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

Optimal. Leaf size=185 \[ -\frac {35 \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3+1}-3 x\right )}{36\ 2^{2/3} \sqrt [3]{3}}+\frac {35 \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2} \sqrt [3]{x^3+1}+\sqrt [3]{3} x}\right )}{12\ 2^{2/3} 3^{5/6}}+\frac {35 \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3+1} x+2^{2/3} \sqrt [3]{3} \left (x^3+1\right )^{2/3}+3 x^2\right )}{72\ 2^{2/3} \sqrt [3]{3}}+\frac {\left (x^3+1\right )^{2/3} \left (-97 x^6+102 x^3+24\right )}{120 x^5 \left (x^3-2\right )} \]

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Rubi [C]  time = 0.50, antiderivative size = 264, normalized size of antiderivative = 1.43, number of steps used = 14, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {6742, 264, 277, 239, 378, 377, 200, 31, 634, 617, 204, 628, 429} \begin {gather*} \frac {3}{8} x F_1\left (\frac {1}{3};1,-\frac {2}{3};\frac {4}{3};\frac {x^3}{2},-x^3\right )+\frac {x \left (x^3+1\right )^{2/3}}{3 \left (2-x^3\right )}-\frac {1}{9} \sqrt [3]{\frac {2}{3}} \log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )-\frac {3}{8} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {2^{2/3} x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )}{3\ 3^{5/6}}-\frac {\left (x^3+1\right )^{5/3}}{10 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{8 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 )}{9\ 2^{2/3} \sqrt [3]{3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(1 + x^3)^(2/3))/(8*x^2) + (x*(1 + x^3)^(2/3))/(3*(2 - x^3)) - (1 + x^3)^(5/3)/(10*x^5) + (3*x*AppellF1[1/
3, 1, -2/3, 4/3, x^3/2, -x^3])/8 + (Sqrt[3]*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/4 + (2^(1/3)*ArcTan[1
/Sqrt[3] + (2^(2/3)*x)/(3^(1/6)*(1 + x^3)^(1/3))])/(3*3^(5/6)) - ((2/3)^(1/3)*Log[2^(1/3) - (3^(1/3)*x)/(1 + x
^3)^(1/3)])/9 + Log[2^(2/3) + (3^(2/3)*x^2)/(1 + x^3)^(2/3) + (6^(1/3)*x)/(1 + x^3)^(1/3)]/(9*2^(2/3)*3^(1/3))
- (3*Log[-x + (1 + x^3)^(1/3)])/8

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 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 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 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 378

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

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 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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.30, size = 162, normalized size = 0.88 \begin {gather*} \frac {35 \left (6 \tan ^{-1}\left (\frac {2^{2/3} x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )+\sqrt {3} \left (\log \left (\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\frac {\sqrt [3]{2} 3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+2\right )-2 \log \left (2-\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )\right )\right )}{72\ 2^{2/3} 3^{5/6}}+\frac {\left (x^3+1\right )^{2/3} \left (-97 x^6+102 x^3+24\right )}{120 x^5 \left (x^3-2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x^3)^(2/3)*(24 + 102*x^3 - 97*x^6))/(120*x^5*(-2 + x^3)) + (35*(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)])))/(72*2^(2/3)*3^(5/6))

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IntegrateAlgebraic [A]  time = 0.55, size = 185, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (24+102 x^3-97 x^6\right )}{120 x^5 \left (-2+x^3\right )}+\frac {35 \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )}{12\ 2^{2/3} 3^{5/6}}-\frac {35 \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1+x^3}\right )}{36\ 2^{2/3} \sqrt [3]{3}}+\frac {35 \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 )}{72\ 2^{2/3} \sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x^3)^(2/3)*(24 + 102*x^3 - 97*x^6))/(120*x^5*(-2 + x^3)) + (35*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)
*(1 + x^3)^(1/3))])/(12*2^(2/3)*3^(5/6)) - (35*Log[-3*x + 2^(1/3)*3^(2/3)*(1 + x^3)^(1/3)])/(36*2^(2/3)*3^(1/3
)) + (35*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)])/(72*2^(2/3)*3^(1/3)
)

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fricas [B]  time = 4.15, size = 319, normalized size = 1.72 \begin {gather*} \frac {350 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{8} - 2 \, x^{5}\right )} \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 ) - 175 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{8} - 2 \, x^{5}\right )} \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 ) - 2100 \cdot 12^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} {\left (x^{8} - 2 \, x^{5}\right )} \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 ) - 108 \, {\left (97 \, x^{6} - 102 \, x^{3} - 24\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{12960 \, {\left (x^{8} - 2 \, x^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12960*(350*12^(2/3)*(-1)^(1/3)*(x^8 - 2*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)) - 175*12^(2/3)*(-1)^(1/3)*(x^8 - 2*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)) - 2100*12^(1/6)*(-1)^(1/3)*(x^8 - 2*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)) - 108*(97*x^6 - 102*x^3 - 24)*(x^3 + 1)^
(2/3))/(x^8 - 2*x^5)

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

[Out]

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

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maple [C]  time = 17.35, size = 901, normalized size = 4.87

method	result	size
risch	\(-\frac {97 x^{9}-5 x^{6}-126 x^{3}-24}{120 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}} \left (x^{3}-2\right )}+\frac {35 \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 (\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 )^{2} 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 )}{216}-\frac {35 \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 (\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 )^{2} 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 )}{216}-\frac {35 \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 (\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 )^{2} 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 )}{12}\)	\(901\)
trager	\(\text {Expression too large to display}\)	\(1108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(97*x^9-5*x^6-126*x^3-24)/x^5/(x^3+1)^(1/3)/(x^3-2)+35/216*RootOf(_Z^3+18)*ln(-(-3*RootOf(RootOf(_Z^3+1
8)^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(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*Ro
otOf(_Z^3+18)^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))-35/216*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*Roo
tOf(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(RootOf(_
Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootOf(_Z^3+18)^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(R
ootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2))/(x^3-2))*RootOf(_Z^3+18)-35/12*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(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootO
f(_Z^3+18)^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)

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

[Out]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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