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

Optimal. Leaf size=150 \[ \frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{1-x^3}-x\right )}{12\ 2^{2/3}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{1-x^3}+x}\right )}{4\ 2^{2/3} \sqrt {3}}+\frac {\left (1-x^3\right )^{2/3} \left (29 x^3-4\right )}{40 x^5}-\frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{1-x^3} x+2^{2/3} \left (1-x^3\right )^{2/3}+x^2\right )}{24\ 2^{2/3}} \]

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Rubi [A]  time = 0.16, antiderivative size = 157, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {5 \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}-\frac {5 \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3}}-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}-\frac {5 \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac {x^2}{\left (1-x^3\right )^{2/3}}+2^{2/3}\right )}{24\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*(1 - x^3)^(2/3)/x^5 + (29*(1 - x^3)^(2/3))/(40*x^2) - (5*ArcTan[(1 + (2^(2/3)*x)/(1 - x^3)^(1/3))/Sqrt[3
]])/(4*2^(2/3)*Sqrt[3]) + (5*Log[2^(1/3) - x/(1 - x^3)^(1/3)])/(12*2^(2/3)) - (5*Log[2^(2/3) + x^2/(1 - x^3)^(
2/3) + (2^(1/3)*x)/(1 - x^3)^(1/3)])/(24*2^(2/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 377

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

Rule 580

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}-\frac {1}{10} \int \frac {-29+31 x^3}{x^3 \sqrt [3]{1-x^3} \left (-2+3 x^3\right )} \, dx\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}-\frac {1}{40} \int -\frac {50}{\sqrt [3]{1-x^3} \left (-2+3 x^3\right )} \, dx\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5}{4} \int \frac {1}{\sqrt [3]{1-x^3} \left (-2+3 x^3\right )} \, dx\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{-2+x^3} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{2}+x} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}+\frac {5 \operatorname {Subst}\left (\int \frac {-2 \sqrt [3]{2}-x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5 \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}-\frac {5 \operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2 x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{24\ 2^{2/3}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{8 \sqrt [3]{2}}\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5 \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}-\frac {5 \log \left (2^{2/3}+\frac {x^2}{\left (1-x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{24\ 2^{2/3}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}\right )}{4\ 2^{2/3}}\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}-\frac {5 \tan ^{-1}\left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3}}+\frac {5 \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}-\frac {5 \log \left (2^{2/3}+\frac {x^2}{\left (1-x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{24\ 2^{2/3}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 129, normalized size = 0.86 \begin {gather*} \left (\frac {29}{40 x^2}-\frac {1}{10 x^5}\right ) \left (1-x^3\right )^{2/3}-\frac {5 \left (-2 \log \left (2-\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+\frac {\sqrt [3]{2} x^2}{\left (x^3-1\right )^{2/3}}+2\right )\right )}{24\ 2^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-1/10*1/x^5 + 29/(40*x^2))*(1 - x^3)^(2/3) - (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 = 0.35, size = 150, normalized size = 1.00 \begin {gather*} \frac {\left (1-x^3\right )^{2/3} \left (-4+29 x^3\right )}{40 x^5}-\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)*(-1 + 4*x^3))/(x^6*(-2 + 3*x^3)),x]

[Out]

((1 - x^3)^(2/3)*(-4 + 29*x^3))/(40*x^5) - (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 = 3.21, size = 278, normalized size = 1.85 \begin {gather*} \frac {100 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (3 \, x^{4} - 2 \, x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \sqrt {3} {\left (27 \, x^{9} - 72 \, x^{6} + 36 \, x^{3} + 8\right )} + 12 \, \sqrt {3} {\left (9 \, x^{8} - 6 \, x^{5} - 4 \, x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (27 \, x^{9} - 36 \, x^{3} + 8\right )}}\right ) + 50 \cdot 4^{\frac {2}{3}} x^{5} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} {\left (3 \, x^{3} - 2\right )} - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{3 \, x^{3} - 2}\right ) - 25 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - 4^{\frac {1}{3}} {\left (9 \, x^{6} - 6 \, x^{3} - 4\right )} - 6 \, {\left (3 \, x^{5} - 4 \, x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{9 \, x^{6} - 12 \, x^{3} + 4}\right ) + 36 \, {\left (29 \, x^{3} - 4\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{1440 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1440*(100*4^(1/6)*sqrt(3)*x^5*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(3*x^4 - 2*x)*(-x^3 + 1)^(2/3) - 4^(1/3
)*sqrt(3)*(27*x^9 - 72*x^6 + 36*x^3 + 8) + 12*sqrt(3)*(9*x^8 - 6*x^5 - 4*x^2)*(-x^3 + 1)^(1/3))/(27*x^9 - 36*x
^3 + 8)) + 50*4^(2/3)*x^5*log(-(6*4^(1/3)*(-x^3 + 1)^(1/3)*x^2 - 4^(2/3)*(3*x^3 - 2) - 12*(-x^3 + 1)^(2/3)*x)/
(3*x^3 - 2)) - 25*4^(2/3)*x^5*log((6*4^(2/3)*(-x^3 + 1)^(2/3)*x - 4^(1/3)*(9*x^6 - 6*x^3 - 4) - 6*(3*x^5 - 4*x
^2)*(-x^3 + 1)^(1/3))/(9*x^6 - 12*x^3 + 4)) + 36*(29*x^3 - 4)*(-x^3 + 1)^(2/3))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 18.04, size = 791, normalized size = 5.27

method	result	size
risch	\(-\frac {29 x^{6}-33 x^{3}+4}{40 x^{5} \left (-x^{3}+1\right )^{\frac {1}{3}}}+\frac {5 \RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (-\frac {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 )^{3} x^{3}+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 )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+9 \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 -9 \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}+2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+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 ) x^{3}+3 \left (-x^{3}+1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}-2\right )-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 )}{3 x^{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}-18 \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 -3 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-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 ) x^{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 )}{3 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}-18 \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 -3 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-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 ) x^{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 )}{3 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}\)	\(791\)
trager	\(\text {Expression too large to display}\)	\(1147\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(29*x^6-33*x^3+4)/x^5/(-x^3+1)^(1/3)+5/24*RootOf(_Z^3-2)*ln(-(3*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3
-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x^3+9*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x^
3+9*(-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-9*(-x^3+1)^(1/3)*Ro
otOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)*x^2+2*RootOf(_Z^3-2)*x^3+6*RootOf(RootOf(_Z^
3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3+3*(-x^3+1)^(2/3)*x-2*RootOf(_Z^3-2)-6*RootOf(RootOf(_Z^3-2)^2+6*_Z*Roo
tOf(_Z^3-2)+36*_Z^2))/(3*x^3-2))-5/24*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^3-18*(-x^3+1)^(2/3)*Root
Of(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2*x-3*RootOf(_Z^3-2)^2*(-x^3+1)^(1/3)*x^2-2*Ro
otOf(_Z^3-2)*x^3-6*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3+4*RootOf(_Z^3-2)+12*RootOf(RootOf(
_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(3*x^3-2))*RootOf(_Z^3-2)-5/4*ln(-(6*RootOf(RootOf(_Z^3-2)^2+6*_Z*Root
Of(_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^3-18*(-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-3*RootOf(_
Z^3-2)^2*(-x^3+1)^(1/3)*x^2-2*RootOf(_Z^3-2)*x^3-6*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3+4*
RootOf(_Z^3-2)+12*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(3*x^3-2))*RootOf(RootOf(_Z^3-2)^2+6*_
Z*RootOf(_Z^3-2)+36*_Z^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-x^3\right )}^{2/3}\,\left (4\,x^3-1\right )}{x^6\,\left (3\,x^3-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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