\(\int \frac {(-1+x^3)^{2/3} (2+x^3)}{x^3 (-4+x^3+x^6)} \, dx\) [1493]

Optimal result

Integrand size = 28, antiderivative size = 104 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{6} \text {RootSum}\left [2-7 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-7 \text {$\#$1}+8 \text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(509\) vs. \(2(104)=208\).

Time = 0.54 (sec) , antiderivative size = 509, normalized size of antiderivative = 4.89, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6860, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\frac {\left (5+\sqrt {17}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt {51}}-\frac {\left (5-\sqrt {17}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt {51}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\sqrt [3]{199+47 \sqrt {17}} \arctan \left (\frac {\frac {\sqrt [3]{7-\sqrt {17}} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {51}}+\frac {\sqrt [3]{199-47 \sqrt {17}} \arctan \left (\frac {\frac {\sqrt [3]{7+\sqrt {17}} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {51}}-\frac {\sqrt [3]{199+47 \sqrt {17}} \log \left (2 x^3-\sqrt {17}+1\right )}{24\ 2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{199-47 \sqrt {17}} \log \left (2 x^3+\sqrt {17}+1\right )}{24\ 2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{199+47 \sqrt {17}} \log \left (\frac {1}{2} \sqrt [3]{7-\sqrt {17}} x-\sqrt [3]{x^3-1}\right )}{8\ 2^{2/3} \sqrt {17}}-\frac {\sqrt [3]{199-47 \sqrt {17}} \log \left (\frac {1}{2} \sqrt [3]{7+\sqrt {17}} x-\sqrt [3]{x^3-1}\right )}{8\ 2^{2/3} \sqrt {17}}-\frac {1}{136} \left (17+5 \sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {1}{136} \left (17-5 \sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{4} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{2/3}}{4 x^2} \]

[In]

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

[Out]

(-1 + x^3)^(2/3)/(4*x^2) - ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) - ((5 - Sqrt[17])*ArcTan[(
1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(4*Sqrt[51]) + ((5 + Sqrt[17])*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[
3]])/(4*Sqrt[51]) - ((199 + 47*Sqrt[17])^(1/3)*ArcTan[(1 + ((7 - Sqrt[17])^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]]
)/(4*2^(2/3)*Sqrt[51]) + ((199 - 47*Sqrt[17])^(1/3)*ArcTan[(1 + ((7 + Sqrt[17])^(1/3)*x)/(-1 + x^3)^(1/3))/Sqr
t[3]])/(4*2^(2/3)*Sqrt[51]) - ((199 + 47*Sqrt[17])^(1/3)*Log[1 - Sqrt[17] + 2*x^3])/(24*2^(2/3)*Sqrt[17]) + ((
199 - 47*Sqrt[17])^(1/3)*Log[1 + Sqrt[17] + 2*x^3])/(24*2^(2/3)*Sqrt[17]) + ((199 + 47*Sqrt[17])^(1/3)*Log[((7
- Sqrt[17])^(1/3)*x)/2 - (-1 + x^3)^(1/3)])/(8*2^(2/3)*Sqrt[17]) - ((199 - 47*Sqrt[17])^(1/3)*Log[((7 + Sqrt[
17])^(1/3)*x)/2 - (-1 + x^3)^(1/3)])/(8*2^(2/3)*Sqrt[17]) + Log[-x + (-1 + x^3)^(1/3)]/4 - ((17 - 5*Sqrt[17])*
Log[-x + (-1 + x^3)^(1/3)])/136 - ((17 + 5*Sqrt[17])*Log[-x + (-1 + x^3)^(1/3)])/136

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 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 {\left (-1+x^3\right )^{2/3}}{2 x^3}+\frac {\left (-1+x^3\right )^{2/3} \left (3+x^3\right )}{2 \left (-4+x^3+x^6\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3} \left (3+x^3\right )}{-4+x^3+x^6} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{2} \int \left (\frac {\left (1+\frac {5}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3}+\frac {\left (1-\frac {5}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3}\right ) \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{34} \left (17-5 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3} \, dx+\frac {1}{34} \left (17+5 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{68} \left (17-5 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{34} \left (17-\sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+\sqrt {17}+2 x^3\right )} \, dx+\frac {1}{34} \left (17+\sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (1-\sqrt {17}+2 x^3\right )} \, dx+\frac {1}{68} \left (17+5 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\left (5-\sqrt {17}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {51}}+\frac {\left (5+\sqrt {17}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {51}}-\frac {\sqrt [3]{3184+752 \sqrt {17}} \arctan \left (\frac {1+\frac {\sqrt [3]{7-\sqrt {17}} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt {51}}+\frac {\sqrt [3]{3184-752 \sqrt {17}} \arctan \left (\frac {1+\frac {\sqrt [3]{7+\sqrt {17}} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt {51}}-\frac {\sqrt [3]{3184+752 \sqrt {17}} \log \left (1-\sqrt {17}+2 x^3\right )}{96 \sqrt {17}}+\frac {\sqrt [3]{3184-752 \sqrt {17}} \log \left (1+\sqrt {17}+2 x^3\right )}{96 \sqrt {17}}+\frac {\sqrt [3]{3184+752 \sqrt {17}} \log \left (\frac {1}{2} \sqrt [3]{7-\sqrt {17}} x-\sqrt [3]{-1+x^3}\right )}{32 \sqrt {17}}-\frac {\sqrt [3]{3184-752 \sqrt {17}} \log \left (\frac {1}{2} \sqrt [3]{7+\sqrt {17}} x-\sqrt [3]{-1+x^3}\right )}{32 \sqrt {17}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{136} \left (17-5 \sqrt {17}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{136} \left (17+5 \sqrt {17}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{6} \text {RootSum}\left [2-7 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-7 \text {$\#$1}+8 \text {$\#$1}^4}\&\right ] \]

[In]

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

[Out]

(-1 + x^3)^(2/3)/(4*x^2) - RootSum[2 - 7*#1^3 + 4*#1^6 & , (-2*Log[x] + 2*Log[(-1 + x^3)^(1/3) - x*#1] + Log[x
]*#1^3 - Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-7*#1 + 8*#1^4) & ]/6

Maple [N/A] (verified)

Time = 201.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.68

method	result	size
pseudoelliptic	\(\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{6}-7 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\left (\textit {\_R}^{3}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )}{8 \textit {\_R}^{4}-7 \textit {\_R}}\right ) x^{2}+3 \left (x^{3}-1\right )^{\frac {2}{3}}}{12 x^{2}}\)	\(71\)
risch	\(\text {Expression too large to display}\)	\(6667\)
trager	\(\text {Expression too large to display}\)	\(10911\)

[In]

int((x^3-1)^(2/3)*(x^3+2)/x^3/(x^6+x^3-4),x,method=_RETURNVERBOSE)

[Out]

1/12*(2*sum((_R^3-2)*ln((-_R*x+(x^3-1)^(1/3))/x)/(8*_R^4-7*_R),_R=RootOf(4*_Z^6-7*_Z^3+2))*x^2+3*(x^3-1)^(2/3)
)/x^2

Fricas [F(-2)]

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

[In]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 4\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 4\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.51 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3+2\right )}{x^3\,\left (x^6+x^3-4\right )} \,d x \]

[In]

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

[Out]

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