\(\int \frac {\sqrt [3]{x+2 x^3} (-1+x^4)}{x^4 (2-x^2+x^4)} \, dx\) [1764]

Optimal result

Integrand size = 32, antiderivative size = 119 \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\frac {3 \left (1+4 x^2\right ) \sqrt [3]{x+2 x^3}}{16 x^3}+\frac {1}{8} \text {RootSum}\left [11-9 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {11 \log (x)-11 \log \left (\sqrt [3]{x+2 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{x+2 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-9 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.55 (sec) , antiderivative size = 893, normalized size of antiderivative = 7.50, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2081, 6860, 270, 477, 476, 486, 597, 12, 503} \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=-\frac {3 \sqrt [3]{2 x^3+x} \left (2 x^2+1\right )}{8 x^3}+\frac {12 \sqrt [3]{2 x^3+x}}{\sqrt {7} \left (3 i+\sqrt {7}\right ) x}-\frac {12 \sqrt [3]{2 x^3+x}}{\sqrt {7} \left (3 i-\sqrt {7}\right ) x}+\frac {3 \left (7 i+5 \sqrt {7}\right ) \sqrt [3]{2 x^3+x}}{56 \left (i+\sqrt {7}\right ) x^3}+\frac {3 \left (7 i-5 \sqrt {7}\right ) \sqrt [3]{2 x^3+x}}{56 \left (i-\sqrt {7}\right ) x^3}-\frac {\sqrt {\frac {3}{7}} \sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} \left (5 i+\sqrt {7}\right ) \sqrt [3]{2 x^3+x} \arctan \left (\frac {\frac {2 \sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} x^{2/3}}{\sqrt [3]{-i+\sqrt {7}} \sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{\left (-i+\sqrt {7}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}+\frac {\sqrt [3]{2} \sqrt {3} \left (5 i-\sqrt {7}\right ) \sqrt [3]{2 x^3+x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {7}}{2 \left (2 i+\sqrt {7}\right )}} \sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{\sqrt {7} \left (i+\sqrt {7}\right )^2 \sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}+\frac {\left (7-5 i \sqrt {7}\right ) \sqrt [3]{2 x^3+x} \log \left (-2 x^2-i \sqrt {7}+1\right )}{7\ 2^{2/3} \left (i+\sqrt {7}\right )^2 \sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}+\frac {\left (7+5 i \sqrt {7}\right ) \sqrt [3]{-2 i+\sqrt {7}} \sqrt [3]{2 x^3+x} \log \left (-2 x^2+i \sqrt {7}+1\right )}{7\ 2^{2/3} \left (-i+\sqrt {7}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}-\frac {3 \left (7+5 i \sqrt {7}\right ) \sqrt [3]{-2 i+\sqrt {7}} \sqrt [3]{2 x^3+x} \log \left (\sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} x^{2/3}-\sqrt [3]{-i+\sqrt {7}} \sqrt [3]{2 x^2+1}\right )}{7\ 2^{2/3} \left (-i+\sqrt {7}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}-\frac {3 \left (7-5 i \sqrt {7}\right ) \sqrt [3]{2 x^3+x} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}} \sqrt [3]{2 x^2+1}\right )}{7\ 2^{2/3} \left (i+\sqrt {7}\right )^2 \sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}} \sqrt [3]{x} \sqrt [3]{2 x^2+1}} \]

[In]

Int[((x + 2*x^3)^(1/3)*(-1 + x^4))/(x^4*(2 - x^2 + x^4)),x]

[Out]

(3*(7*I - 5*Sqrt[7])*(x + 2*x^3)^(1/3))/(56*(I - Sqrt[7])*x^3) + (3*(7*I + 5*Sqrt[7])*(x + 2*x^3)^(1/3))/(56*(
I + Sqrt[7])*x^3) - (12*(x + 2*x^3)^(1/3))/(Sqrt[7]*(3*I - Sqrt[7])*x) + (12*(x + 2*x^3)^(1/3))/(Sqrt[7]*(3*I
+ Sqrt[7])*x) - (3*(1 + 2*x^2)*(x + 2*x^3)^(1/3))/(8*x^3) - (Sqrt[3/7]*(2*(-2*I + Sqrt[7]))^(1/3)*(5*I + Sqrt[
7])*(x + 2*x^3)^(1/3)*ArcTan[(1 + (2*(2*(-2*I + Sqrt[7]))^(1/3)*x^(2/3))/((-I + Sqrt[7])^(1/3)*(1 + 2*x^2)^(1/
3)))/Sqrt[3]])/((-I + Sqrt[7])^(7/3)*x^(1/3)*(1 + 2*x^2)^(1/3)) + (2^(1/3)*Sqrt[3]*(5*I - Sqrt[7])*(x + 2*x^3)
^(1/3)*ArcTan[(1 + (2*x^(2/3))/(((I + Sqrt[7])/(2*(2*I + Sqrt[7])))^(1/3)*(1 + 2*x^2)^(1/3)))/Sqrt[3]])/(Sqrt[
7]*(I + Sqrt[7])^2*((I + Sqrt[7])/(2*I + Sqrt[7]))^(1/3)*x^(1/3)*(1 + 2*x^2)^(1/3)) + ((7 - (5*I)*Sqrt[7])*(x
+ 2*x^3)^(1/3)*Log[1 - I*Sqrt[7] - 2*x^2])/(7*2^(2/3)*(I + Sqrt[7])^2*((I + Sqrt[7])/(2*I + Sqrt[7]))^(1/3)*x^
(1/3)*(1 + 2*x^2)^(1/3)) + ((7 + (5*I)*Sqrt[7])*(-2*I + Sqrt[7])^(1/3)*(x + 2*x^3)^(1/3)*Log[1 + I*Sqrt[7] - 2
*x^2])/(7*2^(2/3)*(-I + Sqrt[7])^(7/3)*x^(1/3)*(1 + 2*x^2)^(1/3)) - (3*(7 + (5*I)*Sqrt[7])*(-2*I + Sqrt[7])^(1
/3)*(x + 2*x^3)^(1/3)*Log[(2*(-2*I + Sqrt[7]))^(1/3)*x^(2/3) - (-I + Sqrt[7])^(1/3)*(1 + 2*x^2)^(1/3)])/(7*2^(
2/3)*(-I + Sqrt[7])^(7/3)*x^(1/3)*(1 + 2*x^2)^(1/3)) - (3*(7 - (5*I)*Sqrt[7])*(x + 2*x^3)^(1/3)*Log[2^(1/3)*x^
(2/3) - ((I + Sqrt[7])/(2*I + Sqrt[7]))^(1/3)*(1 + 2*x^2)^(1/3)])/(7*2^(2/3)*(I + Sqrt[7])^2*((I + Sqrt[7])/(2
*I + Sqrt[7]))^(1/3)*x^(1/3)*(1 + 2*x^2)^(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 270

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

Rule 476

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 486

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/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*
x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x
], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] &&
IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 503

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

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 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x+2 x^3} \int \frac {\sqrt [3]{1+2 x^2} \left (-1+x^4\right )}{x^{11/3} \left (2-x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ & = \frac {\sqrt [3]{x+2 x^3} \int \left (\frac {\sqrt [3]{1+2 x^2}}{x^{11/3}}-\frac {\left (3-x^2\right ) \sqrt [3]{1+2 x^2}}{x^{11/3} \left (2-x^2+x^4\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ & = \frac {\sqrt [3]{x+2 x^3} \int \frac {\sqrt [3]{1+2 x^2}}{x^{11/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {\sqrt [3]{x+2 x^3} \int \frac {\left (3-x^2\right ) \sqrt [3]{1+2 x^2}}{x^{11/3} \left (2-x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ & = -\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\sqrt [3]{x+2 x^3} \int \left (\frac {\left (-1-\frac {5 i}{\sqrt {7}}\right ) \sqrt [3]{1+2 x^2}}{x^{11/3} \left (-1-i \sqrt {7}+2 x^2\right )}+\frac {\left (-1+\frac {5 i}{\sqrt {7}}\right ) \sqrt [3]{1+2 x^2}}{x^{11/3} \left (-1+i \sqrt {7}+2 x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ & = -\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\left (\left (-7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \int \frac {\sqrt [3]{1+2 x^2}}{x^{11/3} \left (-1+i \sqrt {7}+2 x^2\right )} \, dx}{7 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\left (\left (7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \int \frac {\sqrt [3]{1+2 x^2}}{x^{11/3} \left (-1-i \sqrt {7}+2 x^2\right )} \, dx}{7 \sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ & = -\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\left (3 \left (-7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6}}{x^9 \left (-1+i \sqrt {7}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{7 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\left (3 \left (7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6}}{x^9 \left (-1-i \sqrt {7}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{7 \sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ & = -\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\left (3 \left (-7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+2 x^3}}{x^5 \left (-1+i \sqrt {7}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{14 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\left (3 \left (7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+2 x^3}}{x^5 \left (-1-i \sqrt {7}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{14 \sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ & = \frac {3 \left (7 i-5 \sqrt {7}\right ) \sqrt [3]{x+2 x^3}}{56 \left (i-\sqrt {7}\right ) x^3}+\frac {3 \left (7 i+5 \sqrt {7}\right ) \sqrt [3]{x+2 x^3}}{56 \left (i+\sqrt {7}\right ) x^3}-\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {\left (3 \left (-7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \text {Subst}\left (\int \frac {-2 \left (5-i \sqrt {7}\right )-12 x^3}{x^2 \left (1+2 x^3\right )^{2/3} \left (-1+i \sqrt {7}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{56 \left (1-i \sqrt {7}\right ) \sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {\left (3 \left (7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \text {Subst}\left (\int \frac {-2 \left (5+i \sqrt {7}\right )-12 x^3}{x^2 \left (1+2 x^3\right )^{2/3} \left (-1-i \sqrt {7}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{56 \left (1+i \sqrt {7}\right ) \sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ & = \frac {3 \left (7 i-5 \sqrt {7}\right ) \sqrt [3]{x+2 x^3}}{56 \left (i-\sqrt {7}\right ) x^3}+\frac {3 \left (7 i+5 \sqrt {7}\right ) \sqrt [3]{x+2 x^3}}{56 \left (i+\sqrt {7}\right ) x^3}-\frac {12 \sqrt [3]{x+2 x^3}}{\sqrt {7} \left (3 i-\sqrt {7}\right ) x}+\frac {12 \sqrt [3]{x+2 x^3}}{\sqrt {7} \left (3 i+\sqrt {7}\right ) x}-\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {\left (3 \left (-7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \text {Subst}\left (\int -\frac {16 \left (2-i \sqrt {7}\right ) x}{\left (1+2 x^3\right )^{2/3} \left (-1+i \sqrt {7}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{56 \left (1-i \sqrt {7}\right )^2 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {\left (3 \left (7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \text {Subst}\left (\int -\frac {16 \left (2+i \sqrt {7}\right ) x}{\left (1+2 x^3\right )^{2/3} \left (-1-i \sqrt {7}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{56 \left (1+i \sqrt {7}\right )^2 \sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ & = \frac {3 \left (7 i-5 \sqrt {7}\right ) \sqrt [3]{x+2 x^3}}{56 \left (i-\sqrt {7}\right ) x^3}+\frac {3 \left (7 i+5 \sqrt {7}\right ) \sqrt [3]{x+2 x^3}}{56 \left (i+\sqrt {7}\right ) x^3}-\frac {12 \sqrt [3]{x+2 x^3}}{\sqrt {7} \left (3 i-\sqrt {7}\right ) x}+\frac {12 \sqrt [3]{x+2 x^3}}{\sqrt {7} \left (3 i+\sqrt {7}\right ) x}-\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\left (6 \left (2-i \sqrt {7}\right ) \left (-7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+2 x^3\right )^{2/3} \left (-1+i \sqrt {7}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{7 \left (1-i \sqrt {7}\right )^2 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\left (6 \left (2+i \sqrt {7}\right ) \left (7+5 i \sqrt {7}\right ) \sqrt [3]{x+2 x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+2 x^3\right )^{2/3} \left (-1-i \sqrt {7}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{7 \left (1+i \sqrt {7}\right )^2 \sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ & = \frac {3 \left (7 i-5 \sqrt {7}\right ) \sqrt [3]{x+2 x^3}}{56 \left (i-\sqrt {7}\right ) x^3}+\frac {3 \left (7 i+5 \sqrt {7}\right ) \sqrt [3]{x+2 x^3}}{56 \left (i+\sqrt {7}\right ) x^3}-\frac {12 \sqrt [3]{x+2 x^3}}{\sqrt {7} \left (3 i-\sqrt {7}\right ) x}+\frac {12 \sqrt [3]{x+2 x^3}}{\sqrt {7} \left (3 i+\sqrt {7}\right ) x}-\frac {3 \left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\sqrt {\frac {3}{7}} \sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} \left (5 i+\sqrt {7}\right ) \sqrt [3]{x+2 x^3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} x^{2/3}}{\sqrt [3]{-i+\sqrt {7}} \sqrt [3]{1+2 x^2}}}{\sqrt {3}}\right )}{\left (-i+\sqrt {7}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\sqrt {\frac {3}{7}} \left (5 i-\sqrt {7}\right ) \sqrt [3]{2 \left (2 i+\sqrt {7}\right )} \sqrt [3]{x+2 x^3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {7}}{2 \left (2 i+\sqrt {7}\right )}} \sqrt [3]{1+2 x^2}}}{\sqrt {3}}\right )}{\left (i+\sqrt {7}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\left (7-5 i \sqrt {7}\right ) \sqrt [3]{2 i+\sqrt {7}} \sqrt [3]{x+2 x^3} \log \left (1-i \sqrt {7}-2 x^2\right )}{7\ 2^{2/3} \left (i+\sqrt {7}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\left (7+5 i \sqrt {7}\right ) \sqrt [3]{-2 i+\sqrt {7}} \sqrt [3]{x+2 x^3} \log \left (1+i \sqrt {7}-2 x^2\right )}{7\ 2^{2/3} \left (-i+\sqrt {7}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {3 \left (7+5 i \sqrt {7}\right ) \sqrt [3]{-2 i+\sqrt {7}} \sqrt [3]{x+2 x^3} \log \left (\sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} x^{2/3}-\sqrt [3]{-i+\sqrt {7}} \sqrt [3]{1+2 x^2}\right )}{7\ 2^{2/3} \left (-i+\sqrt {7}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {3 \left (7-5 i \sqrt {7}\right ) \sqrt [3]{2 i+\sqrt {7}} \sqrt [3]{x+2 x^3} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}} \sqrt [3]{1+2 x^2}\right )}{7\ 2^{2/3} \left (i+\sqrt {7}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{1+2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.03 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\frac {\sqrt [3]{x+2 x^3} \left (9 \sqrt [3]{1+2 x^2} \left (1+4 x^2\right )+2 x^{8/3} \text {RootSum}\left [11-9 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {22 \log (x)-33 \log \left (\sqrt [3]{1+2 x^2}-x^{2/3} \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-3 \log \left (\sqrt [3]{1+2 x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-9 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ]\right )}{48 x^3 \sqrt [3]{1+2 x^2}} \]

[In]

Integrate[((x + 2*x^3)^(1/3)*(-1 + x^4))/(x^4*(2 - x^2 + x^4)),x]

[Out]

((x + 2*x^3)^(1/3)*(9*(1 + 2*x^2)^(1/3)*(1 + 4*x^2) + 2*x^(8/3)*RootSum[11 - 9*#1^3 + 2*#1^6 & , (22*Log[x] -
33*Log[(1 + 2*x^2)^(1/3) - x^(2/3)*#1] + 2*Log[x]*#1^3 - 3*Log[(1 + 2*x^2)^(1/3) - x^(2/3)*#1]*#1^3)/(-9*#1^2
+ 4*#1^5) & ]))/(48*x^3*(1 + 2*x^2)^(1/3))

Maple [N/A] (verified)

Time = 201.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69

method	result	size
pseudoelliptic	\(\frac {\left (12 x^{2}+3\right ) \left (2 x^{3}+x \right )^{\frac {1}{3}}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+11\right )}{\sum }\frac {\left (\textit {\_R}^{3}+11\right ) \ln \left (\frac {-\textit {\_R} x +\left (2 x^{3}+x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (4 \textit {\_R}^{3}-9\right )}\right ) x^{3}}{16 x^{3}}\)	\(82\)
trager	\(\text {Expression too large to display}\)	\(12391\)
risch	\(\text {Expression too large to display}\)	\(13838\)

[In]

int((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),x,method=_RETURNVERBOSE)

[Out]

1/16*((12*x^2+3)*(2*x^3+x)^(1/3)-2*sum((_R^3+11)*ln((-_R*x+(2*x^3+x)^(1/3))/x)/_R^2/(4*_R^3-9),_R=RootOf(2*_Z^
6-9*_Z^3+11))*x^3)/x^3

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate((2*x**3+x)**(1/3)*(x**4-1)/x**4/(x**4-x**2+2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 1\right )} {\left (2 \, x^{3} + x\right )}^{\frac {1}{3}}}{{\left (x^{4} - x^{2} + 2\right )} x^{4}} \,d x } \]

[In]

integrate((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),x, algorithm="maxima")

[Out]

integrate((x^4 - 1)*(2*x^3 + x)^(1/3)/((x^4 - x^2 + 2)*x^4), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.69 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\int \frac {{\left (2\,x^3+x\right )}^{1/3}\,\left (x^4-1\right )}{x^4\,\left (x^4-x^2+2\right )} \,d x \]

[In]

int(((x + 2*x^3)^(1/3)*(x^4 - 1))/(x^4*(x^4 - x^2 + 2)),x)

[Out]

int(((x + 2*x^3)^(1/3)*(x^4 - 1))/(x^4*(x^4 - x^2 + 2)), x)