \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \\ & = -\frac {(2-x) \left (1+x-x^2\right )}{2 x \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {(-2+x)^{2/3} \left (1+x-x^2\right ) \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-2+x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt {3} (-2+x)^{2/3} \left (1+x-x^2\right ) \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-2+x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} (-2+x)^{2/3} \left (1+x-x^2\right ) \arctan \left (\frac {1-2 \sqrt [3]{\frac {2}{3+\sqrt {5}}} \sqrt [3]{-2+x}}{\sqrt {3}}\right )}{\sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt {\frac {3}{5}} \sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} (-2+x)^{2/3} \left (1+x-x^2\right ) \arctan \left (\frac {1-2^{2/3} \sqrt [3]{3+\sqrt {5}} \sqrt [3]{-2+x}}{\sqrt {3}}\right )}{\sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{2} (-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{2}+\sqrt [3]{-2+x}\right )}{3 \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} (-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{3-\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-2+x}\right )}{\sqrt {5} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} (-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (\sqrt [3]{3+\sqrt {5}}+\sqrt [3]{2} \sqrt [3]{-2+x}\right )}{\sqrt {5} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {(-2+x)^{2/3} \left (1+x-x^2\right ) \log \left (2^{2/3}-\sqrt [3]{2} \sqrt [3]{-2+x}+(-2+x)^{2/3}\right )}{3\ 2^{2/3} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}+\frac {\sqrt [3]{\frac {1}{2} \left (3-\sqrt {5}\right )} (-2+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]{-2+x}+2^{2/3} (-2+x)^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (3+\sqrt {5}\right )} (-2+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]{-2+x}+2^{2/3} (-2+x)^{2/3}\right )}{2 \sqrt {5} \sqrt [3]{-(2-x)^2 \left (1+x-x^2\right )^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=-\frac {\left (-1-x+x^2\right ) \left (-6+3 x+2 \sqrt [3]{2} \sqrt {3} (-2+x)^{2/3} x \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-2+x}}{\sqrt {3}}\right )-2 \sqrt [3]{2} (-2+x)^{2/3} x \log \left (2+2^{2/3} \sqrt [3]{-2+x}\right )+\sqrt [3]{2} (-2+x)^{2/3} x \log \left (-2+2^{2/3} \sqrt [3]{-2+x}-\sqrt [3]{2} (-2+x)^{2/3}\right )+6 (-2+x)^{2/3} x \text {RootSum}\left [1+3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log \left (\sqrt [3]{-2+x}-\text {$\#$1}\right ) \text {$\#$1}}{3+2 \text {$\#$1}^3}\&\right ]\right )}{6 x \sqrt [3]{(-2+x)^2 \left (-1-x+x^2\right )^3}} \]

[In]

Integrate[1/(x^2*(-4 - 8*x + 11*x^2 + 17*x^3 - 20*x^4 - 7*x^5 + 16*x^6 - 7*x^7 + x^8)^(1/3)),x]

[Out]

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

Maple [N/A]

Time = 53.25 (sec) , antiderivative size = 15782, normalized size of antiderivative = 80.52

method	result	size
risch	\(\text {Expression too large to display}\)	\(15782\)
trager	\(\text {Expression too large to display}\)	\(89408\)

[In]

int(1/x^2/(x^8-7*x^7+16*x^6-7*x^5-20*x^4+17*x^3+11*x^2-8*x-4)^(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 = 0.30 (sec) , antiderivative size = 1601, normalized size of antiderivative = 8.17 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\text {Too large to display} \]

[In]

integrate(1/x^2/(x^8-7*x^7+16*x^6-7*x^5-20*x^4+17*x^3+11*x^2-8*x-4)^(1/3),x, algorithm="fricas")

[Out]

-1/300*(3*50^(2/3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - sqrt(-3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x)
- 2*x)*(3*sqrt(5) - 5)^(1/3)*log(-(50^(1/3)*(3*x^3 - 9*x^2 + sqrt(5)*(x^3 - 3*x^2 + sqrt(-3)*(x^3 - 3*x^2 + x
+ 2) + x + 2) + 3*sqrt(-3)*(x^3 - 3*x^2 + x + 2) + 3*x + 6)*(3*sqrt(5) - 5)^(2/3) - 40*(x^8 - 7*x^7 + 16*x^6
- 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3))/(x^3 - 3*x^2 + x + 2)) + 3*50^(2/3)*(x^6 - 4*x^5 + 3*x^4
+ 4*x^3 - 3*x^2 + sqrt(-3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x) - 2*x)*(3*sqrt(5) - 5)^(1/3)*log(-(50^(
1/3)*(3*x^3 - 9*x^2 + sqrt(5)*(x^3 - 3*x^2 - sqrt(-3)*(x^3 - 3*x^2 + x + 2) + x + 2) - 3*sqrt(-3)*(x^3 - 3*x^2
+ x + 2) + 3*x + 6)*(3*sqrt(5) - 5)^(2/3) - 40*(x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x
- 4)^(1/3))/(x^3 - 3*x^2 + x + 2)) - 6*50^(2/3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x)*(3*sqrt(5) - 5)^(
1/3)*log((50^(1/3)*(3*x^3 - 9*x^2 + sqrt(5)*(x^3 - 3*x^2 + x + 2) + 3*x + 6)*(3*sqrt(5) - 5)^(2/3) + 20*(x^8 -
7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3))/(x^3 - 3*x^2 + x + 2)) + 3*50^(2/3)*(x^6
- 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - sqrt(-3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x) - 2*x)*(-3*sqrt(5) - 5)
^(1/3)*log(-(50^(1/3)*(3*x^3 - 9*x^2 - sqrt(5)*(x^3 - 3*x^2 + sqrt(-3)*(x^3 - 3*x^2 + x + 2) + x + 2) + 3*sqrt
(-3)*(x^3 - 3*x^2 + x + 2) + 3*x + 6)*(-3*sqrt(5) - 5)^(2/3) - 40*(x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*
x^3 + 11*x^2 - 8*x - 4)^(1/3))/(x^3 - 3*x^2 + x + 2)) + 3*50^(2/3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 + sqrt
(-3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x) - 2*x)*(-3*sqrt(5) - 5)^(1/3)*log(-(50^(1/3)*(3*x^3 - 9*x^2 -
sqrt(5)*(x^3 - 3*x^2 - sqrt(-3)*(x^3 - 3*x^2 + x + 2) + x + 2) - 3*sqrt(-3)*(x^3 - 3*x^2 + x + 2) + 3*x + 6)*
(-3*sqrt(5) - 5)^(2/3) - 40*(x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3))/(x^3 -
3*x^2 + x + 2)) - 6*50^(2/3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x)*(-3*sqrt(5) - 5)^(1/3)*log((50^(1/3)*
(3*x^3 - 9*x^2 - sqrt(5)*(x^3 - 3*x^2 + x + 2) + 3*x + 6)*(-3*sqrt(5) - 5)^(2/3) + 20*(x^8 - 7*x^7 + 16*x^6 -
7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3))/(x^3 - 3*x^2 + x + 2)) + 100*sqrt(3)*2^(1/3)*(x^6 - 4*x^5 +
3*x^4 + 4*x^3 - 3*x^2 - 2*x)*arctan(-1/3*(sqrt(3)*(x^3 - 3*x^2 + x + 2) - 2*sqrt(3)*2^(1/3)*(x^8 - 7*x^7 + 16
*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3))/(x^3 - 3*x^2 + x + 2)) + 50*2^(1/3)*(x^6 - 4*x^5 + 3
*x^4 + 4*x^3 - 3*x^2 - 2*x)*log(-(2^(2/3)*(x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^
(1/3)*(x^3 - 3*x^2 + x + 2) - 2^(1/3)*(x^6 - 6*x^5 + 11*x^4 - 2*x^3 - 11*x^2 + 4*x + 4) - 2*(x^8 - 7*x^7 + 16*
x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(2/3))/(x^6 - 6*x^5 + 11*x^4 - 2*x^3 - 11*x^2 + 4*x + 4)) -
100*2^(1/3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x)*log((2^(2/3)*(x^3 - 3*x^2 + x + 2) + 2*(x^8 - 7*x^7 +
16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3))/(x^3 - 3*x^2 + x + 2)) + 150*(x^8 - 7*x^7 + 16*x^6
- 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(2/3))/(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x)

Sympy [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\int \frac {1}{x^{2} \sqrt [3]{\left (x - 2\right )^{2} \left (x^{2} - x - 1\right )^{3}}}\, dx \]

[In]

integrate(1/x**2/(x**8-7*x**7+16*x**6-7*x**5-20*x**4+17*x**3+11*x**2-8*x-4)**(1/3),x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} - 7 \, x^{7} + 16 \, x^{6} - 7 \, x^{5} - 20 \, x^{4} + 17 \, x^{3} + 11 \, x^{2} - 8 \, x - 4\right )}^{\frac {1}{3}} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(x^8-7*x^7+16*x^6-7*x^5-20*x^4+17*x^3+11*x^2-8*x-4)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3)*x^2), x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} - 7 \, x^{7} + 16 \, x^{6} - 7 \, x^{5} - 20 \, x^{4} + 17 \, x^{3} + 11 \, x^{2} - 8 \, x - 4\right )}^{\frac {1}{3}} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(x^8-7*x^7+16*x^6-7*x^5-20*x^4+17*x^3+11*x^2-8*x-4)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((x^8 - 7*x^7 + 16*x^6 - 7*x^5 - 20*x^4 + 17*x^3 + 11*x^2 - 8*x - 4)^(1/3)*x^2), x)

Mupad [N/A]

Not integrable

Time = 6.34 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^2 \sqrt [3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx=\int \frac {1}{x^2\,{\left (x^8-7\,x^7+16\,x^6-7\,x^5-20\,x^4+17\,x^3+11\,x^2-8\,x-4\right )}^{1/3}} \,d x \]

[In]

int(1/(x^2*(11*x^2 - 8*x + 17*x^3 - 20*x^4 - 7*x^5 + 16*x^6 - 7*x^7 + x^8 - 4)^(1/3)),x)

[Out]

int(1/(x^2*(11*x^2 - 8*x + 17*x^3 - 20*x^4 - 7*x^5 + 16*x^6 - 7*x^7 + x^8 - 4)^(1/3)), x)