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

Optimal. Leaf size=134 \[ -3^{2/3} \log \left (3^{2/3} \sqrt [3]{x^3+1}-3 x\right )+3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{x^3+1}+\sqrt [3]{3} x}\right )+\frac {\left (x^3+1\right )^{2/3} \left (-19 x^3-4\right )}{10 x^5}+\frac {1}{2} 3^{2/3} \log \left (3^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{3} \left (x^3+1\right )^{2/3}+3 x^2\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 137, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )+3 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}-\frac {19 \left (x^3+1\right )^{2/3}}{10 x^2}+\frac {1}{2} 3^{2/3} \log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 + x^3)^(2/3))/(5*x^5) - (19*(1 + x^3)^(2/3))/(10*x^2) + 3*3^(1/6)*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(1
+ x^3)^(1/3))] - 3^(2/3)*Log[1 - (3^(1/3)*x)/(1 + x^3)^(1/3)] + (3^(2/3)*Log[1 + (3^(2/3)*x^2)/(1 + x^3)^(2/3)
+ (3^(1/3)*x)/(1 + x^3)^(1/3)])/2

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 (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {19+7 x^3}{x^3 \sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {1}{10} \int \frac {90}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-9 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-9 \operatorname {Subst}\left (\int \frac {1}{-1+3 x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-3 \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-3 \operatorname {Subst}\left (\int \frac {-2-\sqrt [3]{3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )+\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{2} 3^{2/3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{3}+2\ 3^{2/3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{2} 3^{2/3} \log \left (1+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )-\left (3\ 3^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}+3 \sqrt [6]{3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{2} 3^{2/3} \log \left (1+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 124, normalized size = 0.93 \begin {gather*} 3 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )-\frac {\left (x^3+1\right )^{2/3} \left (19 x^3+4\right )}{10 x^5}+\frac {1}{2} 3^{2/3} \left (\log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )-2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*((1 + x^3)^(2/3)*(4 + 19*x^3))/x^5 + 3*3^(1/6)*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(1 + x^3)^(1/3))] + (3^
(2/3)*(-2*Log[1 - (3^(1/3)*x)/(1 + x^3)^(1/3)] + Log[1 + (3^(2/3)*x^2)/(1 + x^3)^(2/3) + (3^(1/3)*x)/(1 + x^3)
^(1/3)]))/2

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IntegrateAlgebraic [A]  time = 0.28, size = 134, normalized size = 1.00 \begin {gather*} \frac {\left (-4-19 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )+\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4 - 19*x^3)*(1 + x^3)^(2/3))/(10*x^5) + 3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + x^3)^(1/3))] - 3^(
2/3)*Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3)] + (3^(2/3)*Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3)*(1 + x^3)^
(2/3)])/2

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fricas [B]  time = 2.35, size = 274, normalized size = 2.04 \begin {gather*} -\frac {10 \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {2 \, \sqrt {3} \left (-9\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\right )}}\right ) - 10 \, \left (-9\right )^{\frac {1}{3}} x^{5} \log \left (\frac {3 \, \left (-9\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \left (-9\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )}}{2 \, x^{3} - 1}\right ) + 5 \, \left (-9\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {9 \, \left (-9\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \left (-9\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 27 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) + 3 \, {\left (19 \, x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(10*sqrt(3)*(-9)^(1/3)*x^5*arctan(1/3*(2*sqrt(3)*(-9)^(2/3)*(14*x^7 - 5*x^4 - x)*(x^3 + 1)^(2/3) + 6*sqr
t(3)*(-9)^(1/3)*(31*x^8 + 23*x^5 + x^2)*(x^3 + 1)^(1/3) - sqrt(3)*(127*x^9 + 201*x^6 + 48*x^3 + 1))/(251*x^9 +
231*x^6 + 6*x^3 - 1)) - 10*(-9)^(1/3)*x^5*log((3*(-9)^(2/3)*(x^3 + 1)^(1/3)*x^2 - 9*(x^3 + 1)^(2/3)*x + (-9)^
(1/3)*(2*x^3 - 1))/(2*x^3 - 1)) + 5*(-9)^(1/3)*x^5*log(-(9*(-9)^(1/3)*(7*x^4 + x)*(x^3 + 1)^(2/3) - (-9)^(2/3)
*(31*x^6 + 23*x^3 + 1) - 27*(5*x^5 + 2*x^2)*(x^3 + 1)^(1/3))/(4*x^6 - 4*x^3 + 1)) + 3*(19*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 )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.95, size = 896, normalized size = 6.69 \[-\frac {19 x^{6}+23 x^{3}+4}{10 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}}+\RootOf \left (\textit {\_Z}^{3}+9\right ) \ln \left (-\frac {-3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+9\right )^{3} x^{3}-108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} x^{3}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x -7 \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-18 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{2}-\RootOf \left (\textit {\_Z}^{3}+9\right ) x^{3}-36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{3}+6 x \left (x^{3}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{3}+9\right )-36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )}{2 x^{3}-1}\right )-\ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+9\right )^{3} x^{3}-81 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} x^{3}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x +2 \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+63 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{3}+9\right ) x^{3}+108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{3}+9\right )+27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )}{2 x^{3}-1}\right ) \RootOf \left (\textit {\_Z}^{3}+9\right )-9 \ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+9\right )^{3} x^{3}-81 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} x^{3}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x +2 \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+63 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{3}+9\right ) x^{3}+108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{3}+9\right )+27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )}{2 x^{3}-1}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(19*x^6+23*x^3+4)/x^5/(x^3+1)^(1/3)+RootOf(_Z^3+9)*ln(-(-3*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+8
1*_Z^2)*RootOf(_Z^3+9)^3*x^3-108*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)^2*RootOf(_Z^3+9)^2*x^3+1
5*(x^3+1)^(2/3)*RootOf(_Z^3+9)^2*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x-7*RootOf(_Z^3+9)^2*(x^
3+1)^(1/3)*x^2-18*(x^3+1)^(1/3)*RootOf(_Z^3+9)*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^2-RootOf
(_Z^3+9)*x^3-36*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^3+6*x*(x^3+1)^(2/3)-RootOf(_Z^3+9)-36*R
ootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2))/(2*x^3-1))-ln((3*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3
+9)+81*_Z^2)*RootOf(_Z^3+9)^3*x^3-81*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)^2*RootOf(_Z^3+9)^2*x
^3+15*(x^3+1)^(2/3)*RootOf(_Z^3+9)^2*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x+2*RootOf(_Z^3+9)^2
*(x^3+1)^(1/3)*x^2+63*(x^3+1)^(1/3)*RootOf(_Z^3+9)*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^2-4*
RootOf(_Z^3+9)*x^3+108*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^3-21*x*(x^3+1)^(2/3)-RootOf(_Z^3
+9)+27*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2))/(2*x^3-1))*RootOf(_Z^3+9)-9*ln((3*RootOf(RootOf(_
Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*RootOf(_Z^3+9)^3*x^3-81*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_
Z^2)^2*RootOf(_Z^3+9)^2*x^3+15*(x^3+1)^(2/3)*RootOf(_Z^3+9)^2*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_
Z^2)*x+2*RootOf(_Z^3+9)^2*(x^3+1)^(1/3)*x^2+63*(x^3+1)^(1/3)*RootOf(_Z^3+9)*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootO
f(_Z^3+9)+81*_Z^2)*x^2-4*RootOf(_Z^3+9)*x^3+108*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^3-21*x*
(x^3+1)^(2/3)-RootOf(_Z^3+9)+27*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2))/(2*x^3-1))*RootOf(RootOf
(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^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 )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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