\(\int \frac {(1+x^3)^{2/3} (2+x^3) (4+3 x^3)}{x^6 (4+2 x^3+x^6)} \, dx\) [1669]

Optimal result

Integrand size = 37, antiderivative size = 112 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right ) \left (4+3 x^3\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\frac {\left (-8-23 x^3\right ) \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {1}{12} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-9 \log (x)+9 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+8 \log (x) \text {$\#$1}^3-8 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.60 (sec) , antiderivative size = 587, normalized size of antiderivative = 5.24, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6860, 270, 283, 245, 399, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right ) \left (4+3 x^3\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=-\frac {1}{12} \left (3 \sqrt {3}+i\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\frac {1}{12} \left (-3 \sqrt {3}+i\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3} \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\frac {\left (3 \sqrt {3}+i\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x}{\sqrt [3]{\sqrt {3}-i} \sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{4 \left (3 \left (\sqrt {3}-i\right )\right )^{2/3}}-\frac {\left (-3 \sqrt {3}+i\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x}{\sqrt [3]{\sqrt {3}+i} \sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{4 \left (3 \left (\sqrt {3}+i\right )\right )^{2/3}}-\frac {\left (-3 \sqrt {3}+i\right ) \log \left (2 x^3+2 \left (1-i \sqrt {3}\right )\right )}{24 \sqrt [6]{3} \left (\sqrt {3}+i\right )^{2/3}}+\frac {\left (3 \sqrt {3}+i\right ) \log \left (2 x^3+2 \left (1+i \sqrt {3}\right )\right )}{24 \sqrt [6]{3} \left (\sqrt {3}-i\right )^{2/3}}-\frac {\left (3 \sqrt {3}+i\right ) \log \left (-\sqrt [3]{x^3+1}+\frac {\sqrt [6]{3} x}{\sqrt [3]{\sqrt {3}-i}}\right )}{8 \sqrt [6]{3} \left (\sqrt {3}-i\right )^{2/3}}+\frac {\left (-3 \sqrt {3}+i\right ) \log \left (-\sqrt [3]{x^3+1}+\frac {\sqrt [6]{3} x}{\sqrt [3]{\sqrt {3}+i}}\right )}{8 \sqrt [6]{3} \left (\sqrt {3}+i\right )^{2/3}}+\frac {1}{24} \left (9+i \sqrt {3}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{24} \left (9-i \sqrt {3}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {3}{4} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {2 \left (x^3+1\right )^{5/3}}{5 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{4 x^2} \]

[In]

Int[((1 + x^3)^(2/3)*(2 + x^3)*(4 + 3*x^3))/(x^6*(4 + 2*x^3 + x^6)),x]

[Out]

(-3*(1 + x^3)^(2/3))/(4*x^2) - (2*(1 + x^3)^(5/3))/(5*x^5) + (Sqrt[3]*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[
3]])/2 + ((I - 3*Sqrt[3])*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/12 - ((I + 3*Sqrt[3])*ArcTan[(1 + (2*x)
/(1 + x^3)^(1/3))/Sqrt[3]])/12 + ((I + 3*Sqrt[3])*ArcTan[(1 + (2*3^(1/6)*x)/((-I + Sqrt[3])^(1/3)*(1 + x^3)^(1
/3)))/Sqrt[3]])/(4*(3*(-I + Sqrt[3]))^(2/3)) - ((I - 3*Sqrt[3])*ArcTan[(1 + (2*3^(1/6)*x)/((I + Sqrt[3])^(1/3)
*(1 + x^3)^(1/3)))/Sqrt[3]])/(4*(3*(I + Sqrt[3]))^(2/3)) - ((I - 3*Sqrt[3])*Log[2*(1 - I*Sqrt[3]) + 2*x^3])/(2
4*3^(1/6)*(I + Sqrt[3])^(2/3)) + ((I + 3*Sqrt[3])*Log[2*(1 + I*Sqrt[3]) + 2*x^3])/(24*3^(1/6)*(-I + Sqrt[3])^(
2/3)) - ((I + 3*Sqrt[3])*Log[(3^(1/6)*x)/(-I + Sqrt[3])^(1/3) - (1 + x^3)^(1/3)])/(8*3^(1/6)*(-I + Sqrt[3])^(2
/3)) + ((I - 3*Sqrt[3])*Log[(3^(1/6)*x)/(I + Sqrt[3])^(1/3) - (1 + x^3)^(1/3)])/(8*3^(1/6)*(I + Sqrt[3])^(2/3)
) - (3*Log[-x + (1 + x^3)^(1/3)])/4 + ((9 - I*Sqrt[3])*Log[-x + (1 + x^3)^(1/3)])/24 + ((9 + I*Sqrt[3])*Log[-x
+ (1 + x^3)^(1/3)])/24

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{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 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 399

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

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}& = \int \left (\frac {2 \left (1+x^3\right )^{2/3}}{x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{2 x^3}+\frac {\left (-4-3 x^3\right ) \left (1+x^3\right )^{2/3}}{2 \left (4+2 x^3+x^6\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {\left (-4-3 x^3\right ) \left (1+x^3\right )^{2/3}}{4+2 x^3+x^6} \, dx+\frac {3}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+2 \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{2} \int \left (\frac {\left (-3+\frac {i}{\sqrt {3}}\right ) \left (1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3}+\frac {\left (-3-\frac {i}{\sqrt {3}}\right ) \left (1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3}\right ) \, dx+\frac {3}{2} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{2} \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{4} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \left (-9+i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3} \, dx-\frac {1}{6} \left (9+i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3} \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{2} \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{4} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{12} \left (-9+i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-\frac {1}{12} \left (9+i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\frac {1}{2} \left (-1-3 i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (2-2 i \sqrt {3}+2 x^3\right )} \, dx+\frac {1}{2} \left (-1+3 i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (2+2 i \sqrt {3}+2 x^3\right )} \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{2} \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {1}{12} \left (i-3 \sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {1}{12} \left (i+3 \sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {\left (i+3 \sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x}{\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{4 \left (3 \left (-i+\sqrt {3}\right )\right )^{2/3}}-\frac {\left (i-3 \sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x}{\sqrt [3]{i+\sqrt {3}} \sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{4 \left (3 \left (i+\sqrt {3}\right )\right )^{2/3}}-\frac {\left (i-3 \sqrt {3}\right ) \log \left (2 \left (1-i \sqrt {3}\right )+2 x^3\right )}{24 \sqrt [6]{3} \left (i+\sqrt {3}\right )^{2/3}}+\frac {\left (i+3 \sqrt {3}\right ) \log \left (2 \left (1+i \sqrt {3}\right )+2 x^3\right )}{24 \sqrt [6]{3} \left (-i+\sqrt {3}\right )^{2/3}}-\frac {\left (i+3 \sqrt {3}\right ) \log \left (\frac {\sqrt [6]{3} x}{\sqrt [3]{-i+\sqrt {3}}}-\sqrt [3]{1+x^3}\right )}{8 \sqrt [6]{3} \left (-i+\sqrt {3}\right )^{2/3}}+\frac {\left (i-3 \sqrt {3}\right ) \log \left (\frac {\sqrt [6]{3} x}{\sqrt [3]{i+\sqrt {3}}}-\sqrt [3]{1+x^3}\right )}{8 \sqrt [6]{3} \left (i+\sqrt {3}\right )^{2/3}}-\frac {3}{4} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{24} \left (9-i \sqrt {3}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{24} \left (9+i \sqrt {3}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right ) \left (4+3 x^3\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\frac {\left (-8-23 x^3\right ) \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {1}{12} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-9 \log (x)+9 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+8 \log (x) \text {$\#$1}^3-8 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]

[In]

Integrate[((1 + x^3)^(2/3)*(2 + x^3)*(4 + 3*x^3))/(x^6*(4 + 2*x^3 + x^6)),x]

[Out]

((-8 - 23*x^3)*(1 + x^3)^(2/3))/(20*x^5) + RootSum[3 - 6*#1^3 + 4*#1^6 & , (-9*Log[x] + 9*Log[(1 + x^3)^(1/3)
- x*#1] + 8*Log[x]*#1^3 - 8*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1 + 4*#1^4) & ]/12

Maple [F(-1)]

Timed out.

\[\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (x^{3}+2\right ) \left (3 x^{3}+4\right )}{x^{6} \left (x^{6}+2 x^{3}+4\right )}d x\]

[In]

int((x^3+1)^(2/3)*(x^3+2)*(3*x^3+4)/x^6/(x^6+2*x^3+4),x)

[Out]

int((x^3+1)^(2/3)*(x^3+2)*(3*x^3+4)/x^6/(x^6+2*x^3+4),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right ) \left (4+3 x^3\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((x^3+1)^(2/3)*(x^3+2)*(3*x^3+4)/x^6/(x^6+2*x^3+4),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right ) \left (4+3 x^3\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**3+1)**(2/3)*(x**3+2)*(3*x**3+4)/x**6/(x**6+2*x**3+4),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.33 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right ) \left (4+3 x^3\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\int { \frac {{\left (3 \, x^{3} + 4\right )} {\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}} \,d x } \]

[In]

integrate((x^3+1)^(2/3)*(x^3+2)*(3*x^3+4)/x^6/(x^6+2*x^3+4),x, algorithm="maxima")

[Out]

integrate((3*x^3 + 4)*(x^3 + 2)*(x^3 + 1)^(2/3)/((x^6 + 2*x^3 + 4)*x^6), x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.33 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right ) \left (4+3 x^3\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\int { \frac {{\left (3 \, x^{3} + 4\right )} {\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}} \,d x } \]

[In]

integrate((x^3+1)^(2/3)*(x^3+2)*(3*x^3+4)/x^6/(x^6+2*x^3+4),x, algorithm="giac")

[Out]

integrate((3*x^3 + 4)*(x^3 + 2)*(x^3 + 1)^(2/3)/((x^6 + 2*x^3 + 4)*x^6), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.33 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right ) \left (4+3 x^3\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^3+2\right )\,\left (3\,x^3+4\right )}{x^6\,\left (x^6+2\,x^3+4\right )} \,d x \]

[In]

int(((x^3 + 1)^(2/3)*(x^3 + 2)*(3*x^3 + 4))/(x^6*(2*x^3 + x^6 + 4)),x)

[Out]

int(((x^3 + 1)^(2/3)*(x^3 + 2)*(3*x^3 + 4))/(x^6*(2*x^3 + x^6 + 4)), x)