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

Optimal. Leaf size=143 \[ 3^{2/3} \log \left (3^{2/3} \sqrt [3]{2 x^3+1}-3 x\right )-3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2 x^3+1}+\sqrt [3]{3} x}\right )+\frac {\left (2 x^3+1\right )^{2/3} \left (23 x^3+4\right )}{10 x^5}-\frac {1}{2} 3^{2/3} \log \left (3^{2/3} \sqrt [3]{2 x^3+1} x+\sqrt [3]{3} \left (2 x^3+1\right )^{2/3}+3 x^2\right ) \]

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Rubi [A]  time = 0.17, antiderivative size = 148, normalized size of antiderivative = 1.03, 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]{2 x^3+1}}\right )-3 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{2 x^3+1}}+\frac {1}{\sqrt {3}}\right )+\frac {2 \left (2 x^3+1\right )^{2/3}}{5 x^5}+\frac {23 \left (2 x^3+1\right )^{2/3}}{10 x^2}-\frac {1}{2} 3^{2/3} \log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{2 x^3+1}}+\frac {3^{2/3} x^2}{\left (2 x^3+1\right )^{2/3}}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 + 2*x^3)^(2/3))/(5*x^5) + (23*(1 + 2*x^3)^(2/3))/(10*x^2) - 3*3^(1/6)*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*
(1 + 2*x^3)^(1/3))] + 3^(2/3)*Log[1 - (3^(1/3)*x)/(1 + 2*x^3)^(1/3)] - (3^(2/3)*Log[1 + (3^(2/3)*x^2)/(1 + 2*x
^3)^(2/3) + (3^(1/3)*x)/(1 + 2*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+2 x^3\right )^{2/3}}{x^6 \left (-1+x^3\right )} \, dx &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {-23-22 x^3}{x^3 \left (-1+x^3\right ) \sqrt [3]{1+2 x^3}} \, dx\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}-\frac {1}{10} \int -\frac {90}{\left (-1+x^3\right ) \sqrt [3]{1+2 x^3}} \, dx\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}+9 \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{1+2 x^3}} \, dx\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 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+2 x^3}}\right )\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 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+2 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+2 x^3}}\right )\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}+3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+2 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+2 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+2 x^3}}\right )\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}+3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{2} 3^{2/3} \log \left (1+\frac {3^{2/3} x^2}{\left (1+2 x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+2 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+2 x^3}}\right )\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 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+2 x^3}}}{\sqrt {3}}\right )+3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{2} 3^{2/3} \log \left (1+\frac {3^{2/3} x^2}{\left (1+2 x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+2 x^3}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 128, normalized size = 0.90 \begin {gather*} -3 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+2}}+\frac {1}{\sqrt {3}}\right )+\frac {\left (2 x^3+1\right )^{2/3} \left (23 x^3+4\right )}{10 x^5}+\frac {1}{2} 3^{2/3} \left (2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}\right )-\log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+\frac {3^{2/3} x^2}{\left (x^3+2\right )^{2/3}}+1\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 + 2*x^3)^(2/3)*(4 + 23*x^3))/(10*x^5) - 3*3^(1/6)*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(2 + x^3)^(1/3))] + (3
^(2/3)*(2*Log[1 - (3^(1/3)*x)/(2 + x^3)^(1/3)] - Log[1 + (3^(2/3)*x^2)/(2 + x^3)^(2/3) + (3^(1/3)*x)/(2 + x^3)
^(1/3)]))/2

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IntegrateAlgebraic [A]  time = 0.31, size = 143, normalized size = 1.00 \begin {gather*} \frac {\left (1+2 x^3\right )^{2/3} \left (4+23 x^3\right )}{10 x^5}-3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+2 x^3}}\right )+3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+2 x^3}\right )-\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+2 x^3}+\sqrt [3]{3} \left (1+2 x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 2*x^3)^(2/3)*(4 + 23*x^3))/(10*x^5) - 3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + 2*x^3)^(1/3))] +
3^(2/3)*Log[-3*x + 3^(2/3)*(1 + 2*x^3)^(1/3)] - (3^(2/3)*Log[3*x^2 + 3^(2/3)*x*(1 + 2*x^3)^(1/3) + 3^(1/3)*(1
+ 2*x^3)^(2/3)])/2

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fricas [B]  time = 2.74, size = 281, normalized size = 1.97 \begin {gather*} -\frac {10 \cdot 9^{\frac {1}{3}} \sqrt {3} x^{5} \arctan \left (\frac {2 \cdot 9^{\frac {2}{3}} \sqrt {3} {\left (8 \, x^{7} - 7 \, x^{4} - x\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} - 6 \cdot 9^{\frac {1}{3}} \sqrt {3} {\left (55 \, x^{8} + 25 \, x^{5} + x^{2}\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (377 \, x^{9} + 300 \, x^{6} + 51 \, x^{3} + 1\right )}}{3 \, {\left (487 \, x^{9} + 240 \, x^{6} + 3 \, x^{3} - 1\right )}}\right ) - 10 \cdot 9^{\frac {1}{3}} x^{5} \log \left (\frac {3 \cdot 9^{\frac {2}{3}} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 9 \, {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} x - 9^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) + 5 \cdot 9^{\frac {1}{3}} x^{5} \log \left (\frac {9 \cdot 9^{\frac {1}{3}} {\left (8 \, x^{4} + x\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} + 9^{\frac {2}{3}} {\left (55 \, x^{6} + 25 \, x^{3} + 1\right )} + 27 \, {\left (7 \, x^{5} + 2 \, x^{2}\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) - 3 \, {\left (23 \, x^{3} + 4\right )} {\left (2 \, 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)*(2*x^3+1)^(2/3)/x^6/(x^3-1),x, algorithm="fricas")

[Out]

-1/30*(10*9^(1/3)*sqrt(3)*x^5*arctan(1/3*(2*9^(2/3)*sqrt(3)*(8*x^7 - 7*x^4 - x)*(2*x^3 + 1)^(2/3) - 6*9^(1/3)*
sqrt(3)*(55*x^8 + 25*x^5 + x^2)*(2*x^3 + 1)^(1/3) - sqrt(3)*(377*x^9 + 300*x^6 + 51*x^3 + 1))/(487*x^9 + 240*x
^6 + 3*x^3 - 1)) - 10*9^(1/3)*x^5*log((3*9^(2/3)*(2*x^3 + 1)^(1/3)*x^2 - 9*(2*x^3 + 1)^(2/3)*x - 9^(1/3)*(x^3
- 1))/(x^3 - 1)) + 5*9^(1/3)*x^5*log((9*9^(1/3)*(8*x^4 + x)*(2*x^3 + 1)^(2/3) + 9^(2/3)*(55*x^6 + 25*x^3 + 1)
+ 27*(7*x^5 + 2*x^2)*(2*x^3 + 1)^(1/3))/(x^6 - 2*x^3 + 1)) - 3*(23*x^3 + 4)*(2*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 (2 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 2\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+2)*(2*x^3+1)^(2/3)/x^6/(x^3-1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 19.35, size = 935, normalized size = 6.54

method	result	size
risch	\(\frac {46 x^{6}+31 x^{3}+4}{10 x^{5} \left (2 x^{3}+1\right )^{\frac {1}{3}}}+\RootOf \left (\textit {\_Z}^{3}-9\right ) \ln \left (\frac {6 \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}+135 \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}+21 \left (2 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 +8 \left (2 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}+9 \left (2 x^{3}+1\right )^{\frac {1}{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 ) x^{2}-4 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}-90 \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}-3 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}-9\right )-45 \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 )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )-\ln \left (-\frac {-6 \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}+21 \left (2 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 -\left (2 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}-72 \left (2 x^{3}+1\right )^{\frac {1}{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 ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}+135 \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}+24 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -2 \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 )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )-9 \ln \left (-\frac {-6 \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}+21 \left (2 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 -\left (2 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}-72 \left (2 x^{3}+1\right )^{\frac {1}{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 ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}+135 \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}+24 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -2 \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 )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\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 )\)	\(935\)
trager	\(\text {Expression too large to display}\)	\(1157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(46*x^6+31*x^3+4)/x^5/(2*x^3+1)^(1/3)+RootOf(_Z^3-9)*ln((6*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81
*_Z^2)*RootOf(_Z^3-9)^3*x^3+135*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x^3+21
*(2*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+8*(2*x^3+1)^(1/3)*Roo
tOf(_Z^3-9)^2*x^2+9*(2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)*x^2-4*
RootOf(_Z^3-9)*x^3-90*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^3-3*(2*x^3+1)^(2/3)*x-2*RootOf(_Z
^3-9)-45*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2))/(-1+x)/(x^2+x+1))-ln(-(-6*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+21*(2*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*x^3+1)^(1/3)*RootOf(_Z^3-9)^2*x^2-72*(2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2
)*RootOf(_Z^3-9)*x^2-10*RootOf(_Z^3-9)*x^3+135*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^3+24*(2*
x^3+1)^(2/3)*x-2*RootOf(_Z^3-9)+27*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2))/(-1+x)/(x^2+x+1))*Roo
tOf(_Z^3-9)-9*ln(-(-6*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x^3+81*RootOf(Root
Of(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x^3+21*(2*x^3+1)^(2/3)*RootOf(_Z^3-9)^2*RootOf(Ro
otOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*x-(2*x^3+1)^(1/3)*RootOf(_Z^3-9)^2*x^2-72*(2*x^3+1)^(1/3)*RootOf(R
ootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)*x^2-10*RootOf(_Z^3-9)*x^3+135*RootOf(RootOf(_Z^3-9
)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^3+24*(2*x^3+1)^(2/3)*x-2*RootOf(_Z^3-9)+27*RootOf(RootOf(_Z^3-9)^2+9*_Z*Roo
tOf(_Z^3-9)+81*_Z^2))/(-1+x)/(x^2+x+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 (2 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 2\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+2)*(2*x^3+1)^(2/3)/x^6/(x^3-1),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

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