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

Optimal. Leaf size=186 \[ -\frac {\left (\sqrt [3]{x-1}+1\right ) \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )^2 (x-1)^{2/3} \sqrt [3]{(x-1)^2 \left (x^2-x-1\right )^3} \left (\frac {(x-1)^{2/3} \left (6 x^2-21 x+10\right )}{10 x}-\frac {1}{3} \log \left (\sqrt [3]{x-1}+1\right )+\frac {1}{6} \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{\sqrt {3}}\right )}{\left (-x+(x-1)^{2/3}-\sqrt [3]{x-1}+1\right ) x \left (x^3-2 x^2+1\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.36, antiderivative size = 299, normalized size of antiderivative = 1.61, number of steps used = 12, number of rules used = 11, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.239, Rules used = {6688, 6719, 897, 1482, 1489, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {3 \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3} (1-x)}{5 \left (-x^2+x+1\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}{x \left (-x^2+x+1\right )}+\frac {3 \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}{2 \left (-x^2+x+1\right )}+\frac {\sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3} \log \left (\sqrt [3]{x-1}+1\right )}{3 (x-1)^{2/3} \left (-x^2+x+1\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3} \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )}{6 (x-1)^{2/3} \left (-x^2+x+1\right )}+\frac {\sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{\sqrt {3} (x-1)^{2/3} \left (-x^2+x+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, 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 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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 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 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 1482

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_), x_Symbol] :>
Simp[((-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1))/(n*e^(2*p
+ (m - Mod[m, n])/n)*(q + 1)), x] + Dist[1/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)), Int[x^Mod[m, n]*(d + e*x^n
)^(q + 1)*ExpandToSum[Together[(1*(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)*x^(m - Mod[m, n])*(a + b*x^n + c*x^(2
*n))^p - (-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1)
+ 1)*x^n)))/(d + e*x^n)], x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IG
tQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m, 0]

Rule 1489

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[Expan
dIntegrand[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2, 2*n]
&& IGtQ[n, 0] && IGtQ[p, 0]

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

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 63, normalized size = 0.34 \begin {gather*} \frac {\sqrt [3]{(x-1)^2 \left (x^2-x-1\right )^3} \left (5 x \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};1-x\right )+6 x^2-21 x+10\right )}{10 x \left (x^2-x-1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((-1 + x)^2*(-1 - x + x^2)^3)^(1/3)*(10 - 21*x + 6*x^2 + 5*x*Hypergeometric2F1[2/3, 1, 5/3, 1 - x]))/(10*x*(-
1 - x + x^2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 23.26, size = 190, normalized size = 1.02 \begin {gather*} -\frac {\left (1+\sqrt [3]{-1+x}\right ) \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )^2 (-1+x)^{2/3} \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3} \left (\frac {\left (-5-9 (-1+x)+6 (-1+x)^2\right ) (-1+x)^{2/3}}{10 x}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (1+\sqrt [3]{-1+x}\right )+\frac {1}{6} \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )\right )}{\left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}-x\right ) x \left (1-2 x^2+x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 + (-1 + x)^(1/3))*(1 - (-1 + x)^(1/3) + (-1 + x)^(2/3))^2*(-1 + x)^(2/3)*((-1 + x)^2*(-1 - x + x^2)^3)^(
1/3)*(((-5 - 9*(-1 + x) + 6*(-1 + x)^2)*(-1 + x)^(2/3))/(10*x) - ArcTan[1/Sqrt[3] - (2*(-1 + x)^(1/3))/Sqrt[3]
]/Sqrt[3] - Log[1 + (-1 + x)^(1/3)]/3 + Log[1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)]/6))/((1 - (-1 + x)^(1/3) + (-
1 + x)^(2/3) - x)*x*(1 - 2*x^2 + x^3)))

________________________________________________________________________________________

fricas [B]  time = 0.62, size = 399, normalized size = 2.15 \begin {gather*} -\frac {10 \, \sqrt {3} {\left (x^{3} - x^{2} - x\right )} \arctan \left (-\frac {\sqrt {3} {\left (x^{3} - 2 \, x^{2} + 1\right )} - 2 \, \sqrt {3} {\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x^{3} - 2 \, x^{2} + 1\right )}}\right ) - 5 \, {\left (x^{3} - x^{2} - x\right )} \log \left (\frac {x^{6} - 4 \, x^{5} + 4 \, x^{4} + 2 \, x^{3} - 4 \, x^{2} - {\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} + 1\right )} + {\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {2}{3}} + 1}{x^{6} - 4 \, x^{5} + 4 \, x^{4} + 2 \, x^{3} - 4 \, x^{2} + 1}\right ) + 10 \, {\left (x^{3} - x^{2} - x\right )} \log \left (\frac {x^{3} - 2 \, x^{2} + {\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}} + 1}{x^{3} - 2 \, x^{2} + 1}\right ) - 3 \, {\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}} {\left (6 \, x^{2} - 21 \, x + 10\right )}}{30 \, {\left (x^{3} - x^{2} - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(10*sqrt(3)*(x^3 - x^2 - x)*arctan(-1/3*(sqrt(3)*(x^3 - 2*x^2 + 1) - 2*sqrt(3)*(x^8 - 5*x^7 + 7*x^6 + 2*
x^5 - 10*x^4 + 2*x^3 + 5*x^2 - x - 1)^(1/3))/(x^3 - 2*x^2 + 1)) - 5*(x^3 - x^2 - x)*log((x^6 - 4*x^5 + 4*x^4 +
2*x^3 - 4*x^2 - (x^8 - 5*x^7 + 7*x^6 + 2*x^5 - 10*x^4 + 2*x^3 + 5*x^2 - x - 1)^(1/3)*(x^3 - 2*x^2 + 1) + (x^8
- 5*x^7 + 7*x^6 + 2*x^5 - 10*x^4 + 2*x^3 + 5*x^2 - x - 1)^(2/3) + 1)/(x^6 - 4*x^5 + 4*x^4 + 2*x^3 - 4*x^2 + 1
)) + 10*(x^3 - x^2 - x)*log((x^3 - 2*x^2 + (x^8 - 5*x^7 + 7*x^6 + 2*x^5 - 10*x^4 + 2*x^3 + 5*x^2 - x - 1)^(1/3
) + 1)/(x^3 - 2*x^2 + 1)) - 3*(x^8 - 5*x^7 + 7*x^6 + 2*x^5 - 10*x^4 + 2*x^3 + 5*x^2 - x - 1)^(1/3)*(6*x^2 - 21
*x + 10))/(x^3 - x^2 - x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.78, size = 137, normalized size = 0.74

method	result	size
risch	\(\frac {\left (6 x^{3}-27 x^{2}+31 x -10\right ) \left (\left (-1+x \right )^{2} \left (x^{2}-x -1\right )^{3}\right )^{\frac {1}{3}}}{10 x \left (-1+x \right ) \left (x^{2}-x -1\right )}+\frac {\left (-\frac {\ln \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}{3}+\frac {\ln \left (1-\left (-1+x \right )^{\frac {1}{3}}+\left (-1+x \right )^{\frac {2}{3}}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{3}\right ) \left (\left (-1+x \right )^{2} \left (x^{2}-x -1\right )^{3}\right )^{\frac {1}{3}}}{\left (-1+x \right )^{\frac {2}{3}} \left (x^{2}-x -1\right )}\)	\(137\)
trager	\(\frac {\left (6 x^{2}-21 x +10\right ) \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}}}{10 x \left (x^{2}-x -1\right )}-\frac {\ln \left (-\frac {1+x +6 x^{2} \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}}-x^{5}-3 x^{2}-x^{3}+3 x^{4}-3 \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}}+18 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {2}{3}}+18 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}}+18 \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}-36 \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+72 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{6}-6 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{6}-78 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{4}+504 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{4}+42 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{5}-360 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{5}+72 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x -504 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+78 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-18 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -24 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+72 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}+6 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}-3 \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}} x^{3}}{\left (x^{2}-x -1\right )^{2} x \left (-1+x \right )}\right )}{3}+\frac {\ln \left (\frac {2+6 x^{2} \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}}-8 x^{5}+2 x^{6}-8 x^{2}+4 x^{3}+8 x^{4}-3 \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}}+18 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {2}{3}}+36 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{6}-18 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{6}-108 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{4}+252 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{4}+84 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{5}-180 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{5}+36 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x -252 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+108 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-12 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -30 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+72 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+36 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}-24 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}-3 \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {2}{3}}-3 \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}} x^{3}}{\left (x^{2}-x -1\right )^{2} x \left (-1+x \right )}\right )}{3}-2 \ln \left (\frac {2+6 x^{2} \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}}-8 x^{5}+2 x^{6}-8 x^{2}+4 x^{3}+8 x^{4}-3 \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}}+18 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {2}{3}}+36 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{6}-18 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{6}-108 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{4}+252 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{4}+84 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{5}-180 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{5}+36 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x -252 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+108 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-12 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -30 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+72 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+36 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}-24 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}-3 \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {2}{3}}-3 \left (x^{8}-5 x^{7}+7 x^{6}+2 x^{5}-10 x^{4}+2 x^{3}+5 x^{2}-x -1\right )^{\frac {1}{3}} x^{3}}{\left (x^{2}-x -1\right )^{2} x \left (-1+x \right )}\right ) \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )\)	\(1710\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(6*x^3-27*x^2+31*x-10)/x/(-1+x)*((-1+x)^2*(x^2-x-1)^3)^(1/3)/(x^2-x-1)+(-1/3*ln(1+(-1+x)^(1/3))+1/6*ln(1-
(-1+x)^(1/3)+(-1+x)^(2/3))+1/3*3^(1/2)*arctan(1/3*(2*(-1+x)^(1/3)-1)*3^(1/2)))*((-1+x)^2*(x^2-x-1)^3)^(1/3)/(-
1+x)^(2/3)/(x^2-x-1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________