\(\int \frac {1}{x^6 (1+x^3) \sqrt [3]{x^2+x^3}} \, dx\) [1413]

Optimal result

Integrand size = 22, antiderivative size = 101 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {\left (x^2+x^3\right )^{2/3} \left (-9240+660 x-900 x^2+20985 x^3-6357 x^4+19071 x^5+109573 x^6\right )}{52360 x^7 (1+x)}-\frac {1}{3} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.00 (sec) , antiderivative size = 841, normalized size of antiderivative = 8.33, number of steps used = 29, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {2081, 6857, 21, 47, 37, 129, 491, 597, 12, 384} \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=-\frac {\left (1511+4777 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}-\frac {2187 (x+1)}{1309 x \sqrt [3]{x^3+x^2}}+\frac {\left (2249+153 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (2249-153 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {3645 (x+1)}{2618 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (41+17 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}+\frac {\left (41-17 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}-\frac {1620 (x+1)}{1309 x^3 \sqrt [3]{x^3+x^2}}+\frac {\left (13+17 i \sqrt {3}\right ) (x+1)}{476 x^4 \sqrt [3]{x^3+x^2}}+\frac {\left (13-17 i \sqrt {3}\right ) (x+1)}{476 x^4 \sqrt [3]{x^3+x^2}}+\frac {135 (x+1)}{119 x^4 \sqrt [3]{x^3+x^2}}-\frac {20 (x+1)}{17 x^5 \sqrt [3]{x^3+x^2}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}+\frac {6561 (x+1)}{2618 \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (\sqrt [3]{-1}-x\right ) \sqrt [3]{x+1}}{6 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (-x-(-1)^{2/3}\right ) \sqrt [3]{x+1}}{6 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \log \left (\sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}-\sqrt [3]{x+1}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \log \left (\sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}-\sqrt [3]{x+1}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^3+x^2}}+\frac {1}{x^5 \sqrt [3]{x^3+x^2}} \]

[In]

Int[1/(x^6*(1 + x^3)*(x^2 + x^3)^(1/3)),x]

[Out]

1/(x^5*(x^2 + x^3)^(1/3)) + (6561*(1 + x))/(2618*(x^2 + x^3)^(1/3)) - ((21647 - (11849*I)*Sqrt[3])*(1 + x))/(1
04720*(x^2 + x^3)^(1/3)) - ((21647 + (11849*I)*Sqrt[3])*(1 + x))/(104720*(x^2 + x^3)^(1/3)) - (20*(1 + x))/(17
*x^5*(x^2 + x^3)^(1/3)) + (135*(1 + x))/(119*x^4*(x^2 + x^3)^(1/3)) + ((13 - (17*I)*Sqrt[3])*(1 + x))/(476*x^4
*(x^2 + x^3)^(1/3)) + ((13 + (17*I)*Sqrt[3])*(1 + x))/(476*x^4*(x^2 + x^3)^(1/3)) - (1620*(1 + x))/(1309*x^3*(
x^2 + x^3)^(1/3)) + ((41 - (17*I)*Sqrt[3])*(1 + x))/(2618*x^3*(x^2 + x^3)^(1/3)) + ((41 + (17*I)*Sqrt[3])*(1 +
x))/(2618*x^3*(x^2 + x^3)^(1/3)) + (3645*(1 + x))/(2618*x^2*(x^2 + x^3)^(1/3)) + ((2249 - (153*I)*Sqrt[3])*(1
+ x))/(20944*x^2*(x^2 + x^3)^(1/3)) + ((2249 + (153*I)*Sqrt[3])*(1 + x))/(20944*x^2*(x^2 + x^3)^(1/3)) - (218
7*(1 + x))/(1309*x*(x^2 + x^3)^(1/3)) - ((1511 - (4777*I)*Sqrt[3])*(1 + x))/(52360*x*(x^2 + x^3)^(1/3)) - ((15
11 + (4777*I)*Sqrt[3])*(1 + x))/(52360*x*(x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*ArcTan[(1 + (2*(1 + (-1)^
(1/3))^(1/3)*x^(1/3))/(1 + x)^(1/3))/Sqrt[3]])/(Sqrt[3]*(1 + (-1)^(1/3))^(1/3)*(x^2 + x^3)^(1/3)) + (x^(2/3)*(
1 + x)^(1/3)*ArcTan[(1 + (2*(1 - (-1)^(2/3))^(1/3)*x^(1/3))/(1 + x)^(1/3))/Sqrt[3]])/(Sqrt[3]*(1 - (-1)^(2/3))
^(1/3)*(x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*Log[(-1)^(1/3) - x])/(6*(1 - (-1)^(2/3))^(1/3)*(x^2 + x^3)^
(1/3)) + (x^(2/3)*(1 + x)^(1/3)*Log[-(-1)^(2/3) - x])/(6*(1 + (-1)^(1/3))^(1/3)*(x^2 + x^3)^(1/3)) - (x^(2/3)*
(1 + x)^(1/3)*Log[(1 + (-1)^(1/3))^(1/3)*x^(1/3) - (1 + x)^(1/3)])/(2*(1 + (-1)^(1/3))^(1/3)*(x^2 + x^3)^(1/3)
) - (x^(2/3)*(1 + x)^(1/3)*Log[(1 - (-1)^(2/3))^(1/3)*x^(1/3) - (1 + x)^(1/3)])/(2*(1 - (-1)^(2/3))^(1/3)*(x^2
+ x^3)^(1/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 491

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x} \left (1+x^3\right )} \, dx}{\sqrt [3]{x^2+x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \left (-\frac {1}{3 (-1-x) x^{20/3} \sqrt [3]{1+x}}-\frac {1}{3 x^{20/3} \sqrt [3]{1+x} \left (-1+\sqrt [3]{-1} x\right )}-\frac {1}{3 x^{20/3} \sqrt [3]{1+x} \left (-1-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [3]{x^2+x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{(-1-x) x^{20/3} \sqrt [3]{1+x}} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{x^2+x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} (1+x)^{4/3}} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{x^{18} \sqrt [3]{1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{x^{18} \sqrt [3]{1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}} \\ & = \frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {2 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (13-17 i \sqrt {3}\right )-15 \sqrt [3]{-1} x^3}{x^{15} \sqrt [3]{1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{17 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (13+17 i \sqrt {3}\right )+15 (-1)^{2/3} x^3}{x^{15} \sqrt [3]{1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{17 \sqrt [3]{x^2+x^3}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x}} \, dx}{\sqrt [3]{x^2+x^3}} \\ & = \frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {\left (13-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {41-17 i \sqrt {3}+12 \left (16-i \sqrt {3}\right ) x^3}{x^{12} \sqrt [3]{1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{238 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {41+17 i \sqrt {3}+12 \left (16+i \sqrt {3}\right ) x^3}{x^{12} \sqrt [3]{1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{238 \sqrt [3]{x^2+x^3}}-\frac {\left (90 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{17/3} \sqrt [3]{1+x}} \, dx}{17 \sqrt [3]{x^2+x^3}} \\ & = \frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (1+x)}{119 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {2249-153 i \sqrt {3}+18 \left (23-6 i \sqrt {3}\right ) x^3}{x^9 \sqrt [3]{1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{2618 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {2249+153 i \sqrt {3}+18 \left (23+6 i \sqrt {3}\right ) x^3}{x^9 \sqrt [3]{1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{2618 \sqrt [3]{x^2+x^3}}+\frac {\left (540 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{14/3} \sqrt [3]{1+x}} \, dx}{119 \sqrt [3]{x^2+x^3}} \\ & = \frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (1+x)}{119 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-2 \left (1511+4777 i \sqrt {3}\right )+6 \left (895-1201 i \sqrt {3}\right ) x^3}{x^6 \sqrt [3]{1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{20944 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-2 \left (1511-4777 i \sqrt {3}\right )+6 \left (895+1201 i \sqrt {3}\right ) x^3}{x^6 \sqrt [3]{1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{20944 \sqrt [3]{x^2+x^3}}-\frac {\left (4860 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{1+x}} \, dx}{1309 \sqrt [3]{x^2+x^3}} \\ & = \frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (1+x)}{119 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {3645 (1+x)}{2618 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-2 \left (21647-11849 i \sqrt {3}\right )-6 \left (7921-1633 i \sqrt {3}\right ) x^3}{x^3 \sqrt [3]{1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{104720 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-2 \left (21647+11849 i \sqrt {3}\right )-6 \left (7921+1633 i \sqrt {3}\right ) x^3}{x^3 \sqrt [3]{1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{104720 \sqrt [3]{x^2+x^3}}+\frac {\left (3645 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{8/3} \sqrt [3]{1+x}} \, dx}{1309 \sqrt [3]{x^2+x^3}} \\ & = \frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (1+x)}{119 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {3645 (1+x)}{2618 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {2187 (1+x)}{1309 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int -\frac {209440}{\sqrt [3]{1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{209440 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int -\frac {209440}{\sqrt [3]{1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{209440 \sqrt [3]{x^2+x^3}}-\frac {\left (2187 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{5/3} \sqrt [3]{1+x}} \, dx}{1309 \sqrt [3]{x^2+x^3}} \\ & = \frac {1}{x^5 \sqrt [3]{x^2+x^3}}+\frac {6561 (1+x)}{2618 \sqrt [3]{x^2+x^3}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (1+x)}{119 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {3645 (1+x)}{2618 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {2187 (1+x)}{1309 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}} \\ & = \frac {1}{x^5 \sqrt [3]{x^2+x^3}}+\frac {6561 (1+x)}{2618 \sqrt [3]{x^2+x^3}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (1+x)}{119 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (13+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {3645 (1+x)}{2618 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {2187 (1+x)}{1309 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{-1}-x\right )}{6 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (-(-1)^{2/3}-x\right )}{6 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {9 \left (-9240+660 x-900 x^2+20985 x^3-6357 x^4+19071 x^5+109573 x^6\right )-52360 x^{17/3} \sqrt [3]{1+x} \text {RootSum}\left [1-3 \text {$\#$1}^3+3 \text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{\frac {x}{1+x}}-\text {$\#$1}\right )+3 \log \left (\sqrt [3]{\frac {x}{1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]}{471240 x^5 \sqrt [3]{x^2 (1+x)}} \]

[In]

Integrate[1/(x^6*(1 + x^3)*(x^2 + x^3)^(1/3)),x]

[Out]

(9*(-9240 + 660*x - 900*x^2 + 20985*x^3 - 6357*x^4 + 19071*x^5 + 109573*x^6) - 52360*x^(17/3)*(1 + x)^(1/3)*Ro
otSum[1 - 3*#1^3 + 3*#1^6 & , (-2*Log[(x/(1 + x))^(1/3) - #1] + 3*Log[(x/(1 + x))^(1/3) - #1]*#1^3)/(-#1^2 + 2
*#1^5) & ])/(471240*x^5*(x^2*(1 + x))^(1/3))

Maple [N/A] (verified)

Time = 5.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(\frac {-52360 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-3 \textit {\_Z}^{3}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{5} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+328719 x^{6}+57213 x^{5}-19071 x^{4}+62955 x^{3}-2700 x^{2}+1980 x -27720}{157080 x^{5} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}\)	\(96\)
risch	\(\text {Expression too large to display}\)	\(2060\)
trager	\(\text {Expression too large to display}\)	\(2554\)

[In]

int(1/x^6/(x^3+1)/(x^3+x^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/157080*(-52360*sum(ln((-_R*x+(x^2*(1+x))^(1/3))/x)/_R,_R=RootOf(_Z^6-3*_Z^3+3))*x^5*(x^2*(1+x))^(1/3)+328719
*x^6+57213*x^5-19071*x^4+62955*x^3-2700*x^2+1980*x-27720)/x^5/(x^2*(1+x))^(1/3)

Fricas [C] (verification not implemented)

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

Time = 0.28 (sec) , antiderivative size = 521, normalized size of antiderivative = 5.16 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=-\frac {13090 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7} - \sqrt {-3} {\left (x^{8} + x^{7}\right )}\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} x + i \, x\right )} + 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 13090 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7} + \sqrt {-3} {\left (x^{8} + x^{7}\right )}\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} x + i \, x\right )} - 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 26180 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7}\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (-i \, \sqrt {3} x - 3 \, x\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 12 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 13090 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7} + \sqrt {-3} {\left (x^{8} + x^{7}\right )}\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} x - i \, x\right )} - 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 13090 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7} - \sqrt {-3} {\left (x^{8} + x^{7}\right )}\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} x - i \, x\right )} + 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 26180 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7}\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (i \, \sqrt {3} x - 3 \, x\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 12 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 9 \, {\left (109573 \, x^{6} + 19071 \, x^{5} - 6357 \, x^{4} + 20985 \, x^{3} - 900 \, x^{2} + 660 \, x - 9240\right )} {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{471240 \, {\left (x^{8} + x^{7}\right )}} \]

[In]

integrate(1/x^6/(x^3+1)/(x^3+x^2)^(1/3),x, algorithm="fricas")

[Out]

-1/471240*(13090*6^(2/3)*(x^8 + x^7 - sqrt(-3)*(x^8 + x^7))*(I*sqrt(3) - 3)^(1/3)*log((6^(1/3)*(sqrt(3)*(I*sqr
t(-3)*x + I*x) + 3*sqrt(-3)*x + 3*x)*(I*sqrt(3) - 3)^(2/3) + 24*(x^3 + x^2)^(1/3))/x) + 13090*6^(2/3)*(x^8 + x
^7 + sqrt(-3)*(x^8 + x^7))*(I*sqrt(3) - 3)^(1/3)*log((6^(1/3)*(sqrt(3)*(-I*sqrt(-3)*x + I*x) - 3*sqrt(-3)*x +
3*x)*(I*sqrt(3) - 3)^(2/3) + 24*(x^3 + x^2)^(1/3))/x) - 26180*6^(2/3)*(x^8 + x^7)*(I*sqrt(3) - 3)^(1/3)*log((6
^(1/3)*(-I*sqrt(3)*x - 3*x)*(I*sqrt(3) - 3)^(2/3) + 12*(x^3 + x^2)^(1/3))/x) + 13090*6^(2/3)*(x^8 + x^7 + sqrt
(-3)*(x^8 + x^7))*(-I*sqrt(3) - 3)^(1/3)*log((6^(1/3)*(sqrt(3)*(I*sqrt(-3)*x - I*x) - 3*sqrt(-3)*x + 3*x)*(-I*
sqrt(3) - 3)^(2/3) + 24*(x^3 + x^2)^(1/3))/x) + 13090*6^(2/3)*(x^8 + x^7 - sqrt(-3)*(x^8 + x^7))*(-I*sqrt(3) -
3)^(1/3)*log((6^(1/3)*(sqrt(3)*(-I*sqrt(-3)*x - I*x) + 3*sqrt(-3)*x + 3*x)*(-I*sqrt(3) - 3)^(2/3) + 24*(x^3 +
x^2)^(1/3))/x) - 26180*6^(2/3)*(x^8 + x^7)*(-I*sqrt(3) - 3)^(1/3)*log((6^(1/3)*(I*sqrt(3)*x - 3*x)*(-I*sqrt(3
) - 3)^(2/3) + 12*(x^3 + x^2)^(1/3))/x) - 9*(109573*x^6 + 19071*x^5 - 6357*x^4 + 20985*x^3 - 900*x^2 + 660*x -
9240)*(x^3 + x^2)^(2/3))/(x^8 + x^7)

Sympy [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {1}{x^{6} \sqrt [3]{x^{2} \left (x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

[In]

integrate(1/x**6/(x**3+1)/(x**3+x**2)**(1/3),x)

[Out]

Integral(1/(x**6*(x**2*(x + 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )} x^{6}} \,d x } \]

[In]

integrate(1/x^6/(x^3+1)/(x^3+x^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((x^3 + x^2)^(1/3)*(x^3 + 1)*x^6), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x^6/(x^3+1)/(x^3+x^2)^(1/3),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Invalid _EXT in replace_ext Error: Bad Argument ValueDone

Mupad [N/A]

Not integrable

Time = 6.72 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {1}{x^6\,{\left (x^3+x^2\right )}^{1/3}\,\left (x^3+1\right )} \,d x \]

[In]

int(1/(x^6*(x^2 + x^3)^(1/3)*(x^3 + 1)),x)

[Out]

int(1/(x^6*(x^2 + x^3)^(1/3)*(x^3 + 1)), x)