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

Optimal. Leaf size=173 \[ -\frac {\log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3-1}-3 x\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+\sqrt [3]{3} x}\right )}{4\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3} \left (-x^3-4\right )}{40 x^5}+\frac {\log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3-1} x+2^{2/3} \sqrt [3]{3} \left (x^3-1\right )^{2/3}+3 x^2\right )}{8\ 2^{2/3} \sqrt [3]{3}} \]

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Rubi [A]  time = 0.19, antiderivative size = 168, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3-1}}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {2^{2/3} x}{\sqrt [6]{3} \sqrt [3]{x^3-1}}+\frac {1}{\sqrt {3}}\right )}{4\ 2^{2/3}}-\frac {\left (x^3-1\right )^{2/3}}{10 x^5}-\frac {\left (x^3-1\right )^{2/3}}{40 x^2}+\frac {\log \left (\frac {\sqrt [3]{6} x}{\sqrt [3]{x^3-1}}+\frac {3^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+2^{2/3}\right )}{8\ 2^{2/3} \sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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+x^3\right )}{x^6 \left (2+x^3\right )} \, dx &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {1}{10} \int \frac {-1+7 x^3}{x^3 \sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {1}{40} \int \frac {30}{\sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {3}{4} \int \frac {1}{\sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{2-3 x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}-\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {2 \sqrt [3]{2}+\sqrt [3]{3} x}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}-\frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{8 \sqrt [3]{2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{6}+2\ 3^{2/3} x}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3} \sqrt [3]{3}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}-\frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {\log \left (2^{2/3}+\frac {3^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{6} x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3} \sqrt [3]{3}}-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3}}-\frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {\log \left (2^{2/3}+\frac {3^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{6} x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3} \sqrt [3]{3}}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 156, normalized size = 0.90 \begin {gather*} \frac {1}{240} \left (5 \sqrt [3]{2} \sqrt [6]{3} \left (6 \tan ^{-1}\left (\frac {2^{2/3} x}{\sqrt [6]{3} \sqrt [3]{1-x^3}}+\frac {1}{\sqrt {3}}\right )+\sqrt {3} \left (\log \left (\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{1-x^3}}+\frac {\sqrt [3]{2} 3^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+2\right )-2 \log \left (2-\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{1-x^3}}\right )\right )\right )-\frac {6 \left (x^3-1\right )^{2/3} \left (x^3+4\right )}{x^5}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-6*(-1 + x^3)^(2/3)*(4 + x^3))/x^5 + 5*2^(1/3)*3^(1/6)*(6*ArcTan[1/Sqrt[3] + (2^(2/3)*x)/(3^(1/6)*(1 - x^3)^
(1/3))] + Sqrt[3]*(-2*Log[2 - (2^(2/3)*3^(1/3)*x)/(1 - x^3)^(1/3)] + Log[2 + (2^(1/3)*3^(2/3)*x^2)/(1 - x^3)^(
2/3) + (2^(2/3)*3^(1/3)*x)/(1 - x^3)^(1/3)])))/240

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IntegrateAlgebraic [A]  time = 0.39, size = 173, normalized size = 1.00 \begin {gather*} \frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}-\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )}{8\ 2^{2/3} \sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 4.09, size = 289, normalized size = 1.67 \begin {gather*} \frac {10 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {18 \cdot 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 2\right )} - 36 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 2}\right ) - 5 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (55 \, x^{6} - 50 \, x^{3} + 4\right )} - 18 \, {\left (7 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{3} + 4}\right ) - 60 \cdot 12^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 36 \, \left (-1\right )^{\frac {1}{3}} {\left (55 \, x^{8} - 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{6} + 204 \, x^{3} - 8\right )}\right )}}{6 \, {\left (487 \, x^{9} - 480 \, x^{6} + 12 \, x^{3} + 8\right )}}\right ) - 36 \, {\left (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)*(x^3+1)/x^6/(x^3+2),x, algorithm="fricas")

[Out]

1/1440*(10*12^(2/3)*(-1)^(1/3)*x^5*log(-(18*12^(1/3)*(-1)^(2/3)*(x^3 - 1)^(1/3)*x^2 + 12^(2/3)*(-1)^(1/3)*(x^3
+ 2) - 36*(x^3 - 1)^(2/3)*x)/(x^3 + 2)) - 5*12^(2/3)*(-1)^(1/3)*x^5*log(-(6*12^(2/3)*(-1)^(1/3)*(4*x^4 - x)*(
x^3 - 1)^(2/3) - 12^(1/3)*(-1)^(2/3)*(55*x^6 - 50*x^3 + 4) - 18*(7*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 + 4*x^3
+ 4)) - 60*12^(1/6)*(-1)^(1/3)*x^5*arctan(1/6*12^(1/6)*(12*12^(2/3)*(-1)^(2/3)*(4*x^7 + 7*x^4 - 2*x)*(x^3 - 1)
^(2/3) + 36*(-1)^(1/3)*(55*x^8 - 50*x^5 + 4*x^2)*(x^3 - 1)^(1/3) - 12^(1/3)*(377*x^9 - 600*x^6 + 204*x^3 - 8))
/(487*x^9 - 480*x^6 + 12*x^3 + 8)) - 36*(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 (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (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)*(x^3+1)/x^6/(x^3+2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 16.27, size = 888, normalized size = 5.13

method	result	size
risch	\(-\frac {x^{6}+3 x^{3}-4}{40 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}-\frac {\ln \left (-\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}-162 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+42 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x +\left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}+144 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}+270 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{3}-48 x \left (x^{3}-1\right )^{\frac {2}{3}}+4 \RootOf \left (\textit {\_Z}^{3}+18\right )-108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )}{x^{3}+2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )}{24}-\frac {3 \ln \left (-\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}-162 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+42 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x +\left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}+144 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}+270 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{3}-48 x \left (x^{3}-1\right )^{\frac {2}{3}}+4 \RootOf \left (\textit {\_Z}^{3}+18\right )-108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )}{x^{3}+2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{3}+18\right ) \ln \left (\frac {-3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}-135 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+21 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x -4 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}-9 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}-90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{3}+18\right )+90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )}{x^{3}+2}\right )}{24}\)	\(888\)
trager	\(\text {Expression too large to display}\)	\(1113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(x^6+3*x^3-4)/x^5/(x^3-1)^(1/3)-1/24*ln(-(6*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*Roo
tOf(_Z^3+18)^3*x^3-162*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)^2*RootOf(_Z^3+18)^2*x^3+42*(x^
3-1)^(2/3)*RootOf(_Z^3+18)^2*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x+(x^3-1)^(1/3)*RootOf(_
Z^3+18)^2*x^2+144*(x^3-1)^(1/3)*RootOf(_Z^3+18)*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^2-1
0*RootOf(_Z^3+18)*x^3+270*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^3-48*x*(x^3-1)^(2/3)+4*Ro
otOf(_Z^3+18)-108*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2))/(x^3+2))*RootOf(_Z^3+18)-3/4*ln(-(
6*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootOf(_Z^3+18)^3*x^3-162*RootOf(RootOf(_Z^3+18)^2+
18*_Z*RootOf(_Z^3+18)+324*_Z^2)^2*RootOf(_Z^3+18)^2*x^3+42*(x^3-1)^(2/3)*RootOf(_Z^3+18)^2*RootOf(RootOf(_Z^3+
18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x+(x^3-1)^(1/3)*RootOf(_Z^3+18)^2*x^2+144*(x^3-1)^(1/3)*RootOf(_Z^3+18)*
RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^2-10*RootOf(_Z^3+18)*x^3+270*RootOf(RootOf(_Z^3+18)
^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^3-48*x*(x^3-1)^(2/3)+4*RootOf(_Z^3+18)-108*RootOf(RootOf(_Z^3+18)^2+18*_Z
*RootOf(_Z^3+18)+324*_Z^2))/(x^3+2))*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)+1/24*RootOf(_Z^3
+18)*ln((-3*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootOf(_Z^3+18)^3*x^3-135*RootOf(RootOf(_
Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)^2*RootOf(_Z^3+18)^2*x^3+21*(x^3-1)^(2/3)*RootOf(_Z^3+18)^2*RootOf(Ro
otOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x-4*(x^3-1)^(1/3)*RootOf(_Z^3+18)^2*x^2-9*(x^3-1)^(1/3)*RootOf
(_Z^3+18)*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^2-2*RootOf(_Z^3+18)*x^3-90*RootOf(RootOf(
_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^3+3*x*(x^3-1)^(2/3)+2*RootOf(_Z^3+18)+90*RootOf(RootOf(_Z^3+18)^2
+18*_Z*RootOf(_Z^3+18)+324*_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^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (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)*(x^3+1)/x^6/(x^3+2),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

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