3.15.40 \(\int \frac {(-2+x^3)^{2/3} (4+x^3)}{x^6 (-1+x^3)} \, dx\) [1440]

Optimal. Leaf size=102 \[ \frac {\left (-2+x^3\right )^{2/3} \left (8+21 x^3\right )}{10 x^5}+\frac {5 \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-2+x^3}}\right )}{\sqrt {3}}+\frac {5}{3} \log \left (x+\sqrt [3]{-2+x^3}\right )-\frac {5}{6} \log \left (x^2-x \sqrt [3]{-2+x^3}+\left (-2+x^3\right )^{2/3}\right ) \]

[Out]

1/10*(x^3-2)^(2/3)*(21*x^3+8)/x^5+5/3*arctan(3^(1/2)*x/(-x+2*(x^3-2)^(1/3)))*3^(1/2)+5/3*ln(x+(x^3-2)^(1/3))-5
/6*ln(x^2-x*(x^3-2)^(1/3)+(x^3-2)^(2/3))

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Rubi [A]
time = 0.05, antiderivative size = 91, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {594, 597, 12, 384} \begin {gather*} -\frac {5 \text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-2}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5}{6} \log \left (x^3-1\right )+\frac {5}{2} \log \left (-\sqrt [3]{x^3-2}-x\right )+\frac {4 \left (x^3-2\right )^{2/3}}{5 x^5}+\frac {21 \left (x^3-2\right )^{2/3}}{10 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-2 + x^3)^(2/3))/(5*x^5) + (21*(-2 + x^3)^(2/3))/(10*x^2) - (5*ArcTan[(1 - (2*x)/(-2 + x^3)^(1/3))/Sqrt[3]
])/Sqrt[3] - (5*Log[-1 + x^3])/6 + (5*Log[-x - (-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 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 594

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 597

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]

Rubi steps

\begin {align*} \int \frac {\left (-2+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-1+x^3\right )} \, dx &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {42-17 x^3}{x^3 \sqrt [3]{-2+x^3} \left (-1+x^3\right )} \, dx\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}+\frac {1}{20} \int -\frac {100}{\sqrt [3]{-2+x^3} \left (-1+x^3\right )} \, dx\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}-5 \int \frac {1}{\sqrt [3]{-2+x^3} \left (-1+x^3\right )} \, dx\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}-5 \text {Subst}\left (\int \frac {1}{-1-x^3} \, dx,x,\frac {x}{\sqrt [3]{-2+x^3}}\right )\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}-\frac {5}{3} \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,\frac {x}{\sqrt [3]{-2+x^3}}\right )-\frac {5}{3} \text {Subst}\left (\int \frac {-2+x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-2+x^3}}\right )\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}+\frac {5}{3} \log \left (1+\frac {x}{\sqrt [3]{-2+x^3}}\right )-\frac {5}{6} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-2+x^3}}\right )+\frac {5}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-2+x^3}}\right )\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}-\frac {5}{6} \log \left (1+\frac {x^2}{\left (-2+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-2+x^3}}\right )+\frac {5}{3} \log \left (1+\frac {x}{\sqrt [3]{-2+x^3}}\right )-5 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x}{\sqrt [3]{-2+x^3}}\right )\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}-\frac {5 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{-2+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5}{6} \log \left (1+\frac {x^2}{\left (-2+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-2+x^3}}\right )+\frac {5}{3} \log \left (1+\frac {x}{\sqrt [3]{-2+x^3}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 102, normalized size = 1.00 \begin {gather*} \frac {\left (-2+x^3\right )^{2/3} \left (8+21 x^3\right )}{10 x^5}+\frac {5 \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-2+x^3}}\right )}{\sqrt {3}}+\frac {5}{3} \log \left (x+\sqrt [3]{-2+x^3}\right )-\frac {5}{6} \log \left (x^2-x \sqrt [3]{-2+x^3}+\left (-2+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2 + x^3)^(2/3)*(8 + 21*x^3))/(10*x^5) + (5*ArcTan[(Sqrt[3]*x)/(-x + 2*(-2 + x^3)^(1/3))])/Sqrt[3] + (5*Log[
x + (-2 + x^3)^(1/3)])/3 - (5*Log[x^2 - x*(-2 + x^3)^(1/3) + (-2 + x^3)^(2/3)])/6

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 4.37, size = 405, normalized size = 3.97

method	result	size
risch	\(\frac {21 x^{6}-34 x^{3}-16}{10 x^{5} \left (x^{3}-2\right )^{\frac {1}{3}}}+\frac {5 \ln \left (-\frac {3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x +6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 \left (x^{3}-2\right )^{\frac {2}{3}} x +\left (x^{3}-2\right )^{\frac {1}{3}} x^{2}+x^{3}-2}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}-\frac {5 \ln \left (\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-\left (x^{3}-2\right )^{\frac {2}{3}} x -2 \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}-5 \ln \left (\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-\left (x^{3}-2\right )^{\frac {2}{3}} x -2 \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\)	\(405\)
trager	\(\frac {\left (x^{3}-2\right )^{\frac {2}{3}} \left (21 x^{3}+8\right )}{10 x^{5}}-\frac {5 \ln \left (\frac {100353024 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+1075968 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x -1075968 \left (x^{3}-2\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-2991072 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+68673 \left (x^{3}-2\right )^{\frac {2}{3}} x -68673 \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-4530 x^{3}-802824192 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-3250368 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-3020}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}-160 \ln \left (\frac {100353024 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+1075968 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x -1075968 \left (x^{3}-2\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-2991072 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+68673 \left (x^{3}-2\right )^{\frac {2}{3}} x -68673 \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-4530 x^{3}-802824192 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-3250368 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-3020}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+160 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \ln \left (-\frac {-100353024 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+1075968 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x -1075968 \left (x^{3}-2\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-5081760 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}-57465 \left (x^{3}-2\right )^{\frac {2}{3}} x +57465 \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-37516 x^{3}[phone] \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+56274}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )\)	\(500\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(21*x^6-34*x^3-16)/x^5/(x^3-2)^(1/3)+5/3*ln(-(3*RootOf(9*_Z^2+3*_Z+1)*(x^3-2)^(2/3)*x+6*RootOf(9*_Z^2+3*_
Z+1)*(x^3-2)^(1/3)*x^2+3*RootOf(9*_Z^2+3*_Z+1)*x^3+2*(x^3-2)^(2/3)*x+(x^3-2)^(1/3)*x^2+x^3-2)/(-1+x)/(x^2+x+1)
)-5/3*ln((-9*RootOf(9*_Z^2+3*_Z+1)^2*x^3+3*RootOf(9*_Z^2+3*_Z+1)*(x^3-2)^(2/3)*x-3*RootOf(9*_Z^2+3*_Z+1)*(x^3-
2)^(1/3)*x^2-3*RootOf(9*_Z^2+3*_Z+1)*x^3-(x^3-2)^(2/3)*x-2*(x^3-2)^(1/3)*x^2-6*RootOf(9*_Z^2+3*_Z+1)-2)/(-1+x)
/(x^2+x+1))-5*ln((-9*RootOf(9*_Z^2+3*_Z+1)^2*x^3+3*RootOf(9*_Z^2+3*_Z+1)*(x^3-2)^(2/3)*x-3*RootOf(9*_Z^2+3*_Z+
1)*(x^3-2)^(1/3)*x^2-3*RootOf(9*_Z^2+3*_Z+1)*x^3-(x^3-2)^(2/3)*x-2*(x^3-2)^(1/3)*x^2-6*RootOf(9*_Z^2+3*_Z+1)-2
)/(-1+x)/(x^2+x+1))*RootOf(9*_Z^2+3*_Z+1)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.74, size = 122, normalized size = 1.20 \begin {gather*} -\frac {50 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} - 2\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} - 2\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} - 2\right )}}{7 \, x^{3} + 2}\right ) - 25 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} - 2\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 2\right )}^{\frac {2}{3}} x - 2}{x^{3} - 1}\right ) - 3 \, {\left (21 \, x^{3} + 8\right )} {\left (x^{3} - 2\right )}^{\frac {2}{3}}}{30 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(50*sqrt(3)*x^5*arctan((4*sqrt(3)*(x^3 - 2)^(1/3)*x^2 + 2*sqrt(3)*(x^3 - 2)^(2/3)*x + sqrt(3)*(x^3 - 2))
/(7*x^3 + 2)) - 25*x^5*log((2*x^3 + 3*(x^3 - 2)^(1/3)*x^2 + 3*(x^3 - 2)^(2/3)*x - 2)/(x^3 - 1)) - 3*(21*x^3 +
8)*(x^3 - 2)^(2/3))/x^5

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^3 + 4)*(x^3 - 2)^(2/3)/((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 )}^{2/3}\,\left (x^3+4\right )}{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)^(2/3)*(x^3 + 4))/(x^6*(x^3 - 1)),x)

[Out]

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

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