3.15.65 \(\int \frac {(-2+x^3) (1+x^3)^{2/3}}{x^3 (-2+x^3+2 x^6)} \, dx\) [1465]

Optimal. Leaf size=103 \[ -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{3} \text {RootSum}\left [1-5 \text {$\#$1}^3+2 \text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(327\) vs. \(2(103)=206\).
time = 0.43, antiderivative size = 327, normalized size of antiderivative = 3.17, number of steps used = 10, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6860, 283, 245, 1532, 384} \begin {gather*} \frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \text {ArcTan}\left (\frac {\frac {\sqrt [3]{2 \left (5-\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \text {ArcTan}\left (\frac {\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (4 x^3-\sqrt {17}+1\right )}{6 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \log \left (4 x^3+\sqrt {17}+1\right )}{6 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \log \left (\frac {\sqrt [3]{5-\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{x^3+1}\right )}{2 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\frac {\sqrt [3]{5+\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{x^3+1}\right )}{2 \sqrt {17}}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 + (((-19 + 5*Sqrt[17])/2)^(1/3)*ArcTan[(1 + ((2*(5 - Sqrt[17]))^(1/3)*x)/(1 + x^3)^(1
/3))/Sqrt[3]])/Sqrt[51] + (((19 + 5*Sqrt[17])/2)^(1/3)*ArcTan[(1 + ((2*(5 + Sqrt[17]))^(1/3)*x)/(1 + x^3)^(1/3
))/Sqrt[3]])/Sqrt[51] + (((19 + 5*Sqrt[17])/2)^(1/3)*Log[1 - Sqrt[17] + 4*x^3])/(6*Sqrt[17]) + (((-19 + 5*Sqrt
[17])/2)^(1/3)*Log[1 + Sqrt[17] + 4*x^3])/(6*Sqrt[17]) - (((-19 + 5*Sqrt[17])/2)^(1/3)*Log[((5 - Sqrt[17])^(1/
3)*x)/2^(2/3) - (1 + x^3)^(1/3)])/(2*Sqrt[17]) - (((19 + 5*Sqrt[17])/2)^(1/3)*Log[((5 + Sqrt[17])^(1/3)*x)/2^(
2/3) - (1 + x^3)^(1/3)])/(2*Sqrt[17])

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 1532

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol
] :> Dist[e*(f^n/c), Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^n/c, Int[(f*x)^(m - n)*(d + e*x^n)
^(q - 1)*(Simp[a*e - (c*d - b*e)*x^n, x]/(a + b*x^n + c*x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && E
qQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2*n
- 1]

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*} \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{x^3}-\frac {2 x^3 \left (1+x^3\right )^{2/3}}{-2+x^3+2 x^6}\right ) \, dx\\ &=-\left (2 \int \frac {x^3 \left (1+x^3\right )^{2/3}}{-2+x^3+2 x^6} \, dx\right )+\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \frac {-2-x^3}{\sqrt [3]{1+x^3} \left (-2+x^3+2 x^6\right )} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \left (\frac {-1-\frac {7}{\sqrt {17}}}{\sqrt [3]{1+x^3} \left (1-\sqrt {17}+4 x^3\right )}+\frac {-1+\frac {7}{\sqrt {17}}}{\sqrt [3]{1+x^3} \left (1+\sqrt {17}+4 x^3\right )}\right ) \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{17} \left (-17+7 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (1+\sqrt {17}+4 x^3\right )} \, dx-\frac {1}{17} \left (17+7 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (1-\sqrt {17}+4 x^3\right )} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{17} \left (-17+7 \sqrt {17}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {17}-\left (-3+\sqrt {17}\right ) x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{17} \left (17+7 \sqrt {17}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {17}-\left (-3-\sqrt {17}\right ) x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\sqrt [3]{71+17 \sqrt {17}} \text {Subst}\left (\int \frac {1}{-\sqrt [3]{-1+\sqrt {17}}+\sqrt [3]{3+\sqrt {17}} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}-\frac {\sqrt [3]{71+17 \sqrt {17}} \text {Subst}\left (\int \frac {-2 \sqrt [3]{-1+\sqrt {17}}-\sqrt [3]{3+\sqrt {17}} x}{\left (-1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )} x+\left (3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{-289+71 \sqrt {17}} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{-3+\sqrt {17}} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}+\frac {\sqrt [3]{-289+71 \sqrt {17}} \text {Subst}\left (\int \frac {2 \sqrt [3]{1+\sqrt {17}}+\sqrt [3]{-3+\sqrt {17}} x}{\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )} x+\left (-3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\frac {\sqrt [3]{-3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\sqrt [3]{-1+\sqrt {17}}-\frac {\sqrt [3]{3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{459-109 \sqrt {17}} \text {Subst}\left (\int \frac {1}{\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )} x+\left (-3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{34^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \text {Subst}\left (\int \frac {\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )}+2 \left (-3+\sqrt {17}\right )^{2/3} x}{\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )} x+\left (-3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{6\ 17^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \text {Subst}\left (\int \frac {\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}+2 \left (3+\sqrt {17}\right )^{2/3} x}{\left (-1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )} x+\left (3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{6 \sqrt {17}}+\frac {\sqrt [3]{109+27 \sqrt {17}} \text {Subst}\left (\int \frac {1}{\left (-1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )} x+\left (3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{2^{2/3} \sqrt {17}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\left (1+\sqrt {17}\right )^{2/3}+\frac {\left (-3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7-\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\frac {\sqrt [3]{-3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\sqrt [3]{-1+\sqrt {17}}-\frac {\sqrt [3]{3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\left (-1+\sqrt {17}\right )^{2/3}+\frac {\left (3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {\sqrt [3]{10-2 \sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt {17}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2 \left (5-\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3} 17^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\left (1+\sqrt {17}\right )^{2/3}+\frac {\left (-3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7-\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\frac {\sqrt [3]{-3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\sqrt [3]{-1+\sqrt {17}}-\frac {\sqrt [3]{3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\left (-1+\sqrt {17}\right )^{2/3}+\frac {\left (3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6 \sqrt {17}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 103, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{3} \text {RootSum}\left [1-5 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 256.96, size = 7525, normalized size = 73.06

method	result	size
trager	\(\text {Expression too large to display}\)	\(7525\)
risch	\(\text {Expression too large to display}\)	\(8344\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^3-2\right )}{x^3\,\left (2\,x^6+x^3-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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