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

Optimal. Leaf size=196 \[ \frac {(x-2)^{2/3} \left (x^2-x-1\right ) \left (-\text {RootSum}\left [\text {$\#$1}^6+3 \text {$\#$1}^3+1\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{x-2}-\text {$\#$1}\right )}{2 \text {$\#$1}^3+3}\& \right ]-\frac {\sqrt [3]{x-2}}{2 x}+\frac {1}{3} \sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x-2}+2\right )-\frac {\log \left (-\sqrt [3]{2} (x-2)^{2/3}+2^{2/3} \sqrt [3]{x-2}-2\right )}{3\ 2^{2/3}}-\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \sqrt [3]{x-2}}{\sqrt {3}}\right )}{\sqrt {3}}\right )}{\sqrt [3]{(x-2)^2 \left (x^2-x-1\right )^3}} \]

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Rubi [B]  time = 0.91, antiderivative size = 845, normalized size of antiderivative = 4.31, number of steps used = 31, number of rules used = 12, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6688, 6719, 897, 1424, 199, 200, 31, 634, 617, 204, 628, 1374} \begin {gather*} \frac {\sqrt [3]{2} (x-2)^{2/3} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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}} \end {gather*}

Warning: Unable to verify antiderivative.

[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 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 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 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 1374

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*(b/q + 1))/2, Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n*(b/q - 1))/2, 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 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]{-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]{(-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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \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]{-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 ) \operatorname {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 ) \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]{-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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \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]{-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 ) \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]{-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 ) \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]{-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 ) \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]{-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 ) \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \tan ^{-1}\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 ) \tan ^{-1}\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 ) \tan ^{-1}\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 ) \tan ^{-1}\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*}

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Mathematica [A]  time = 0.26, size = 208, normalized size = 1.06 \begin {gather*} \frac {(x-2)^{2/3} \left (x^2-x-1\right ) \left (\sqrt [3]{2} \left (2 \log \left (2^{2/3} \sqrt [3]{x-2}+2\right )-\log \left (\sqrt [3]{2} (x-2)^{2/3}-2^{2/3} \sqrt [3]{x-2}+2\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x-2}-1}{\sqrt {3}}\right )\right )-6 \text {RootSum}\left [\text {$\#$1}^6+3 \text {$\#$1}^3+1\&,\frac {\text {$\#$1} \log \left (\sqrt [3]{x-2}-\text {$\#$1}\right )}{2 \text {$\#$1}^3+3}\&\right ]\right )}{6 \sqrt [3]{(x-2)^2 \left (x^2-x-1\right )^3}}-\frac {(x-2) \left (x^2-x-1\right )}{2 x \sqrt [3]{(x-2)^2 \left (x^2-x-1\right )^3}} \end {gather*}

Antiderivative was successfully verified.

[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]

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

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IntegrateAlgebraic [A]  time = 27.30, size = 196, normalized size = 1.00 \begin {gather*} \frac {(-2+x)^{2/3} \left (-1-x+x^2\right ) \left (-\frac {\sqrt [3]{-2+x}}{2 x}-\frac {\sqrt [3]{2} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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]

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

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fricas [B]  time = 0.68, size = 2143, normalized size = 10.93

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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*(12*50^(2/3)*sqrt(3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*x)*(3*sqrt(5) - 5)^(1/3)*arctan(1/750*(50^
(2/3)*sqrt(5)*sqrt(3)*(x^3 - 3*x^2 + x + 2)*(3*sqrt(5) - 5)^(1/3)*sqrt(-(5*50^(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)*(3*x^3 - 9*x^2 + sqrt(5)*(x^3 - 3*x^2 + x + 2) + 3*x + 6)*(3*s
qrt(5) - 5)^(2/3) - 50^(2/3)*(5*x^6 - 30*x^5 + 55*x^4 - 10*x^3 - 55*x^2 + 3*sqrt(5)*(x^6 - 6*x^5 + 11*x^4 - 2*
x^3 - 11*x^2 + 4*x + 4) + 20*x + 20)*(3*sqrt(5) - 5)^(1/3) - 100*(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)) - 10*50^(2/3)*sqrt(5)*sqrt(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)*(3*sqrt(5) - 5)^(1/3) + 250*sqrt(3)
*(x^3 - 3*x^2 + x + 2))/(x^3 - 3*x^2 + x + 2)) - 12*50^(2/3)*sqrt(3)*(x^6 - 4*x^5 + 3*x^4 + 4*x^3 - 3*x^2 - 2*
x)*(-3*sqrt(5) - 5)^(1/3)*arctan(1/750*(50^(2/3)*sqrt(5)*sqrt(3)*(x^3 - 3*x^2 + x + 2)*(-3*sqrt(5) - 5)^(1/3)*
sqrt(-(5*50^(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)*(3*x^3 - 9*x^2 - s
qrt(5)*(x^3 - 3*x^2 + x + 2) + 3*x + 6)*(-3*sqrt(5) - 5)^(2/3) - 50^(2/3)*(5*x^6 - 30*x^5 + 55*x^4 - 10*x^3 -
55*x^2 - 3*sqrt(5)*(x^6 - 6*x^5 + 11*x^4 - 2*x^3 - 11*x^2 + 4*x + 4) + 20*x + 20)*(-3*sqrt(5) - 5)^(1/3) - 100
*(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)) - 10*50^(2/3)*sqrt(5)*sqrt(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)*(-3*sqrt(5) - 5)^(1/3) - 250*sqrt(3)*(x^3 - 3*x^2 + x + 2))/(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 - 2*x)*(3*sqrt(5) - 5)^(1/3)*log(-16*(5*50^(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)*(3*x^3 - 9*x^2 + sqrt(5)*(x^3 - 3*x^2 + x + 2) + 3*x + 6)*(3*sq
rt(5) - 5)^(2/3) - 50^(2/3)*(5*x^6 - 30*x^5 + 55*x^4 - 10*x^3 - 55*x^2 + 3*sqrt(5)*(x^6 - 6*x^5 + 11*x^4 - 2*x
^3 - 11*x^2 + 4*x + 4) + 20*x + 20)*(3*sqrt(5) - 5)^(1/3) - 100*(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)) - 3*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(-16*(5*50^(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)*(3*x^3 - 9*x^2 - sqrt(5)*(x^3 - 3*x^2 + x + 2) + 3*x + 6)*(-3*sqrt(5) - 5)^(
2/3) - 50^(2/3)*(5*x^6 - 30*x^5 + 55*x^4 - 10*x^3 - 55*x^2 - 3*sqrt(5)*(x^6 - 6*x^5 + 11*x^4 - 2*x^3 - 11*x^2
+ 4*x + 4) + 20*x + 20)*(-3*sqrt(5) - 5)^(1/3) - 100*(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)) + 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)) + 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)) + 1
00*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 + 1
6*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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 48.83, size = 15767, normalized size = 80.44

method	result	size
risch	\(\text {Expression too large to display}\)	\(15767\)
trager	\(\text {Expression too large to display}\)	\(87802\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^2\,{\left (x^8-7\,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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 - 2\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**8-7*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)

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