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

Optimal. Leaf size=149 \[ -\frac {2}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{3 x^3+1}-2 x\right )+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{3 x^3+1}+x}\right )}{\sqrt {3}}+\frac {\left (3 x^3+1\right )^{2/3} \left (1-2 x^3\right )}{5 x^5}+\frac {1}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{3 x^3+1} x+\sqrt [3]{2} \left (3 x^3+1\right )^{2/3}+2 x^2\right ) \]

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Rubi [A]  time = 0.18, antiderivative size = 158, normalized size of antiderivative = 1.06, 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*} -\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{3 x^3+1}}\right )+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{3 x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (3 x^3+1\right )^{2/3}}{5 x^5}-\frac {2 \left (3 x^3+1\right )^{2/3}}{5 x^2}+\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{3 x^3+1}}+\frac {2^{2/3} x^2}{\left (3 x^3+1\right )^{2/3}}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.27, size = 130, normalized size = 0.87 \begin {gather*} \frac {1}{3} 2^{2/3} \left (-2 \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+3}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+3}}+\frac {2^{2/3} x^2}{\left (x^3+3\right )^{2/3}}+1\right )\right )+\left (3 x^3+1\right )^{2/3} \left (\frac {1}{5 x^5}-\frac {2}{5 x^2}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(1/(5*x^5) - 2/(5*x^2))*(1 + 3*x^3)^(2/3) + (2^(2/3)*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*x)/(3 + x^3)^(1/3))/Sqr
t[3]] - 2*Log[1 - (2^(1/3)*x)/(3 + x^3)^(1/3)] + Log[1 + (2^(2/3)*x^2)/(3 + x^3)^(2/3) + (2^(1/3)*x)/(3 + x^3)
^(1/3)]))/3

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IntegrateAlgebraic [A]  time = 0.38, size = 149, normalized size = 1.00 \begin {gather*} \frac {\left (1-2 x^3\right ) \left (1+3 x^3\right )^{2/3}}{5 x^5}+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+3 x^3}}\right )}{\sqrt {3}}-\frac {2}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+3 x^3}\right )+\frac {1}{3} 2^{2/3} \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+3 x^3}+\sqrt [3]{2} \left (1+3 x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 2.92, size = 279, normalized size = 1.87 \begin {gather*} \frac {10 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (7 \, x^{7} + 8 \, x^{4} + x\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} - 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (55 \, x^{8} + 20 \, x^{5} + x^{2}\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (433 \, x^{9} + 255 \, x^{6} + 39 \, x^{3} + 1\right )}}{3 \, {\left (323 \, x^{9} + 105 \, x^{6} - 3 \, x^{3} - 1\right )}}\right ) + 10 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} x - \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) - 5 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (55 \, x^{6} + 20 \, x^{3} + 1\right )} - 24 \, {\left (4 \, x^{5} + x^{2}\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 9 \, {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{3} - 1\right )}}{45 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(10*sqrt(3)*(-4)^(1/3)*x^5*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(7*x^7 + 8*x^4 + x)*(3*x^3 + 1)^(2/3) - 6*sqr
t(3)*(-4)^(1/3)*(55*x^8 + 20*x^5 + x^2)*(3*x^3 + 1)^(1/3) + sqrt(3)*(433*x^9 + 255*x^6 + 39*x^3 + 1))/(323*x^9
+ 105*x^6 - 3*x^3 - 1)) + 10*(-4)^(1/3)*x^5*log(-(3*(-4)^(2/3)*(3*x^3 + 1)^(1/3)*x^2 - 6*(3*x^3 + 1)^(2/3)*x
- (-4)^(1/3)*(x^3 + 1))/(x^3 + 1)) - 5*(-4)^(1/3)*x^5*log(-(6*(-4)^(1/3)*(7*x^4 + x)*(3*x^3 + 1)^(2/3) - (-4)^
(2/3)*(55*x^6 + 20*x^3 + 1) - 24*(4*x^5 + x^2)*(3*x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) - 9*(3*x^3 + 1)^(2/3)*(2*
x^3 - 1))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 18.03, size = 643, normalized size = 4.32

method	result	size
risch	\(-\frac {6 x^{6}-x^{3}-1}{5 x^{5} \left (3 x^{3}+1\right )^{\frac {1}{3}}}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {15 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+24 \left (3 x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +8 \left (3 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+42 \left (3 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{2}+15 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-2 \left (3 x^{3}+1\right )^{\frac {2}{3}} x +5 \RootOf \left (\textit {\_Z}^{3}+4\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )+\frac {2 \RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+45 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+12 \left (3 x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +4 \left (3 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+3 \left (3 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{2}-5 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-75 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-7 \left (3 x^{3}+1\right )^{\frac {2}{3}} x -\RootOf \left (\textit {\_Z}^{3}+4\right )-15 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}\)	\(643\)
trager	\(\text {Expression too large to display}\)	\(1164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(6*x^6-x^3-1)/x^5/(3*x^3+1)^(1/3)+4*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*ln((15*RootOf(Ro
otOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3+36*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4
)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+24*(3*x^3+1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*Root
Of(_Z^3+4)^2*x+8*(3*x^3+1)^(1/3)*RootOf(_Z^3+4)^2*x^2+42*(3*x^3+1)^(1/3)*RootOf(_Z^3+4)*RootOf(RootOf(_Z^3+4)^
2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^2+15*RootOf(_Z^3+4)*x^3+36*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^
2)*x^3-2*(3*x^3+1)^(2/3)*x+5*RootOf(_Z^3+4)+12*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(1+x)/(x^
2-x+1))+2/3*RootOf(_Z^3+4)*ln(-(3*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3+45
*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+12*(3*x^3+1)^(2/3)*RootOf(RootOf(
_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*x+4*(3*x^3+1)^(1/3)*RootOf(_Z^3+4)^2*x^2+3*(3*x^3+1)^(
1/3)*RootOf(_Z^3+4)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^2-5*RootOf(_Z^3+4)*x^3-75*RootOf(Ro
otOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^3-7*(3*x^3+1)^(2/3)*x-RootOf(_Z^3+4)-15*RootOf(RootOf(_Z^3+4)^2+
6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(1+x)/(x^2-x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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