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

Optimal. Leaf size=150 \[ \frac {2}{3} \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{2 x^3-1}-2 x\right )-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt [3]{2} \sqrt [3]{2 x^3-1}+x}\right )}{\sqrt {3}}+\frac {\left (2 x^3-1\right )^{2/3} \left (9 x^3-2\right )}{10 x^5}-\frac {1}{3} \sqrt [3]{2} \log \left (2 \sqrt [3]{2} \sqrt [3]{2 x^3-1} x+2^{2/3} \left (2 x^3-1\right )^{2/3}+4 x^2\right ) \]

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Rubi [A]  time = 0.18, antiderivative size = 159, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {2}{3} \sqrt [3]{2} \log \left (1-\frac {2^{2/3} x}{\sqrt [3]{2 x^3-1}}\right )-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\frac {2\ 2^{2/3} x}{\sqrt [3]{2 x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (2 x^3-1\right )^{2/3}}{5 x^5}+\frac {9 \left (2 x^3-1\right )^{2/3}}{10 x^2}-\frac {1}{3} \sqrt [3]{2} \log \left (\frac {2^{2/3} x}{\sqrt [3]{2 x^3-1}}+\frac {2 \sqrt [3]{2} x^2}{\left (2 x^3-1\right )^{2/3}}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(-1 + 2*x^3)^(2/3)/x^5 + (9*(-1 + 2*x^3)^(2/3))/(10*x^2) - (2*2^(1/3)*ArcTan[(1 + (2*2^(2/3)*x)/(-1 + 2*x
^3)^(1/3))/Sqrt[3]])/Sqrt[3] + (2*2^(1/3)*Log[1 - (2^(2/3)*x)/(-1 + 2*x^3)^(1/3)])/3 - (2^(1/3)*Log[1 + (2*2^(
1/3)*x^2)/(-1 + 2*x^3)^(2/3) + (2^(2/3)*x)/(-1 + 2*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+2 x^3\right )^{2/3}}{x^6 \left (1+2 x^3\right )} \, dx &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {1}{5} \int \frac {9-2 x^3}{x^3 \sqrt [3]{-1+2 x^3} \left (1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}+\frac {1}{10} \int -\frac {40}{\sqrt [3]{-1+2 x^3} \left (1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}-4 \int \frac {1}{\sqrt [3]{-1+2 x^3} \left (1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}-4 \operatorname {Subst}\left (\int \frac {1}{1-4 x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1-2^{2/3} x} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {2+2^{2/3} x}{1+2^{2/3} x+2 \sqrt [3]{2} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}+\frac {2}{3} \sqrt [3]{2} \log \left (1-\frac {2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{1+2^{2/3} x+2 \sqrt [3]{2} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {1}{3} \sqrt [3]{2} \operatorname {Subst}\left (\int \frac {2^{2/3}+4 \sqrt [3]{2} x}{1+2^{2/3} x+2 \sqrt [3]{2} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}+\frac {2}{3} \sqrt [3]{2} \log \left (1-\frac {2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {2 \sqrt [3]{2} x^2}{\left (-1+2 x^3\right )^{2/3}}+\frac {2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )+\left (2 \sqrt [3]{2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2\ 2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {1+\frac {2\ 2^{2/3} x}{\sqrt [3]{-1+2 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2}{3} \sqrt [3]{2} \log \left (1-\frac {2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {2 \sqrt [3]{2} x^2}{\left (-1+2 x^3\right )^{2/3}}+\frac {2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 139, normalized size = 0.93 \begin {gather*} \left (\frac {9}{10 x^2}-\frac {1}{5 x^5}\right ) \left (2 x^3-1\right )^{2/3}-\frac {1}{3} \sqrt [3]{2} \left (-2 \log \left (1-\frac {2^{2/3} x}{\sqrt [3]{2-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2\ 2^{2/3} x}{\sqrt [3]{2-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{2-x^3}}+\frac {2 \sqrt [3]{2} x^2}{\left (2-x^3\right )^{2/3}}+1\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-1/5*1/x^5 + 9/(10*x^2))*(-1 + 2*x^3)^(2/3) - (2^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*2^(2/3)*x)/(2 - x^3)^(1/3))/
Sqrt[3]] - 2*Log[1 - (2^(2/3)*x)/(2 - x^3)^(1/3)] + Log[1 + (2*2^(1/3)*x^2)/(2 - x^3)^(2/3) + (2^(2/3)*x)/(2 -
x^3)^(1/3)]))/3

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IntegrateAlgebraic [A]  time = 0.37, size = 150, normalized size = 1.00 \begin {gather*} \frac {\left (-1+2 x^3\right )^{2/3} \left (-2+9 x^3\right )}{10 x^5}-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+\sqrt [3]{2} \sqrt [3]{-1+2 x^3}}\right )}{\sqrt {3}}+\frac {2}{3} \sqrt [3]{2} \log \left (-2 x+\sqrt [3]{2} \sqrt [3]{-1+2 x^3}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (4 x^2+2 \sqrt [3]{2} x \sqrt [3]{-1+2 x^3}+2^{2/3} \left (-1+2 x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 2.84, size = 290, normalized size = 1.93 \begin {gather*} -\frac {20 \, \sqrt {3} 2^{\frac {1}{3}} x^{5} \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (20 \, x^{7} + 8 \, x^{4} - x\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} - 12 \, \sqrt {3} 2^{\frac {1}{3}} {\left (76 \, x^{8} - 32 \, x^{5} + x^{2}\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (568 \, x^{9} - 444 \, x^{6} + 66 \, x^{3} - 1\right )}}{3 \, {\left (872 \, x^{9} - 420 \, x^{6} + 6 \, x^{3} + 1\right )}}\right ) - 20 \cdot 2^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 2^{\frac {2}{3}} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} x - 2^{\frac {1}{3}} {\left (2 \, x^{3} + 1\right )}}{2 \, x^{3} + 1}\right ) + 10 \cdot 2^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (10 \, x^{4} - x\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (76 \, x^{6} - 32 \, x^{3} + 1\right )} + 24 \, {\left (4 \, x^{5} - x^{2}\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{4 \, x^{6} + 4 \, x^{3} + 1}\right ) - 9 \, {\left (9 \, x^{3} - 2\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*(20*sqrt(3)*2^(1/3)*x^5*arctan(1/3*(6*sqrt(3)*2^(2/3)*(20*x^7 + 8*x^4 - x)*(2*x^3 - 1)^(2/3) - 12*sqrt(3
)*2^(1/3)*(76*x^8 - 32*x^5 + x^2)*(2*x^3 - 1)^(1/3) - sqrt(3)*(568*x^9 - 444*x^6 + 66*x^3 - 1))/(872*x^9 - 420
*x^6 + 6*x^3 + 1)) - 20*2^(1/3)*x^5*log(-(6*2^(2/3)*(2*x^3 - 1)^(1/3)*x^2 - 6*(2*x^3 - 1)^(2/3)*x - 2^(1/3)*(2
*x^3 + 1))/(2*x^3 + 1)) + 10*2^(1/3)*x^5*log((6*2^(1/3)*(10*x^4 - x)*(2*x^3 - 1)^(2/3) + 2^(2/3)*(76*x^6 - 32*
x^3 + 1) + 24*(4*x^5 - x^2)*(2*x^3 - 1)^(1/3))/(4*x^6 + 4*x^3 + 1)) - 9*(9*x^3 - 2)*(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} + 1\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+1)*(2*x^3-1)^(2/3)/x^6/(2*x^3+1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 17.26, size = 631, normalized size = 4.21

method	result	size
risch	\(\frac {18 x^{6}-13 x^{3}+2}{10 x^{5} \left (2 x^{3}-1\right )^{\frac {1}{3}}}+\frac {2 \RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (-\frac {12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-12 \left (2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +4 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (\textit {\_Z}^{3}-2\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+54 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-5 \left (2 x^{3}-1\right )^{\frac {2}{3}} x -\RootOf \left (\textit {\_Z}^{3}-2\right )-9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{2 x^{3}+1}\right )}{3}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+144 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-24 \left (2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +8 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+60 \RootOf \left (\textit {\_Z}^{3}-2\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{2}-6 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}+2 \left (2 x^{3}-1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{3}-2\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{2 x^{3}+1}\right )\)	\(631\)
trager	\(\frac {\left (2 x^{3}-1\right )^{\frac {2}{3}} \left (9 x^{3}-2\right )}{10 x^{5}}+\frac {2 \RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (\frac {204 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+396 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-720 \left (2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +240 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+1422 \RootOf \left (\textit {\_Z}^{3}-2\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{2}-204 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3}-396 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2}-374 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-726 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-3 \left (2 x^{3}-1\right )^{\frac {2}{3}} x +17 \RootOf \left (\textit {\_Z}^{3}-2\right )+33 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{2 x^{3}+1}\right )}{3}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {2676 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+96912 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-64656 \left (2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +21552 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+24372 \RootOf \left (\textit {\_Z}^{3}-2\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{2}-2676 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3}-96912 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+5798 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+209976 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-17490 \left (2 x^{3}-1\right )^{\frac {2}{3}} x -1115 \RootOf \left (\textit {\_Z}^{3}-2\right )-40380 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{2 x^{3}+1}\right )\)	\(765\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(18*x^6-13*x^3+2)/x^5/(2*x^3-1)^(1/3)+2/3*RootOf(_Z^3-2)*ln(-(12*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3
-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x^3+108*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*
x^3-12*(2*x^3-1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2*x+4*RootOf(_Z^3-2
)^2*(2*x^3-1)^(1/3)*x^2-6*RootOf(_Z^3-2)*(2*x^3-1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*
x^2+6*RootOf(_Z^3-2)*x^3+54*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3-5*(2*x^3-1)^(2/3)*x-RootO
f(_Z^3-2)-9*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(2*x^3+1))+4*RootOf(RootOf(_Z^3-2)^2+6*_Z*Ro
otOf(_Z^3-2)+36*_Z^2)*ln((36*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x^3+144*Roo
tOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-24*(2*x^3-1)^(2/3)*RootOf(RootOf(_Z^3
-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2*x+8*RootOf(_Z^3-2)^2*(2*x^3-1)^(1/3)*x^2+60*RootOf(_Z^3-2)
*(2*x^3-1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^2-6*RootOf(_Z^3-2)*x^3-24*RootOf(RootO
f(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3+2*(2*x^3-1)^(2/3)*x+3*RootOf(_Z^3-2)+12*RootOf(RootOf(_Z^3-2)^2+6
*_Z*RootOf(_Z^3-2)+36*_Z^2))/(2*x^3+1))

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

[Out]

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

[Out]

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

[Out]

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

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