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

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

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Rubi [B]  time = 0.62, antiderivative size = 211, normalized size of antiderivative = 2.03, number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6728, 277, 239, 430, 429} \begin {gather*} \frac {\left (17+5 \sqrt {17}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (17-5 \sqrt {17}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {1}{4} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\left (x^3-1\right )^{2/3}}{4 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[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) + ((17 + 5*Sqrt[17])*x*(-1 + x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, x^3, (-2*x^3)/(1
- Sqrt[17])])/(34*(1 - Sqrt[17])*(1 - x^3)^(2/3)) + ((17 - 5*Sqrt[17])*x*(-1 + x^3)^(2/3)*AppellF1[1/3, -2/3,
1, 4/3, x^3, (-2*x^3)/(1 + Sqrt[17])])/(34*(1 + Sqrt[17])*(1 - x^3)^(2/3)) - ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3
))/Sqrt[3]]/(2*Sqrt[3]) + Log[-x + (-1 + x^3)^(1/3)]/4

Rule 239

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 277

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 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 6728

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 (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx &=\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 {\tan ^{-1}\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 {\tan ^{-1}\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 {\left (\left (17-5 \sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}+\frac {\left (\left (17+5 \sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (17+5 \sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (17-5 \sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\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 )\\ \end {align*}

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Mathematica [B]  time = 0.75, size = 387, normalized size = 3.72 \begin {gather*} \frac {\left (x^3-1\right )^{2/3}}{4 x^2}+\frac {-2 \sqrt [3]{199-47 \sqrt {17}} \log \left (\sqrt [3]{7-\sqrt {17}}-\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}\right )+2 \sqrt [3]{199+47 \sqrt {17}} \log \left (\sqrt [3]{7+\sqrt {17}}-\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}\right )-2 \sqrt {3} \sqrt [3]{199+47 \sqrt {17}} \tan ^{-1}\left (\frac {\frac {2\ 2^{2/3} x}{\sqrt [3]{7+\sqrt {17}} \sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+2 \sqrt {3} \sqrt [3]{199-47 \sqrt {17}} \tan ^{-1}\left (\frac {\frac {2\ 2^{2/3} x}{\sqrt [3]{-\left (\left (\sqrt {17}-7\right ) \left (x^3-1\right )\right )}}+1}{\sqrt {3}}\right )+\sqrt [3]{199-47 \sqrt {17}} \log \left (\frac {2^{2/3} \sqrt [3]{7-\sqrt {17}} x}{\sqrt [3]{x^3-1}}+\frac {2 \sqrt [3]{2} x^2}{\left (x^3-1\right )^{2/3}}+\left (7-\sqrt {17}\right )^{2/3}\right )-\sqrt [3]{199+47 \sqrt {17}} \log \left (\frac {2^{2/3} \sqrt [3]{7+\sqrt {17}} x}{\sqrt [3]{x^3-1}}+\frac {2 \sqrt [3]{2} x^2}{\left (x^3-1\right )^{2/3}}+\left (7+\sqrt {17}\right )^{2/3}\right )}{24\ 2^{2/3} \sqrt {17}} \end {gather*}

Warning: Unable to verify antiderivative.

[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) + (-2*Sqrt[3]*(199 + 47*Sqrt[17])^(1/3)*ArcTan[(1 + (2*2^(2/3)*x)/((7 + Sqrt[17])^(1/
3)*(-1 + x^3)^(1/3)))/Sqrt[3]] + 2*Sqrt[3]*(199 - 47*Sqrt[17])^(1/3)*ArcTan[(1 + (2*2^(2/3)*x)/(-((-7 + Sqrt[1
7])*(-1 + x^3)))^(1/3))/Sqrt[3]] - 2*(199 - 47*Sqrt[17])^(1/3)*Log[(7 - Sqrt[17])^(1/3) - (2^(2/3)*x)/(-1 + x^
3)^(1/3)] + 2*(199 + 47*Sqrt[17])^(1/3)*Log[(7 + Sqrt[17])^(1/3) - (2^(2/3)*x)/(-1 + x^3)^(1/3)] + (199 - 47*S
qrt[17])^(1/3)*Log[(7 - Sqrt[17])^(2/3) + (2*2^(1/3)*x^2)/(-1 + x^3)^(2/3) + (2^(2/3)*(7 - Sqrt[17])^(1/3)*x)/
(-1 + x^3)^(1/3)] - (199 + 47*Sqrt[17])^(1/3)*Log[(7 + Sqrt[17])^(2/3) + (2*2^(1/3)*x^2)/(-1 + x^3)^(2/3) + (2
^(2/3)*(7 + Sqrt[17])^(1/3)*x)/(-1 + x^3)^(1/3)])/(24*2^(2/3)*Sqrt[17])

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IntegrateAlgebraic [A]  time = 0.20, size = 104, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-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

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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^3-1)^(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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 275.84, size = 7989, normalized size = 76.82

method	result	size
risch	\(\text {Expression too large to display}\)	\(7989\)
trager	\(\text {Expression too large to display}\)	\(10402\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(2/3)*(x^3+2)/x^3/(x^6+x^3-4),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*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 (x^6+x^3-4\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 + x^6 - 4)),x)

[Out]

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

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sympy [F(-1)]  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**3-1)**(2/3)*(x**3+2)/x**3/(x**6+x**3-4),x)

[Out]

Timed out

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