3.21.62 \(\int \frac {(1+x^3)^{2/3} (2+x^3)}{x^6 (4+x^3)} \, dx\)

Optimal. Leaf size=149 \[ -\frac {\log \left (6^{2/3} \sqrt [3]{x^3+1}-3 x\right )}{16 \sqrt [3]{6}}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2\ 2^{2/3} \sqrt [3]{x^3+1}+\sqrt [3]{3} x}\right )}{16 \sqrt [3]{2}}+\frac {\left (x^3+1\right )^{2/3} \left (-13 x^3-8\right )}{80 x^5}+\frac {\log \left (6^{2/3} \sqrt [3]{x^3+1} x+2 \sqrt [3]{6} \left (x^3+1\right )^{2/3}+3 x^2\right )}{32 \sqrt [3]{6}} \]

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Rubi [A]  time = 0.20, antiderivative size = 162, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )}{16 \sqrt [3]{6}}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{6} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{16 \sqrt [3]{2}}-\frac {\left (x^3+1\right )^{2/3}}{10 x^5}-\frac {13 \left (x^3+1\right )^{2/3}}{80 x^2}+\frac {\log \left (\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+2 \sqrt [3]{2}\right )}{32 \sqrt [3]{6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^3)),x]

[Out]

-1/10*(1 + x^3)^(2/3)/x^5 - (13*(1 + x^3)^(2/3))/(80*x^2) + (3^(1/6)*ArcTan[(1 + (6^(1/3)*x)/(1 + x^3)^(1/3))/
Sqrt[3]])/(16*2^(1/3)) - Log[2^(2/3) - (3^(1/3)*x)/(1 + x^3)^(1/3)]/(16*6^(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)]/(32*6^(1/3))

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 )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^3\right )} \, dx &=-\frac {\left (1+x^3\right )^{2/3}}{10 x^5}+\frac {1}{20} \int \frac {26+14 x^3}{x^3 \sqrt [3]{1+x^3} \left (4+x^3\right )} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{10 x^5}-\frac {13 \left (1+x^3\right )^{2/3}}{80 x^2}-\frac {1}{160} \int -\frac {60}{\sqrt [3]{1+x^3} \left (4+x^3\right )} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{10 x^5}-\frac {13 \left (1+x^3\right )^{2/3}}{80 x^2}+\frac {3}{8} \int \frac {1}{\sqrt [3]{1+x^3} \left (4+x^3\right )} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{10 x^5}-\frac {13 \left (1+x^3\right )^{2/3}}{80 x^2}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{4-3 x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\left (1+x^3\right )^{2/3}}{10 x^5}-\frac {13 \left (1+x^3\right )^{2/3}}{80 x^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{16 \sqrt [3]{2}}+\frac {\operatorname {Subst}\left (\int \frac {2\ 2^{2/3}+\sqrt [3]{3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{16 \sqrt [3]{2}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{10 x^5}-\frac {13 \left (1+x^3\right )^{2/3}}{80 x^2}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{16 \sqrt [3]{6}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{16\ 2^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{3}+2\ 3^{2/3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{32 \sqrt [3]{6}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{10 x^5}-\frac {13 \left (1+x^3\right )^{2/3}}{80 x^2}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{16 \sqrt [3]{6}}+\frac {\log \left (2 \sqrt [3]{2}+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{32 \sqrt [3]{6}}-\frac {3^{2/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {\sqrt [3]{6} x}{\sqrt [3]{1+x^3}}\right )}{16 \sqrt [3]{2}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{10 x^5}-\frac {13 \left (1+x^3\right )^{2/3}}{80 x^2}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{6} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt [3]{2}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{16 \sqrt [3]{6}}+\frac {\log \left (2 \sqrt [3]{2}+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{32 \sqrt [3]{6}}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 135, normalized size = 0.91 \begin {gather*} \frac {1}{192} \left (6\ 2^{2/3} \sqrt [6]{3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{6} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+6^{2/3} \left (\log \left (\frac {2 \sqrt [3]{6} x}{\sqrt [3]{x^3+1}}+\frac {6^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+4\right )-2 \log \left (2-\frac {\sqrt [3]{6} x}{\sqrt [3]{x^3+1}}\right )\right )\right )+\left (x^3+1\right )^{2/3} \left (-\frac {1}{10 x^5}-\frac {13}{80 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^3)),x]

[Out]

(-1/10*1/x^5 - 13/(80*x^2))*(1 + x^3)^(2/3) + (6*2^(2/3)*3^(1/6)*ArcTan[(1 + (6^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt
[3]] + 6^(2/3)*(-2*Log[2 - (6^(1/3)*x)/(1 + x^3)^(1/3)] + Log[4 + (6^(2/3)*x^2)/(1 + x^3)^(2/3) + (2*6^(1/3)*x
)/(1 + x^3)^(1/3)]))/192

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IntegrateAlgebraic [A]  time = 0.39, size = 149, normalized size = 1.00 \begin {gather*} \frac {\left (-8-13 x^3\right ) \left (1+x^3\right )^{2/3}}{80 x^5}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2\ 2^{2/3} \sqrt [3]{1+x^3}}\right )}{16 \sqrt [3]{2}}-\frac {\log \left (-3 x+6^{2/3} \sqrt [3]{1+x^3}\right )}{16 \sqrt [3]{6}}+\frac {\log \left (3 x^2+6^{2/3} x \sqrt [3]{1+x^3}+2 \sqrt [3]{6} \left (1+x^3\right )^{2/3}\right )}{32 \sqrt [3]{6}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^3)),x]

[Out]

((-8 - 13*x^3)*(1 + x^3)^(2/3))/(80*x^5) + (3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(2/3)*(1 + x^3)^(1/3))
])/(16*2^(1/3)) - Log[-3*x + 6^(2/3)*(1 + x^3)^(1/3)]/(16*6^(1/3)) + Log[3*x^2 + 6^(2/3)*x*(1 + x^3)^(1/3) + 2
*6^(1/3)*(1 + x^3)^(2/3)]/(32*6^(1/3))

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fricas [B]  time = 3.25, size = 303, normalized size = 2.03 \begin {gather*} \frac {30 \cdot 6^{\frac {1}{6}} \sqrt {2} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {6^{\frac {1}{6}} {\left (24 \cdot 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 22 \, x^{4} + 8 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 36 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (109 \, x^{8} + 116 \, x^{5} + 16 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 6^{\frac {1}{3}} \sqrt {2} {\left (1189 \, x^{9} + 2064 \, x^{6} + 912 \, x^{3} + 64\right )}\right )}}{6 \, {\left (971 \, x^{9} + 960 \, x^{6} - 48 \, x^{3} - 64\right )}}\right ) + 10 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {18 \cdot 6^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 4\right )} - 36 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 4}\right ) - 5 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {12 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} + 2 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 6^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (109 \, x^{6} + 116 \, x^{3} + 16\right )} - 18 \, {\left (11 \, x^{5} + 8 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 8 \, x^{3} + 16}\right ) - 36 \, {\left (13 \, x^{3} + 8\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2880 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^3+2)/x^6/(x^3+4),x, algorithm="fricas")

[Out]

1/2880*(30*6^(1/6)*sqrt(2)*(-1)^(1/3)*x^5*arctan(1/6*6^(1/6)*(24*6^(2/3)*sqrt(2)*(-1)^(2/3)*(5*x^7 + 22*x^4 +
8*x)*(x^3 + 1)^(2/3) - 36*sqrt(2)*(-1)^(1/3)*(109*x^8 + 116*x^5 + 16*x^2)*(x^3 + 1)^(1/3) + 6^(1/3)*sqrt(2)*(1
189*x^9 + 2064*x^6 + 912*x^3 + 64))/(971*x^9 + 960*x^6 - 48*x^3 - 64)) + 10*6^(2/3)*(-1)^(1/3)*x^5*log(-(18*6^
(1/3)*(-1)^(2/3)*(x^3 + 1)^(1/3)*x^2 - 6^(2/3)*(-1)^(1/3)*(x^3 + 4) - 36*(x^3 + 1)^(2/3)*x)/(x^3 + 4)) - 5*6^(
2/3)*(-1)^(1/3)*x^5*log(-(12*6^(2/3)*(-1)^(1/3)*(5*x^4 + 2*x)*(x^3 + 1)^(2/3) - 6^(1/3)*(-1)^(2/3)*(109*x^6 +
116*x^3 + 16) - 18*(11*x^5 + 8*x^2)*(x^3 + 1)^(1/3))/(x^6 + 8*x^3 + 16)) - 36*(13*x^3 + 8)*(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} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 4\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^3+2)/x^6/(x^3+4),x, algorithm="giac")

[Out]

integrate((x^3 + 2)*(x^3 + 1)^(2/3)/((x^3 + 4)*x^6), x)

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maple [C]  time = 17.28, size = 609, normalized size = 4.09

method	result	size
risch	\(-\frac {13 x^{6}+21 x^{3}+8}{80 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {\RootOf \left (\textit {\_Z}^{3}+36\right ) \ln \left (-\frac {756 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+36\right )^{2} x^{3}+9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+36\right )^{3} x^{3}+132 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+36\right )^{2} x +36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right ) \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+36\right ) x^{2}+11 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+36\right )^{2} x^{2}-1764 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right ) x^{3}-21 \RootOf \left (\textit {\_Z}^{3}+36\right ) x^{3}-120 x \left (x^{3}+1\right )^{\frac {2}{3}}-1008 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right )-12 \RootOf \left (\textit {\_Z}^{3}+36\right )}{x^{3}+4}\right )}{96}+\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right ) \ln \left (\frac {324 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+36\right )^{2} x^{3}+21 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+36\right )^{3} x^{3}+132 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+36\right )^{2} x +360 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right ) \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+36\right ) x^{2}+11 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+36\right )^{2} x^{2}+432 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right ) x^{3}+28 \RootOf \left (\textit {\_Z}^{3}+36\right ) x^{3}-12 x \left (x^{3}+1\right )^{\frac {2}{3}}+432 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+36\right )^{2}+36 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+36\right )+1296 \textit {\_Z}^{2}\right )+28 \RootOf \left (\textit {\_Z}^{3}+36\right )}{x^{3}+4}\right )}{8}\)	\(609\)
trager	\(\text {Expression too large to display}\)	\(1113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(x^3+2)/x^6/(x^3+4),x,method=_RETURNVERBOSE)

[Out]

-1/80*(13*x^6+21*x^3+8)/x^5/(x^3+1)^(1/3)+1/96*RootOf(_Z^3+36)*ln(-(756*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(
_Z^3+36)+1296*_Z^2)^2*RootOf(_Z^3+36)^2*x^3+9*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)*RootOf
(_Z^3+36)^3*x^3+132*(x^3+1)^(2/3)*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)*RootOf(_Z^3+36)^2*
x+36*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)*(x^3+1)^(1/3)*RootOf(_Z^3+36)*x^2+11*(x^3+1)^(1
/3)*RootOf(_Z^3+36)^2*x^2-1764*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)*x^3-21*RootOf(_Z^3+36
)*x^3-120*x*(x^3+1)^(2/3)-1008*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)-12*RootOf(_Z^3+36))/(
x^3+4))+3/8*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)*ln((324*RootOf(RootOf(_Z^3+36)^2+36*_Z*R
ootOf(_Z^3+36)+1296*_Z^2)^2*RootOf(_Z^3+36)^2*x^3+21*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)
*RootOf(_Z^3+36)^3*x^3+132*(x^3+1)^(2/3)*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)*RootOf(_Z^3
+36)^2*x+360*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)*(x^3+1)^(1/3)*RootOf(_Z^3+36)*x^2+11*(x
^3+1)^(1/3)*RootOf(_Z^3+36)^2*x^2+432*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)*x^3+28*RootOf(
_Z^3+36)*x^3-12*x*(x^3+1)^(2/3)+432*RootOf(RootOf(_Z^3+36)^2+36*_Z*RootOf(_Z^3+36)+1296*_Z^2)+28*RootOf(_Z^3+3
6))/(x^3+4))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 4\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^3+2)/x^6/(x^3+4),x, algorithm="maxima")

[Out]

integrate((x^3 + 2)*(x^3 + 1)^(2/3)/((x^3 + 4)*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 (x^3+4\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*(x^3 + 4)),x)

[Out]

int(((x^3 + 1)^(2/3)*(x^3 + 2))/(x^6*(x^3 + 4)), 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 (x^{3} + 4\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+1)**(2/3)*(x**3+2)/x**6/(x**3+4),x)

[Out]

Integral(((x + 1)*(x**2 - x + 1))**(2/3)*(x**3 + 2)/(x**6*(x**3 + 4)), x)

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