\(\int \frac {1}{x^6 (-1+x^3) \sqrt [3]{x^2+x^3}} \, dx\) [2543]

Optimal result

Integrand size = 22, antiderivative size = 214 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {3 \left (x^2+x^3\right )^{2/3} \left (3080-3300 x+3600 x^2+2495 x^3-2994 x^4+4491 x^5\right )}{52360 x^7}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^3}+\sqrt [3]{2} \left (x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.15 (sec) , antiderivative size = 979, normalized size of antiderivative = 4.57, number of steps used = 30, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2081, 6857, 129, 491, 597, 12, 384} \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=-\frac {\left (8689+731 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}-\frac {\left (8689-731 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}+\frac {2099 (x+1)}{13090 x \sqrt [3]{x^3+x^2}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {173 (x+1)}{5236 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (163+221 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}+\frac {\left (163-221 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}+\frac {107 (x+1)}{1309 x^3 \sqrt [3]{x^3+x^2}}-\frac {\left (47+17 i \sqrt {3}\right ) (x+1)}{476 x^4 \sqrt [3]{x^3+x^2}}-\frac {\left (47-17 i \sqrt {3}\right ) (x+1)}{476 x^4 \sqrt [3]{x^3+x^2}}+\frac {x+1}{119 x^4 \sqrt [3]{x^3+x^2}}+\frac {3 (x+1)}{17 x^5 \sqrt [3]{x^3+x^2}}-\frac {\left (113+23987 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}-\frac {\left (113-23987 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}+\frac {6793 (x+1)}{26180 \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x+1}}{\sqrt [3]{2} \sqrt {3} \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 (1-x) \sqrt [3]{x+1}}{6 \sqrt [3]{2} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \log \left (x+\sqrt [3]{-1}\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]{2} \sqrt [3]{x}-\sqrt [3]{x+1}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{2} \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}} \]

[In]

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

[Out]

(6793*(1 + x))/(26180*(x^2 + x^3)^(1/3)) - ((113 - (23987*I)*Sqrt[3])*(1 + x))/(104720*(x^2 + x^3)^(1/3)) - ((
113 + (23987*I)*Sqrt[3])*(1 + x))/(104720*(x^2 + x^3)^(1/3)) + (3*(1 + x))/(17*x^5*(x^2 + x^3)^(1/3)) + (1 + x
)/(119*x^4*(x^2 + x^3)^(1/3)) - ((47 - (17*I)*Sqrt[3])*(1 + x))/(476*x^4*(x^2 + x^3)^(1/3)) - ((47 + (17*I)*Sq
rt[3])*(1 + x))/(476*x^4*(x^2 + x^3)^(1/3)) + (107*(1 + x))/(1309*x^3*(x^2 + x^3)^(1/3)) + ((163 - (221*I)*Sqr
t[3])*(1 + x))/(2618*x^3*(x^2 + x^3)^(1/3)) + ((163 + (221*I)*Sqrt[3])*(1 + x))/(2618*x^3*(x^2 + x^3)^(1/3)) +
(173*(1 + x))/(5236*x^2*(x^2 + x^3)^(1/3)) + ((1151 - (1989*I)*Sqrt[3])*(1 + x))/(20944*x^2*(x^2 + x^3)^(1/3)
) + ((1151 + (1989*I)*Sqrt[3])*(1 + x))/(20944*x^2*(x^2 + x^3)^(1/3)) + (2099*(1 + x))/(13090*x*(x^2 + x^3)^(1
/3)) - ((8689 - (731*I)*Sqrt[3])*(1 + x))/(52360*x*(x^2 + x^3)^(1/3)) - ((8689 + (731*I)*Sqrt[3])*(1 + x))/(52
360*x*(x^2 + x^3)^(1/3)) - (x^(2/3)*(1 + x)^(1/3)*ArcTan[(1 + (2*2^(1/3)*x^(1/3))/(1 + x)^(1/3))/Sqrt[3]])/(2^
(1/3)*Sqrt[3]*(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 - x])/(6*2^(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[2^(1/3)*x^(1/3) - (1 + x)^(1/3)])/(2*2^(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 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 ) \text {Subst}\left (\int \frac {1}{x^{18} \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, 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+\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 {3 (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 {2+15 x^3}{x^{15} \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, 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 (-47-17 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 (-47+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 {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+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 {-214-24 x^3}{x^{12} \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, 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 {-163-221 i \sqrt {3}+12 \left (1-16 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 {-163+221 i \sqrt {3}+12 \left (1+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 {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 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 {692+1926 x^3}{x^9 \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, 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 {1151-1989 i \sqrt {3}+18 \left (125-96 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 (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1151+1989 i \sqrt {3}+18 \left (125+96 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 {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 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 {-16792-4152 x^3}{x^6 \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, 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 (8689-731 i \sqrt {3}\right )+6 \left (3559-419 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 (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {2 \left (8689+731 i \sqrt {3}\right )+6 \left (3559+419 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 {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 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 {54344+50376 x^3}{x^3 \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, 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 (113+23987 i \sqrt {3}\right )+6 \left (5441-3979 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 (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-2 \left (113-23987 i \sqrt {3}\right )+6 \left (5441+3979 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 {6793 (1+x)}{26180 \sqrt [3]{x^2+x^3}}-\frac {\left (113-23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (113+23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}+\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 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}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, 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+\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 {6793 (1+x)}{26180 \sqrt [3]{x^2+x^3}}-\frac {\left (113-23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (113+23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}+\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 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}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, 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+\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 {6793 (1+x)}{26180 \sqrt [3]{x^2+x^3}}-\frac {\left (113-23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (113+23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}+\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 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]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} \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 (1-x)}{6 \sqrt [3]{2} \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]{2} \sqrt [3]{x}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{2} \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 = 0.28 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {27720-1980 x+2700 x^2+54855 x^3-4491 x^4+13473 x^5+40419 x^6-26180\ 2^{2/3} \sqrt {3} x^{17/3} \sqrt [3]{1+x} \arctan \left (\frac {1+2 \sqrt [3]{2} \sqrt [3]{\frac {x}{1+x}}}{\sqrt {3}}\right )+26180\ 2^{2/3} x^{17/3} \sqrt [3]{1+x} \log \left (-1+\sqrt [3]{2} \sqrt [3]{\frac {x}{1+x}}\right )-13090\ 2^{2/3} x^{17/3} \sqrt [3]{1+x} \log \left (1+\sqrt [3]{2} \sqrt [3]{\frac {x}{1+x}}+2^{2/3} \left (\frac {x}{1+x}\right )^{2/3}\right )+52360 x^{17/3} \sqrt [3]{1+x} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{\frac {x}{1+x}}-\text {$\#$1}\right )+\log \left (\sqrt [3]{\frac {x}{1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]}{157080 x^5 \sqrt [3]{x^2 (1+x)}} \]

[In]

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

[Out]

(27720 - 1980*x + 2700*x^2 + 54855*x^3 - 4491*x^4 + 13473*x^5 + 40419*x^6 - 26180*2^(2/3)*Sqrt[3]*x^(17/3)*(1
+ x)^(1/3)*ArcTan[(1 + 2*2^(1/3)*(x/(1 + x))^(1/3))/Sqrt[3]] + 26180*2^(2/3)*x^(17/3)*(1 + x)^(1/3)*Log[-1 + 2
^(1/3)*(x/(1 + x))^(1/3)] - 13090*2^(2/3)*x^(17/3)*(1 + x)^(1/3)*Log[1 + 2^(1/3)*(x/(1 + x))^(1/3) + 2^(2/3)*(
x/(1 + x))^(2/3)] + 52360*x^(17/3)*(1 + x)^(1/3)*RootSum[1 - #1^3 + #1^6 & , (-2*Log[(x/(1 + x))^(1/3) - #1] +
Log[(x/(1 + x))^(1/3) - #1]*#1^3)/(-#1^2 + 2*#1^5) & ])/(157080*x^5*(x^2*(1 + x))^(1/3))

Maple [N/A] (verified)

Time = 28.86 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {-13090 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{7}+52360 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{7}+26180 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x^{7}+26180 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right ) x^{7}+40419 \left (1+x \right ) \left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}} \left (x^{4}-\frac {5}{3} x^{3}+\frac {20}{9} x^{2}-\frac {6380}{4491} x +\frac {3080}{4491}\right )}{157080 x^{7}}\)	\(189\)
risch	\(\text {Expression too large to display}\)	\(3671\)
trager	\(\text {Expression too large to display}\)	\(4304\)

[In]

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

[Out]

1/157080*(-13090*2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*(x^2*(1+x))^(1/3)*x+(x^2*(1+x))^(2/3))/x^2)*x^7+52360*sum(ln(
(-_R*x+(x^2*(1+x))^(1/3))/x)/_R,_R=RootOf(_Z^6-_Z^3+1))*x^7+26180*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*
(x^2*(1+x))^(1/3)+x)/x)*x^7+26180*2^(2/3)*ln((-2^(1/3)*x+(x^2*(1+x))^(1/3))/x)*x^7+40419*(1+x)*(x^2*(1+x))^(2/
3)*(x^4-5/3*x^3+20/9*x^2-6380/4491*x+3080/4491))/x^7

Fricas [C] (verification not implemented)

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

Time = 0.30 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.87 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {26180 \cdot 2^{\frac {2}{3}} x^{7} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 26180 \cdot 2^{\frac {2}{3}} x^{7} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (-i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 26180 \, \sqrt {6} 2^{\frac {1}{6}} x^{7} \arctan \left (\frac {2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} x + 2 \, \sqrt {6} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + 26180 \cdot 2^{\frac {2}{3}} x^{7} \log \left (-\frac {2^{\frac {1}{3}} x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 13090 \cdot 2^{\frac {2}{3}} x^{7} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 13090 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{7} + x^{7}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x - i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 13090 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{7} - x^{7}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x - i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 13090 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{7} - x^{7}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x + i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 13090 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{7} + x^{7}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x + i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 9 \, {\left (4491 \, x^{5} - 2994 \, x^{4} + 2495 \, x^{3} + 3600 \, x^{2} - 3300 \, x + 3080\right )} {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{157080 \, x^{7}} \]

[In]

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

[Out]

1/157080*(26180*2^(2/3)*x^7*(I*sqrt(3) + 1)^(1/3)*log(((I*sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*(I*sqrt(3) + 1)^(2/3)
+ 4*(x^3 + x^2)^(1/3))/x) + 26180*2^(2/3)*x^7*(-I*sqrt(3) + 1)^(1/3)*log(((-I*sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*
(-I*sqrt(3) + 1)^(2/3) + 4*(x^3 + x^2)^(1/3))/x) + 26180*sqrt(6)*2^(1/6)*x^7*arctan(1/6*2^(1/6)*(sqrt(6)*2^(1/
3)*x + 2*sqrt(6)*(x^3 + x^2)^(1/3))/x) + 26180*2^(2/3)*x^7*log(-(2^(1/3)*x - (x^3 + x^2)^(1/3))/x) - 13090*2^(
2/3)*x^7*log((2^(2/3)*x^2 + 2^(1/3)*(x^3 + x^2)^(1/3)*x + (x^3 + x^2)^(2/3))/x^2) - 13090*2^(2/3)*(sqrt(-3)*x^
7 + x^7)*(I*sqrt(3) + 1)^(1/3)*log(((sqrt(3)*2^(1/3)*(I*sqrt(-3)*x - I*x) - 2^(1/3)*(sqrt(-3)*x - x))*(I*sqrt(
3) + 1)^(2/3) + 8*(x^3 + x^2)^(1/3))/x) + 13090*2^(2/3)*(sqrt(-3)*x^7 - x^7)*(I*sqrt(3) + 1)^(1/3)*log(((sqrt(
3)*2^(1/3)*(-I*sqrt(-3)*x - I*x) + 2^(1/3)*(sqrt(-3)*x + x))*(I*sqrt(3) + 1)^(2/3) + 8*(x^3 + x^2)^(1/3))/x) +
13090*2^(2/3)*(sqrt(-3)*x^7 - x^7)*(-I*sqrt(3) + 1)^(1/3)*log(((sqrt(3)*2^(1/3)*(I*sqrt(-3)*x + I*x) + 2^(1/3
)*(sqrt(-3)*x + x))*(-I*sqrt(3) + 1)^(2/3) + 8*(x^3 + x^2)^(1/3))/x) - 13090*2^(2/3)*(sqrt(-3)*x^7 + x^7)*(-I*
sqrt(3) + 1)^(1/3)*log(((sqrt(3)*2^(1/3)*(-I*sqrt(-3)*x + I*x) - 2^(1/3)*(sqrt(-3)*x - x))*(-I*sqrt(3) + 1)^(2
/3) + 8*(x^3 + x^2)^(1/3))/x) + 9*(4491*x^5 - 2994*x^4 + 2495*x^3 + 3600*x^2 - 3300*x + 3080)*(x^3 + x^2)^(2/3
))/x^7

Sympy [N/A]

Not integrable

Time = 0.86 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {1}{x^{6} \sqrt [3]{x^{2} \left (x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]

[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 = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.10 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )} x^{6}} \,d x } \]

[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 [C] (verification not implemented)

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

Time = 41.10 (sec) , antiderivative size = 986, normalized size of antiderivative = 4.61 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

3/17*(1/x + 1)^(17/3) - 15/14*(1/x + 1)^(14/3) + 30/11*(1/x + 1)^(11/3) - 27/8*(1/x + 1)^(8/3) + 1/6*sqrt(3)*2
^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(1/x + 1)^(1/3))) - 1/3*(sqrt(3)*cos(4/9*pi)^5 - 10*sqrt(3)*cos
(4/9*pi)^3*sin(4/9*pi)^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 5*cos(4/9*pi)^4*sin(4/9*pi) + 10*cos(4/9*pi)^
2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3)*cos(4/9*pi)^2 - sqrt(3)*sin(4/9*pi)^2 - 2*cos(4/9*pi)*sin(4/9*pi))*a
rctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi))) - 1/3*(sqrt(
3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^4 - 5*cos(2/9*pi
)^4*sin(2/9*pi) + 10*cos(2/9*pi)^2*sin(2/9*pi)^3 - sin(2/9*pi)^5 + sqrt(3)*cos(2/9*pi)^2 - sqrt(3)*sin(2/9*pi)
^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(2/9*pi) + 2*(1/x + 1)^(1/3))/((1/2*I*sqrt(3)
+ 1/2)*sin(2/9*pi))) + 1/3*(sqrt(3)*cos(1/9*pi)^5 - 10*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi)^2 + 5*sqrt(3)*cos(1/9
*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*pi) - 10*cos(1/9*pi)^2*sin(1/9*pi)^3 + sin(1/9*pi)^5 - sqrt(3)*co
s(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi))*arctan(-1/2*((-I*sqrt(3) - 1)*cos(1/9*pi) - 2
*(1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) - 1/6*(5*sqrt(3)*cos(4/9*pi)^4*sin(4/9*pi) - 10*sqrt(3)
*cos(4/9*pi)^2*sin(4/9*pi)^3 + sqrt(3)*sin(4/9*pi)^5 + cos(4/9*pi)^5 - 10*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*cos(
4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) + cos(4/9*pi)^2 - sin(4/9*pi)^2)*log((-I*sqrt(3)*cos
(4/9*pi) - cos(4/9*pi))*(1/x + 1)^(1/3) + (1/x + 1)^(2/3) + 1) - 1/6*(5*sqrt(3)*cos(2/9*pi)^4*sin(2/9*pi) - 10
*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^3 + sqrt(3)*sin(2/9*pi)^5 + cos(2/9*pi)^5 - 10*cos(2/9*pi)^3*sin(2/9*pi)^2
+ 5*cos(2/9*pi)*sin(2/9*pi)^4 + 2*sqrt(3)*cos(2/9*pi)*sin(2/9*pi) + cos(2/9*pi)^2 - sin(2/9*pi)^2)*log((-I*sqr
t(3)*cos(2/9*pi) - cos(2/9*pi))*(1/x + 1)^(1/3) + (1/x + 1)^(2/3) + 1) - 1/6*(5*sqrt(3)*cos(1/9*pi)^4*sin(1/9*
pi) - 10*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^3 + sqrt(3)*sin(1/9*pi)^5 - cos(1/9*pi)^5 + 10*cos(1/9*pi)^3*sin(1/
9*pi)^2 - 5*cos(1/9*pi)*sin(1/9*pi)^4 - 2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) + cos(1/9*pi)^2 - sin(1/9*pi)^2)*log
((I*sqrt(3)*cos(1/9*pi) + cos(1/9*pi))*(1/x + 1)^(1/3) + (1/x + 1)^(2/3) + 1) + 9/5*(1/x + 1)^(5/3) - 1/12*2^(
2/3)*log(2^(2/3) + 2^(1/3)*(1/x + 1)^(1/3) + (1/x + 1)^(2/3)) + 1/6*2^(2/3)*log(abs(-2^(1/3) + (1/x + 1)^(1/3)
))

Mupad [N/A]

Not integrable

Time = 6.99 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=-\int \frac {1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^6-x^9\right )} \,d x \]

[In]

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

[Out]

-int(1/((x^2 + x^3)^(1/3)*(x^6 - x^9)), x)