3.26.44 \(\int \frac {1}{x^6 (-1+x^3) \sqrt [3]{x^2+x^3}} \, dx\) [2544]

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

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [C] Result contains complex when optimal does not.
time = 1.03, antiderivative size = 979, normalized size of antiderivative = 4.57, number of steps used = 30, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2081, 6857, 129, 491, 597, 12, 384} \begin {gather*} -\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} \text {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} \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 (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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {2}{3}-5 x}{(1-x) x^{17/3} \sqrt [3]{1+x}} \, 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 \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 {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 (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 {214}{9}+\frac {8 x}{3}}{(1-x) x^{14/3} \sqrt [3]{1+x}} \, 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 (163+221 i \sqrt {3}\right )-\frac {4}{3} \left (1-16 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 (163-221 i \sqrt {3}\right )-\frac {4}{3} \left (1+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 {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 (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 {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 (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {692}{27}-\frac {214 x}{3}}{(1-x) x^{11/3} \sqrt [3]{1+x}} \, 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 (-1151+1989 i \sqrt {3}\right )-\frac {2}{3} \left (125-96 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 (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{27} \left (-1151-1989 i \sqrt {3}\right )-\frac {2}{3} \left (125+96 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 {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 (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 {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 (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {16792}{81}+\frac {1384 x}{27}}{(1-x) x^{8/3} \sqrt [3]{1+x}} \, 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 (8689-731 i \sqrt {3}\right )-\frac {2}{27} \left (3559-419 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 (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {2}{81} \left (8689+731 i \sqrt {3}\right )-\frac {2}{27} \left (3559+419 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 {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 (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 {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 (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {54344}{243}-\frac {16792 x}{81}}{(1-x) x^{5/3} \sqrt [3]{1+x}} \, 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 (113+23987 i \sqrt {3}\right )-\frac {2}{81} \left (5441-3979 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 (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{243} \left (113-23987 i \sqrt {3}\right )-\frac {2}{81} \left (5441+3979 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 {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 (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 {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 (243 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {209440}{729 (1-x) x^{2/3} \sqrt [3]{1+x}} \, 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+\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 {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 (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 {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 ) \int \frac {1}{(1-x) x^{2/3} \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^{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 {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 (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 {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} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{2} \sqrt {3} \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 (1-x)}{6 \sqrt [3]{2} \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]{2}}\right )}{2 \sqrt [3]{2} \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 = 0.00, size = 281, normalized size = 1.31 \begin {gather*} \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} \text {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)}} \end {gather*}

Antiderivative was successfully verified.

[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 [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 43.59, size = 3953, normalized size = 18.47

method	result	size
risch	\(\text {Expression too large to display}\)	\(3953\)
trager	\(\text {Expression too large to display}\)	\(4935\)

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]

3/52360*(4491*x^6+1497*x^5-499*x^4+6095*x^3+300*x^2-220*x+3080)/x^5/(x^2*(1+x))^(1/3)+1/6*RootOf(_Z^3-4)*ln(-(
30*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x^2-9*RootOf(RootOf(_Z^3-4)^2+3*_Z*Roo
tOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x^2-60*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-
4)^3*x+360*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2+18*RootOf(Root
Of(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x-240*(x^3+x^2)^(1/3)*RootOf(_Z^3-4)^2*x-594*(x^3+
x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)*x+340*RootOf(_Z^3-4)*x^2-102*Roo
tOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x^2+60*RootOf(_Z^3-4)*x+84*(x^3+x^2)^(2/3)-18*RootOf(RootOf(_
Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x)/x/(-1+x))+1/2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*ln((
-3*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x^2+90*RootOf(RootOf(_Z^3-4)^2+3*_Z*Ro
otOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+360*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*
_Z^2)*RootOf(_Z^3-4)^2+6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x-180*RootOf(Roo
tOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x-240*(x^3+x^2)^(1/3)*RootOf(_Z^3-4)^2*x-126*(x^3
+x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)*x+30*RootOf(_Z^3-4)*x^2-900*Roo
tOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x^2+396*(x^3+x^2)^(2/3)+14*RootOf(_Z^3-4)*x-420*RootOf(RootOf
(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x)/x/(-1+x))+1/6*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3
-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*ln((-45*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*Root
Of(_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*(x^3+x^2)^(1/3)*x+24*Ro
otOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*RootOf(RootOf(_Z^3-4)^2+3*
_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x^2+108*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)
+9*_Z^2)*RootOf(_Z^3-4)^2+12*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^
2-8)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x-96*RootOf(_Z^3-6*RootOf(RootOf(_Z^
3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*(x^3+x^2)^(1/3)*x+208*RootOf(_Z^3-6*RootOf(RootOf(_Z^
3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*x^2-240*(x^3+x^2)^(2/3)+104*RootOf(_Z^3-6*RootOf(RootOf
(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*x)/(3*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z
*RootOf(_Z^3-4)+9*_Z^2)*x-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2+12*x+4)/x)+1/
8*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3
*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*ln(-(45*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-
4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x^2-90*R
ootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*RootOf(RootOf(_Z^3-4)^2+3
*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x+126*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+
9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*(x^3+x^2)^(
1/3)*x-108*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*RootOf(RootOf
(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x^2+504*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*R
ootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-204*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*R
ootOf(_Z^3-4)^2-8)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x-32*RootOf(_Z^3-6*Roo
tOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*x^2+672*(x^3+x^2)^(2/3)-48*RootOf(_Z^3-6*
RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*x)/(3*RootOf(_Z^3-4)^2*RootOf(RootOf(_
Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2+
12*x+4)/x)+1/16*ln((45*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2
*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x^2-90*RootOf(_Z^3-6*RootOf(RootOf(_Z^
3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2
*RootOf(_Z^3-4)^4*x+228*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^
2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x^2-36*RootOf(_Z^3-6*RootOf(RootOf(_Z^3
-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*Ro
otOf(_Z^3-4)^2*x+504*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*Roo
tOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2...

________________________________________________________________________________________

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 = 0.49, size = 870, normalized size = 4.07 \begin {gather*} \frac {52360 \, x^{7} \cos \left (\frac {1}{9} \, \pi \right ) \log \left (\frac {16 \, {\left (x^{2} - {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) + 2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} - x\right )} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}\right )}}{x^{2}}\right ) - 209440 \, x^{7} \arctan \left (\frac {8 \, {\left (2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{3} - x \cos \left (\frac {1}{9} \, \pi \right )\right )} \sin \left (\frac {1}{9} \, \pi \right ) + \sqrt {3} x + 2 \, {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3} x\right )} \sqrt {\frac {x^{2} - {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) + 2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} - x\right )} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}} - 2 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3}\right )}}{16 \, x \cos \left (\frac {1}{9} \, \pi \right )^{4} - 16 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} + 3 \, x}\right ) \sin \left (\frac {1}{9} \, \pi \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 ) + 104720 \, {\left (\sqrt {3} x^{7} \cos \left (\frac {1}{9} \, \pi \right ) + x^{7} \sin \left (\frac {1}{9} \, \pi \right )\right )} \arctan \left (\frac {8 \, {\left (2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{3} - x \cos \left (\frac {1}{9} \, \pi \right )\right )} \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3} x - 2 \, {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right )^{2} - 2 \, x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3} x\right )} \sqrt {\frac {x^{2} + {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - 2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} + x\right )} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}} + 2 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} - 2 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3}\right )}}{16 \, x \cos \left (\frac {1}{9} \, \pi \right )^{4} - 16 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} + 3 \, x}\right ) - 104720 \, {\left (\sqrt {3} x^{7} \cos \left (\frac {1}{9} \, \pi \right ) - x^{7} \sin \left (\frac {1}{9} \, \pi \right )\right )} \arctan \left (-\frac {2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} - x \sqrt {\frac {x^{2} + 2 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} - x\right )} + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}} - x + {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{2 \, x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )}\right ) + 26180 \, {\left (\sqrt {3} x^{7} \sin \left (\frac {1}{9} \, \pi \right ) - x^{7} \cos \left (\frac {1}{9} \, \pi \right )\right )} \log \left (\frac {64 \, {\left (x^{2} + {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - 2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} + x\right )} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}\right )}}{x^{2}}\right ) - 26180 \, {\left (\sqrt {3} x^{7} \sin \left (\frac {1}{9} \, \pi \right ) + x^{7} \cos \left (\frac {1}{9} \, \pi \right )\right )} \log \left (\frac {64 \, {\left (x^{2} + 2 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} - x\right )} + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}\right )}}{x^{2}}\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}} \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]

1/157080*(52360*x^7*cos(1/9*pi)*log(16*(x^2 - (2*sqrt(3)*x*cos(1/9*pi)*sin(1/9*pi) + 2*x*cos(1/9*pi)^2 - x)*(x
^3 + x^2)^(1/3) + (x^3 + x^2)^(2/3))/x^2) - 209440*x^7*arctan((8*(2*x*cos(1/9*pi)^3 - x*cos(1/9*pi))*sin(1/9*p
i) + sqrt(3)*x + 2*(2*sqrt(3)*x*cos(1/9*pi)^2 + 2*x*cos(1/9*pi)*sin(1/9*pi) - sqrt(3)*x)*sqrt((x^2 - (2*sqrt(3
)*x*cos(1/9*pi)*sin(1/9*pi) + 2*x*cos(1/9*pi)^2 - x)*(x^3 + x^2)^(1/3) + (x^3 + x^2)^(2/3))/x^2) - 2*(x^3 + x^
2)^(1/3)*(2*sqrt(3)*cos(1/9*pi)^2 + 2*cos(1/9*pi)*sin(1/9*pi) - sqrt(3)))/(16*x*cos(1/9*pi)^4 - 16*x*cos(1/9*p
i)^2 + 3*x))*sin(1/9*pi) + 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) + 104720*(sqrt(3)*x^7*cos(1/9*pi) + x^7*sin(1/9*pi))
*arctan((8*(2*x*cos(1/9*pi)^3 - x*cos(1/9*pi))*sin(1/9*pi) - sqrt(3)*x - 2*(2*sqrt(3)*x*cos(1/9*pi)^2 - 2*x*co
s(1/9*pi)*sin(1/9*pi) - sqrt(3)*x)*sqrt((x^2 + (2*sqrt(3)*x*cos(1/9*pi)*sin(1/9*pi) - 2*x*cos(1/9*pi)^2 + x)*(
x^3 + x^2)^(1/3) + (x^3 + x^2)^(2/3))/x^2) + 2*(x^3 + x^2)^(1/3)*(2*sqrt(3)*cos(1/9*pi)^2 - 2*cos(1/9*pi)*sin(
1/9*pi) - sqrt(3)))/(16*x*cos(1/9*pi)^4 - 16*x*cos(1/9*pi)^2 + 3*x)) - 104720*(sqrt(3)*x^7*cos(1/9*pi) - x^7*s
in(1/9*pi))*arctan(-1/2*(2*x*cos(1/9*pi)^2 - x*sqrt((x^2 + 2*(x^3 + x^2)^(1/3)*(2*x*cos(1/9*pi)^2 - x) + (x^3
+ x^2)^(2/3))/x^2) - x + (x^3 + x^2)^(1/3))/(x*cos(1/9*pi)*sin(1/9*pi))) + 26180*(sqrt(3)*x^7*sin(1/9*pi) - x^
7*cos(1/9*pi))*log(64*(x^2 + (2*sqrt(3)*x*cos(1/9*pi)*sin(1/9*pi) - 2*x*cos(1/9*pi)^2 + x)*(x^3 + x^2)^(1/3) +
(x^3 + x^2)^(2/3))/x^2) - 26180*(sqrt(3)*x^7*sin(1/9*pi) + x^7*cos(1/9*pi))*log(64*(x^2 + 2*(x^3 + x^2)^(1/3)
*(2*x*cos(1/9*pi)^2 - x) + (x^3 + x^2)^(2/3))/x^2) + 9*(4491*x^5 - 2994*x^4 + 2495*x^3 + 3600*x^2 - 3300*x + 3
080)*(x^3 + x^2)^(2/3))/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 [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 34.48, size = 986, normalized size = 4.61 \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="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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^6-x^9\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^2 + x^3)^(1/3)*(x^6 - x^9)), x)

________________________________________________________________________________________