3.21.35 \(\int \frac {(-1+x^3)^{2/3} (4+x^3+x^6)}{x^9 (-2+x^3)} \, dx\)

Optimal. Leaf size=145 \[ \frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )}{12\ 2^{2/3}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{4\ 2^{2/3} \sqrt {3}}-\frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{24\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3} \left (7 x^6+8 x^3+10\right )}{40 x^8} \]

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Rubi [C]  time = 0.35, antiderivative size = 143, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6725, 271, 264, 277, 239, 430, 429} \begin {gather*} -\frac {5 x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{8 \left (1-x^3\right )^{2/3}}+\frac {5}{8} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {5 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {\left (x^3-1\right )^{5/3}}{4 x^8}-\frac {9 \left (x^3-1\right )^{5/3}}{20 x^5}+\frac {5 \left (x^3-1\right )^{2/3}}{8 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((-1 + x^3)^(2/3)*(4 + x^3 + x^6))/(x^9*(-2 + x^3)),x]

[Out]

(5*(-1 + x^3)^(2/3))/(8*x^2) - (-1 + x^3)^(5/3)/(4*x^8) - (9*(-1 + x^3)^(5/3))/(20*x^5) - (5*x*(-1 + x^3)^(2/3
)*AppellF1[1/3, -2/3, 1, 4/3, x^3, x^3/2])/(8*(1 - x^3)^(2/3)) - (5*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3
]])/(4*Sqrt[3]) + (5*Log[-x + (-1 + x^3)^(1/3)])/8

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 264

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 271

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

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 6725

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*} \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3+x^6\right )}{x^9 \left (-2+x^3\right )} \, dx &=\int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{x^9}-\frac {3 \left (-1+x^3\right )^{2/3}}{2 x^6}-\frac {5 \left (-1+x^3\right )^{2/3}}{4 x^3}+\frac {5 \left (-1+x^3\right )^{2/3}}{4 \left (-2+x^3\right )}\right ) \, dx\\ &=-\left (\frac {5}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )+\frac {5}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx-\frac {3}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx-2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^9} \, dx\\ &=\frac {5 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}-\frac {3 \left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {3}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx-\frac {5}{4} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {\left (5 \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-2+x^3} \, dx}{4 \left (1-x^3\right )^{2/3}}\\ &=\frac {5 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}-\frac {9 \left (-1+x^3\right )^{5/3}}{20 x^5}-\frac {5 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{8 \left (1-x^3\right )^{2/3}}-\frac {5 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {5}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 138, normalized size = 0.95 \begin {gather*} \frac {\left (x^3-1\right )^{2/3} \left (7 x^6+8 x^3+10\right )}{40 x^8}-\frac {5 \left (-2 \log \left (2-\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+\frac {\sqrt [3]{2} x^2}{\left (1-x^3\right )^{2/3}}+2\right )\right )}{24\ 2^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 + x^3)^(2/3)*(4 + x^3 + x^6))/(x^9*(-2 + x^3)),x]

[Out]

((-1 + x^3)^(2/3)*(10 + 8*x^3 + 7*x^6))/(40*x^8) - (5*(2*Sqrt[3]*ArcTan[(1 + (2^(2/3)*x)/(1 - x^3)^(1/3))/Sqrt
[3]] - 2*Log[2 - (2^(2/3)*x)/(1 - x^3)^(1/3)] + Log[2 + (2^(1/3)*x^2)/(1 - x^3)^(2/3) + (2^(2/3)*x)/(1 - x^3)^
(1/3)]))/(24*2^(2/3))

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IntegrateAlgebraic [A]  time = 0.34, size = 145, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (10+8 x^3+7 x^6\right )}{40 x^8}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3} \sqrt {3}}+\frac {5 \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{12\ 2^{2/3}}-\frac {5 \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{24\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 + x^3 + x^6))/(x^9*(-2 + x^3)),x]

[Out]

((-1 + x^3)^(2/3)*(10 + 8*x^3 + 7*x^6))/(40*x^8) - (5*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))])/(4
*2^(2/3)*Sqrt[3]) + (5*Log[-x + 2^(1/3)*(-1 + x^3)^(1/3)])/(12*2^(2/3)) - (5*Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1
/3) + 2^(2/3)*(-1 + x^3)^(2/3)])/(24*2^(2/3))

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fricas [B]  time = 3.36, size = 277, normalized size = 1.91 \begin {gather*} \frac {100 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{8} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, \sqrt {3} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 50 \cdot 4^{\frac {2}{3}} x^{8} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 25 \cdot 4^{\frac {2}{3}} x^{8} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) + 36 \, {\left (7 \, x^{6} + 8 \, x^{3} + 10\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{1440 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+x^3+4)/x^9/(x^3-2),x, algorithm="fricas")

[Out]

1/1440*(100*4^(1/6)*sqrt(3)*x^8*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(2*x^7 - 5*x^4 + 2*x)*(x^3 - 1)^(2/3) +
4^(1/3)*sqrt(3)*(91*x^9 - 168*x^6 + 84*x^3 - 8) + 12*sqrt(3)*(19*x^8 - 22*x^5 + 4*x^2)*(x^3 - 1)^(1/3))/(53*x
^9 - 48*x^6 - 12*x^3 + 8)) + 50*4^(2/3)*x^8*log((6*4^(1/3)*(x^3 - 1)^(1/3)*x^2 + 4^(2/3)*(x^3 - 2) - 12*(x^3 -
1)^(2/3)*x)/(x^3 - 2)) - 25*4^(2/3)*x^8*log((6*4^(2/3)*(2*x^4 - x)*(x^3 - 1)^(2/3) + 4^(1/3)*(19*x^6 - 22*x^3
+ 4) + 6*(5*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 - 4*x^3 + 4)) + 36*(7*x^6 + 8*x^3 + 10)*(x^3 - 1)^(2/3))/x^8

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+x^3+4)/x^9/(x^3-2),x, algorithm="giac")

[Out]

integrate((x^6 + x^3 + 4)*(x^3 - 1)^(2/3)/((x^3 - 2)*x^9), x)

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maple [C]  time = 16.63, size = 744, normalized size = 5.13

method	result	size
risch	\(\frac {7 x^{9}+x^{6}+2 x^{3}-10}{40 x^{8} \left (x^{3}-1\right )^{\frac {1}{3}}}-\frac {5 \ln \left (\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-6 x \left (x^{3}-1\right )^{\frac {2}{3}}+4 \RootOf \left (\textit {\_Z}^{3}-2\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )}{24}-\frac {5 \ln \left (\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-6 x \left (x^{3}-1\right )^{\frac {2}{3}}+4 \RootOf \left (\textit {\_Z}^{3}-2\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{4}+\frac {5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \ln \left (-\frac {9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+30 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +4 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+30 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}+9 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}+8 x \left (x^{3}-1\right )^{\frac {2}{3}}-6 \RootOf \left (\textit {\_Z}^{3}-2\right )-24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-2}\right )}{4}\)	\(744\)
trager	\(\text {Expression too large to display}\)	\(1120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(2/3)*(x^6+x^3+4)/x^9/(x^3-2),x,method=_RETURNVERBOSE)

[Out]

1/40*(7*x^9+x^6+2*x^3-10)/x^8/(x^3-1)^(1/3)-5/24*ln((6*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*Ro
otOf(_Z^3-2)^3*x^3+18*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-3*RootOf(_Z^
3-2)^2*(x^3-1)^(1/3)*x^2-4*RootOf(_Z^3-2)*x^3-12*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3-6*x*
(x^3-1)^(2/3)+4*RootOf(_Z^3-2)+12*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^3-2))*RootOf(_Z^3-2
)-5/4*ln((6*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x^3+18*RootOf(RootOf(_Z^3-2)
^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-3*RootOf(_Z^3-2)^2*(x^3-1)^(1/3)*x^2-4*RootOf(_Z^3-2)*x
^3-12*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3-6*x*(x^3-1)^(2/3)+4*RootOf(_Z^3-2)+12*RootOf(Ro
otOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^3-2))*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)+5/4
*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*ln(-(9*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z
^2)*RootOf(_Z^3-2)^3*x^3+36*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+30*(x^
3-1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2*x+4*RootOf(_Z^3-2)^2*(x^3-1)^
(1/3)*x^2+30*(x^3-1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)*x^2+9*RootOf(_Z
^3-2)*x^3+36*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3+8*x*(x^3-1)^(2/3)-6*RootOf(_Z^3-2)-24*Ro
otOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^3-2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+x^3+4)/x^9/(x^3-2),x, algorithm="maxima")

[Out]

integrate((x^6 + x^3 + 4)*(x^3 - 1)^(2/3)/((x^3 - 2)*x^9), 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^6+x^3+4\right )}{x^9\,\left (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 + x^6 + 4))/(x^9*(x^3 - 2)),x)

[Out]

int(((x^3 - 1)^(2/3)*(x^3 + x^6 + 4))/(x^9*(x^3 - 2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + x^{3} + 4\right )}{x^{9} \left (x^{3} - 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)**(2/3)*(x**6+x**3+4)/x**9/(x**3-2),x)

[Out]

Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x**6 + x**3 + 4)/(x**9*(x**3 - 2)), x)

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