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

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

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [C] Result contains complex when optimal does not.
time = 1.03, antiderivative size = 841, normalized size of antiderivative = 8.33, number of steps used = 29, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2081, 6857, 21, 47, 37, 129, 491, 597, 12, 384} \begin {gather*} -\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} \text {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} \text {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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx &=\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 {2 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{3} \left (-15+17 \sqrt [3]{-1}\right )+5 \sqrt [3]{-1} x}{x^{17/3} \sqrt [3]{1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{17 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{3} \left (-15-17 (-1)^{2/3}\right )-5 (-1)^{2/3} x}{x^{17/3} \sqrt [3]{1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{17 \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 {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {2 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {\left (15-17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (15+17 (-1)^{2/3}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{9} \left (-41+17 i \sqrt {3}\right )-\frac {4}{3} \left (16-i \sqrt {3}\right ) x}{x^{14/3} \sqrt [3]{1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{238 \sqrt [3]{x^2+x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{9} \left (-41-17 i \sqrt {3}\right )-\frac {4}{3} \left (16+i \sqrt {3}\right ) x}{x^{14/3} \sqrt [3]{1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{238 \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 (15-17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (15+17 (-1)^{2/3}\right ) (1+x)}{238 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 (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{27} \left (-2249+153 i \sqrt {3}\right )-\frac {2}{3} \left (23-6 i \sqrt {3}\right ) x}{x^{11/3} \sqrt [3]{1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{2618 \sqrt [3]{x^2+x^3}}-\frac {\left (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{27} \left (-2249-153 i \sqrt {3}\right )-\frac {2}{3} \left (23+6 i \sqrt {3}\right ) x}{x^{11/3} \sqrt [3]{1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{2618 \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 (15-17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (15+17 (-1)^{2/3}\right ) (1+x)}{238 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 (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 (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{81} \left (1511+4777 i \sqrt {3}\right )-\frac {2}{27} \left (895-1201 i \sqrt {3}\right ) x}{x^{8/3} \sqrt [3]{1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{20944 \sqrt [3]{x^2+x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{81} \left (1511-4777 i \sqrt {3}\right )-\frac {2}{27} \left (895+1201 i \sqrt {3}\right ) x}{x^{8/3} \sqrt [3]{1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{20944 \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 (15-17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (15+17 (-1)^{2/3}\right ) (1+x)}{238 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 (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 (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{243} \left (21647-11849 i \sqrt {3}\right )+\frac {2}{81} \left (7921-1633 i \sqrt {3}\right ) x}{x^{5/3} \sqrt [3]{1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{243} \left (21647+11849 i \sqrt {3}\right )+\frac {2}{81} \left (7921+1633 i \sqrt {3}\right ) x}{x^{5/3} \sqrt [3]{1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{104720 \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 {\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 (15-17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (15+17 (-1)^{2/3}\right ) (1+x)}{238 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 (243 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {209440}{729 x^{2/3} \sqrt [3]{1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{209440 \sqrt [3]{x^2+x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {209440}{729 x^{2/3} \sqrt [3]{1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{209440 \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 (15-17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (15+17 (-1)^{2/3}\right ) (1+x)}{238 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 ) \int \frac {1}{x^{2/3} \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^{2/3} \sqrt [3]{1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \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 (15-17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (15+17 (-1)^{2/3}\right ) (1+x)}{238 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} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}}\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 (-1+\sqrt [3]{-1} 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 (-1-(-1)^{2/3} 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 (-\sqrt [3]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1+\sqrt [3]{-1}}}\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]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-(-1)^{2/3}}}\right )}{2 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^2+x^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.11, size = 141, normalized size = 1.40 \begin {gather*} \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)}} \end {gather*}

Antiderivative was successfully verified.

[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 [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 8.16, size = 2060, normalized size = 20.40

method	result	size
risch	\(\text {Expression too large to display}\)	\(2060\)
trager	\(\text {Expression too large to display}\)	\(2119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/52360*(109573*x^6+19071*x^5-6357*x^4+20985*x^3-900*x^2+660*x-9240)/x^5/(x^2*(1+x))^(1/3)+1/3*RootOf(RootOf(3
*_Z^6+3*_Z^3+1)^3+_Z^3+1)*ln(-(-6*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x^2+6*Roo
tOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x-39*(x^3+x^2)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z
^3+1)^3+_Z^3+1)^2*RootOf(3*_Z^6+3*_Z^3+1)^3*x-10*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3
+1)^3*x^2+24*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3+17*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z
^6+3*_Z^3+1)^3*x-15*(x^3+x^2)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*x+4*RootOf(RootOf(3*_Z^6+3*_Z^3
+1)^3+_Z^3+1)*x^2+13*(x^3+x^2)^(2/3)+10*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x)/(3*x*RootOf(3*_Z^6+3*_Z^3+
1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^3+2*x-1)/x)-RootOf(3*_Z^6+3*_Z^3+1)^3*ln((21*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_
Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x^2-21*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x+3
*(x^3+x^2)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*RootOf(3*_Z^6+3*_Z^3+1)^3*x+89*RootOf(RootOf(3*_Z^
6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x^2+21*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3+2*RootOf(Root
Of(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x+24*(x^3+x^2)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+
_Z^3+1)^2*x+44*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x^2-(x^3+x^2)^(2/3)+8*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3
+_Z^3+1)*x)/(3*x*RootOf(3*_Z^6+3*_Z^3+1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^3+2*x-1)/x)*RootOf(RootOf(3*_Z^6+3*_Z^3+1
)^3+_Z^3+1)-2/3*ln((21*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x^2-21*RootOf(RootOf
(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^6*x+3*(x^3+x^2)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^
3+1)^2*RootOf(3*_Z^6+3*_Z^3+1)^3*x+89*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)^3*x^2+2
1*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3+2*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*RootOf(3*_Z^6+3*_Z^3+1)
^3*x+24*(x^3+x^2)^(1/3)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)^2*x+44*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+
1)*x^2-(x^3+x^2)^(2/3)+8*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)*x)/(3*x*RootOf(3*_Z^6+3*_Z^3+1)^3-3*RootOf(3
*_Z^6+3*_Z^3+1)^3+2*x-1)/x)*RootOf(RootOf(3*_Z^6+3*_Z^3+1)^3+_Z^3+1)-RootOf(3*_Z^6+3*_Z^3+1)^4*ln((24*RootOf(3
*_Z^6+3*_Z^3+1)^7*x^2-24*RootOf(3*_Z^6+3*_Z^3+1)^7*x+6*(x^3+x^2)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^5*x+50*RootOf(3
*_Z^6+3*_Z^3+1)^4*x^2+24*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3-22*RootOf(3*_Z^6+3*_Z^3+1)^4*x+15*(x^3+x^2)
^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^2*x+24*RootOf(3*_Z^6+3*_Z^3+1)*x^2+11*(x^3+x^2)^(2/3)-3*RootOf(3*_Z^6+3*_Z^3+1)
*x)/(3*x*RootOf(3*_Z^6+3*_Z^3+1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^3+x-2)/x)-2/3*RootOf(3*_Z^6+3*_Z^3+1)*ln((24*Root
Of(3*_Z^6+3*_Z^3+1)^7*x^2-24*RootOf(3*_Z^6+3*_Z^3+1)^7*x+6*(x^3+x^2)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^5*x+50*Root
Of(3*_Z^6+3*_Z^3+1)^4*x^2+24*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3-22*RootOf(3*_Z^6+3*_Z^3+1)^4*x+15*(x^3+
x^2)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^2*x+24*RootOf(3*_Z^6+3*_Z^3+1)*x^2+11*(x^3+x^2)^(2/3)-3*RootOf(3*_Z^6+3*_Z^
3+1)*x)/(3*x*RootOf(3*_Z^6+3*_Z^3+1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^3+x-2)/x)+1/3*RootOf(3*_Z^6+3*_Z^3+1)*ln(-(6*
RootOf(3*_Z^6+3*_Z^3+1)^7*x^2-6*RootOf(3*_Z^6+3*_Z^3+1)^7*x-69*(x^3+x^2)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^5*x-19*
RootOf(3*_Z^6+3*_Z^3+1)^4*x^2+21*(x^3+x^2)^(2/3)*RootOf(3*_Z^6+3*_Z^3+1)^3-7*RootOf(3*_Z^6+3*_Z^3+1)^4*x-24*(x
^3+x^2)^(1/3)*RootOf(3*_Z^6+3*_Z^3+1)^2*x+8*RootOf(3*_Z^6+3*_Z^3+1)*x^2+22*(x^3+x^2)^(2/3)+5*RootOf(3*_Z^6+3*_
Z^3+1)*x)/(3*x*RootOf(3*_Z^6+3*_Z^3+1)^3-3*RootOf(3*_Z^6+3*_Z^3+1)^3+x-2)/x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.48, size = 1412, normalized size = 13.98 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/471240*(26180*12^(1/6)*6^(2/3)*(x^8 + x^7)*cos(2/3*arctan(sqrt(3) + 2))*log(12*(4*12^(1/3)*6^(1/3)*sqrt(3)*(
x^3 + x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) - 12*12^(1/3)*6^(1/3)*(x^3 + x^2)
^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 + 12^(2/3)*6^(2/3)*x^2 + 6*12^(1/3)*6^(1/3)*(x^3 + x^2)^(1/3)*x + 12*(
x^3 + x^2)^(2/3))/x^2) - 104720*12^(1/6)*6^(2/3)*(x^8 + x^7)*arctan(1/108*(12*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 +
x^2)^(1/3)*cos(2/3*arctan(sqrt(3) + 2))^2 - 6*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 + x^2)^(1/3) - sqrt(3)*(2*12^(2/3)
*6^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 - 6*12^(2/3)*6^(2/3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*
arctan(sqrt(3) + 2)) - 12^(2/3)*6^(2/3)*sqrt(3)*x)*sqrt((4*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 + x^2)^(1/3)*x*cos(2/
3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) - 12*12^(1/3)*6^(1/3)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(s
qrt(3) + 2))^2 + 12^(2/3)*6^(2/3)*x^2 + 6*12^(1/3)*6^(1/3)*(x^3 + x^2)^(1/3)*x + 12*(x^3 + x^2)^(2/3))/x^2) +
36*(48*x*cos(2/3*arctan(sqrt(3) + 2))^3 - (12^(2/3)*6^(2/3)*(x^3 + x^2)^(1/3) + 24*x)*cos(2/3*arctan(sqrt(3) +
2)))*sin(2/3*arctan(sqrt(3) + 2)) - 108*sqrt(3)*x)/(16*x*cos(2/3*arctan(sqrt(3) + 2))^4 - 16*x*cos(2/3*arctan
(sqrt(3) + 2))^2 + x))*sin(2/3*arctan(sqrt(3) + 2)) - 52360*(12^(1/6)*6^(2/3)*sqrt(3)*(x^8 + x^7)*cos(2/3*arct
an(sqrt(3) + 2)) + 12^(1/6)*6^(2/3)*(x^8 + x^7)*sin(2/3*arctan(sqrt(3) + 2)))*arctan(-1/108*(12*12^(2/3)*6^(2/
3)*sqrt(3)*(x^3 + x^2)^(1/3)*cos(2/3*arctan(sqrt(3) + 2))^2 - 6*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 + x^2)^(1/3) - s
qrt(3)*(2*12^(2/3)*6^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 + 6*12^(2/3)*6^(2/3)*x*cos(2/3*arctan(sqrt
(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) - 12^(2/3)*6^(2/3)*sqrt(3)*x)*sqrt((4*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 + x
^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) + 12*12^(1/3)*6^(1/3)*(x^3 + x^2)^(1/3)*
x*cos(2/3*arctan(sqrt(3) + 2))^2 + 12^(2/3)*6^(2/3)*x^2 - 6*12^(1/3)*6^(1/3)*(x^3 + x^2)^(1/3)*x + 12*(x^3 + x
^2)^(2/3))/x^2) + 36*(48*x*cos(2/3*arctan(sqrt(3) + 2))^3 + (12^(2/3)*6^(2/3)*(x^3 + x^2)^(1/3) - 24*x)*cos(2/
3*arctan(sqrt(3) + 2)))*sin(2/3*arctan(sqrt(3) + 2)) + 108*sqrt(3)*x)/(16*x*cos(2/3*arctan(sqrt(3) + 2))^4 - 1
6*x*cos(2/3*arctan(sqrt(3) + 2))^2 + x)) - 52360*(12^(1/6)*6^(2/3)*sqrt(3)*(x^8 + x^7)*cos(2/3*arctan(sqrt(3)
+ 2)) - 12^(1/6)*6^(2/3)*(x^8 + x^7)*sin(2/3*arctan(sqrt(3) + 2)))*arctan(1/72*(144*x*cos(2/3*arctan(sqrt(3) +
2))*sin(2/3*arctan(sqrt(3) + 2)) + 12^(2/3)*6^(2/3)*x*sqrt(-(8*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 + x^2)^(1/3)*x*c
os(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) - 12^(2/3)*6^(2/3)*x^2 - 12*(x^3 + x^2)^(2/3))/x^2) -
2*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 + x^2)^(1/3))/(2*x*cos(2/3*arctan(sqrt(3) + 2))^2 - x)) - 13090*(12^(1/6)*6^(
2/3)*sqrt(3)*(x^8 + x^7)*sin(2/3*arctan(sqrt(3) + 2)) + 12^(1/6)*6^(2/3)*(x^8 + x^7)*cos(2/3*arctan(sqrt(3) +
2)))*log(-48*(8*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(
3) + 2)) - 12^(2/3)*6^(2/3)*x^2 - 12*(x^3 + x^2)^(2/3))/x^2) + 13090*(12^(1/6)*6^(2/3)*sqrt(3)*(x^8 + x^7)*sin
(2/3*arctan(sqrt(3) + 2)) - 12^(1/6)*6^(2/3)*(x^8 + x^7)*cos(2/3*arctan(sqrt(3) + 2)))*log(48*(4*12^(1/3)*6^(1
/3)*sqrt(3)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) + 12*12^(1/3)*6^(1/3
)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 + 12^(2/3)*6^(2/3)*x^2 - 6*12^(1/3)*6^(1/3)*(x^3 + x^2)^(
1/3)*x + 12*(x^3 + x^2)^(2/3))/x^2) + 9*(109573*x^6 + 19071*x^5 - 6357*x^4 + 20985*x^3 - 900*x^2 + 660*x - 924
0)*(x^3 + x^2)^(2/3))/(x^8 + x^7)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{6} \sqrt [3]{x^{2} \left (x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________