\(\int \frac {(1+2 x^2) \sqrt [3]{x+2 x^3}}{x^4 (1+2 x^4)} \, dx\) [1711]

Optimal result

Integrand size = 31, antiderivative size = 115 \[ \int \frac {\left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{x^4 \left (1+2 x^4\right )} \, dx=-\frac {3 \left (1+10 x^2\right ) \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {1}{2} \text {RootSum}\left [6-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-6 \log (x)+6 \log \left (\sqrt [3]{x+2 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{x+2 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.11 (sec) , antiderivative size = 1018, normalized size of antiderivative = 8.85, number of steps used = 22, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2081, 1284, 1505, 1535, 270, 6857, 283, 337, 21, 1543, 488, 598, 503} \[ \int \frac {\left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{x^4 \left (1+2 x^4\right )} \, dx=-\frac {3 \sqrt [3]{2 x^3+x} \left (2 x^2+1\right )}{8 x^3}-\frac {3 \sqrt [3]{2 x^3+x}}{x}+\frac {\left (3+2 i \sqrt {2}\right ) \sqrt [3]{2 x^3+x} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}+\frac {\left (3-2 i \sqrt {2}\right ) \sqrt [3]{2 x^3+x} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}-\frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{2 x^3+x} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{x} \sqrt [3]{2 x^2+1}}-\frac {\sqrt {3} \left (1-2 i \sqrt {2}\right ) \sqrt [3]{2 x^3+x} \arctan \left (\frac {\frac {2 \sqrt [3]{2-i \sqrt {2}} x^{2/3}}{\sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{2 \left (2-i \sqrt {2}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}-\frac {\sqrt {3} \left (1+2 i \sqrt {2}\right ) \sqrt [3]{2 x^3+x} \arctan \left (\frac {\frac {2 \sqrt [3]{2+i \sqrt {2}} x^{2/3}}{\sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{2 \left (2+i \sqrt {2}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}+\frac {\left (1-2 i \sqrt {2}\right ) \sqrt [3]{2 x^3+x} \log \left (i-\sqrt {2} x^2\right )}{4 \left (2-i \sqrt {2}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}+\frac {\left (1+2 i \sqrt {2}\right ) \sqrt [3]{2 x^3+x} \log \left (\sqrt {2} x^2+i\right )}{4 \left (2+i \sqrt {2}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}+\frac {\left (3+2 i \sqrt {2}\right ) \sqrt [3]{2 x^3+x} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{2 x^2+1}\right )}{2\ 2^{2/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}+\frac {\left (3-2 i \sqrt {2}\right ) \sqrt [3]{2 x^3+x} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{2 x^2+1}\right )}{2\ 2^{2/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}-\frac {3 \sqrt [3]{2 x^3+x} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{2 x^2+1}\right )}{2^{2/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}-\frac {3 \left (1-2 i \sqrt {2}\right ) \sqrt [3]{2 x^3+x} \log \left (\sqrt [3]{2-i \sqrt {2}} x^{2/3}-\sqrt [3]{2 x^2+1}\right )}{4 \left (2-i \sqrt {2}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}}-\frac {3 \left (1+2 i \sqrt {2}\right ) \sqrt [3]{2 x^3+x} \log \left (\sqrt [3]{2+i \sqrt {2}} x^{2/3}-\sqrt [3]{2 x^2+1}\right )}{4 \left (2+i \sqrt {2}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{2 x^2+1}} \]

[In]

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

[Out]

(-3*(x + 2*x^3)^(1/3))/x - (3*(1 + 2*x^2)*(x + 2*x^3)^(1/3))/(8*x^3) - (2^(1/3)*Sqrt[3]*(x + 2*x^3)^(1/3)*ArcT
an[(1 + (2*2^(1/3)*x^(2/3))/(1 + 2*x^2)^(1/3))/Sqrt[3]])/(x^(1/3)*(1 + 2*x^2)^(1/3)) + ((3 - (2*I)*Sqrt[2])*(x
+ 2*x^3)^(1/3)*ArcTan[(1 + (2*2^(1/3)*x^(2/3))/(1 + 2*x^2)^(1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]*x^(1/3)*(1 + 2*x
^2)^(1/3)) + ((3 + (2*I)*Sqrt[2])*(x + 2*x^3)^(1/3)*ArcTan[(1 + (2*2^(1/3)*x^(2/3))/(1 + 2*x^2)^(1/3))/Sqrt[3]
])/(2^(2/3)*Sqrt[3]*x^(1/3)*(1 + 2*x^2)^(1/3)) - (Sqrt[3]*(1 - (2*I)*Sqrt[2])*(x + 2*x^3)^(1/3)*ArcTan[(1 + (2
*(2 - I*Sqrt[2])^(1/3)*x^(2/3))/(1 + 2*x^2)^(1/3))/Sqrt[3]])/(2*(2 - I*Sqrt[2])^(2/3)*x^(1/3)*(1 + 2*x^2)^(1/3
)) - (Sqrt[3]*(1 + (2*I)*Sqrt[2])*(x + 2*x^3)^(1/3)*ArcTan[(1 + (2*(2 + I*Sqrt[2])^(1/3)*x^(2/3))/(1 + 2*x^2)^
(1/3))/Sqrt[3]])/(2*(2 + I*Sqrt[2])^(2/3)*x^(1/3)*(1 + 2*x^2)^(1/3)) + ((1 - (2*I)*Sqrt[2])*(x + 2*x^3)^(1/3)*
Log[I - Sqrt[2]*x^2])/(4*(2 - I*Sqrt[2])^(2/3)*x^(1/3)*(1 + 2*x^2)^(1/3)) + ((1 + (2*I)*Sqrt[2])*(x + 2*x^3)^(
1/3)*Log[I + Sqrt[2]*x^2])/(4*(2 + I*Sqrt[2])^(2/3)*x^(1/3)*(1 + 2*x^2)^(1/3)) - (3*(x + 2*x^3)^(1/3)*Log[2^(1
/3)*x^(2/3) - (1 + 2*x^2)^(1/3)])/(2^(2/3)*x^(1/3)*(1 + 2*x^2)^(1/3)) + ((3 - (2*I)*Sqrt[2])*(x + 2*x^3)^(1/3)
*Log[2^(1/3)*x^(2/3) - (1 + 2*x^2)^(1/3)])/(2*2^(2/3)*x^(1/3)*(1 + 2*x^2)^(1/3)) + ((3 + (2*I)*Sqrt[2])*(x + 2
*x^3)^(1/3)*Log[2^(1/3)*x^(2/3) - (1 + 2*x^2)^(1/3)])/(2*2^(2/3)*x^(1/3)*(1 + 2*x^2)^(1/3)) - (3*(1 - (2*I)*Sq
rt[2])*(x + 2*x^3)^(1/3)*Log[(2 - I*Sqrt[2])^(1/3)*x^(2/3) - (1 + 2*x^2)^(1/3)])/(4*(2 - I*Sqrt[2])^(2/3)*x^(1
/3)*(1 + 2*x^2)^(1/3)) - (3*(1 + (2*I)*Sqrt[2])*(x + 2*x^3)^(1/3)*Log[(2 + I*Sqrt[2])^(1/3)*x^(2/3) - (1 + 2*x
^2)^(1/3)])/(4*(2 + I*Sqrt[2])^(2/3)*x^(1/3)*(1 + 2*x^2)^(1/3))

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 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 283

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

Rule 337

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

Rule 488

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Dist[1/(b*(m + n*(p + q) + 1
)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d
)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && N
eQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 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 598

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 1284

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominat
or[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + e*(x^(2*k)/f))^q*(a + c*(x^(4*k)/f))^p, x], x, (f*x)^(1/k)]
, x]] /; FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]

Rule 1505

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = GCD[m +
1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + c*x^(2*(n/k)))^p, x], x, x^k], x] /; k !=
1] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[m]

Rule 1535

Int[(((f_.)*(x_))^(m_)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Dist[d/a, Int[
(f*x)^m*(d + e*x^n)^(q - 1), x], x] + Dist[1/(a*f^n), Int[(f*x)^(m + n)*(d + e*x^n)^(q - 1)*(Simp[a*e - c*d*x^
n, x]/(a + c*x^(2*n))), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] &&  !IntegerQ[q] &&
GtQ[q, 0] && LtQ[m, 0]

Rule 1543

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Int[ExpandInte
grand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGt
Q[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

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

Mathematica [A] (verified)

Time = 3.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.31 \[ \int \frac {\left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{x^4 \left (1+2 x^4\right )} \, dx=\frac {\sqrt [3]{x+2 x^3} \left (-9 \sqrt [3]{1+2 x^2} \left (1+10 x^2\right )+4 x^{8/3} \text {RootSum}\left [6-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-12 \log (x)+18 \log \left (\sqrt [3]{1+2 x^2}-x^{2/3} \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-3 \log \left (\sqrt [3]{1+2 x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ]\right )}{24 x^3 \sqrt [3]{1+2 x^2}} \]

[In]

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

[Out]

((x + 2*x^3)^(1/3)*(-9*(1 + 2*x^2)^(1/3)*(1 + 10*x^2) + 4*x^(8/3)*RootSum[6 - 4*#1^3 + #1^6 & , (-12*Log[x] +
18*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)/(-2*#1^2
+ #1^5) & ]))/(24*x^3*(1 + 2*x^2)^(1/3))

Maple [N/A] (verified)

Time = 160.70 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.68

method	result	size
pseudoelliptic	\(\frac {\left (-30 x^{2}-3\right ) \left (2 x^{3}+x \right )^{\frac {1}{3}}-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-4 \textit {\_Z}^{3}+6\right )}{\sum }\frac {\left (\textit {\_R}^{3}-6\right ) \ln \left (\frac {-\textit {\_R} x +\left (2 x^{3}+x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (\textit {\_R}^{3}-2\right )}\right ) x^{3}}{8 x^{3}}\)	\(78\)
trager	\(\text {Expression too large to display}\)	\(10612\)
risch	\(\text {Expression too large to display}\)	\(19706\)

[In]

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

[Out]

1/8*((-30*x^2-3)*(2*x^3+x)^(1/3)-4*sum((_R^3-6)*ln((-_R*x+(2*x^3+x)^(1/3))/x)/_R^2/(_R^3-2),_R=RootOf(_Z^6-4*_
Z^3+6))*x^3)/x^3

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.37 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {\left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{x^4 \left (1+2 x^4\right )} \, dx=\int \frac {\sqrt [3]{x \left (2 x^{2} + 1\right )} \left (2 x^{2} + 1\right )}{x^{4} \cdot \left (2 x^{4} + 1\right )}\, dx \]

[In]

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

[Out]

Integral((x*(2*x**2 + 1))**(1/3)*(2*x**2 + 1)/(x**4*(2*x**4 + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.27 \[ \int \frac {\left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{x^4 \left (1+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{2} + 1\right )}}{{\left (2 \, x^{4} + 1\right )} x^{4}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((2*x^2+1)*(2*x^3+x)^(1/3)/x^4/(2*x^4+1),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 = 5.88 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.27 \[ \int \frac {\left (1+2 x^2\right ) \sqrt [3]{x+2 x^3}}{x^4 \left (1+2 x^4\right )} \, dx=\int \frac {{\left (2\,x^3+x\right )}^{1/3}\,\left (2\,x^2+1\right )}{x^4\,\left (2\,x^4+1\right )} \,d x \]

[In]

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

[Out]

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