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

Optimal result

Integrand size = 25, antiderivative size = 102 \[ \int \frac {\left (-2+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-1+x^3\right )} \, dx=\frac {\left (-2+x^3\right )^{2/3} \left (8+21 x^3\right )}{10 x^5}+\frac {5 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-2+x^3}}\right )}{\sqrt {3}}+\frac {5}{3} \log \left (x+\sqrt [3]{-2+x^3}\right )-\frac {5}{6} \log \left (x^2-x \sqrt [3]{-2+x^3}+\left (-2+x^3\right )^{2/3}\right ) \]

[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))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {594, 597, 12, 384} \[ \int \frac {\left (-2+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-1+x^3\right )} \, dx=-\frac {5 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-2}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5}{6} \log \left (x^3-1\right )+\frac {5}{2} \log \left (-\sqrt [3]{x^3-2}-x\right )+\frac {4 \left (x^3-2\right )^{2/3}}{5 x^5}+\frac {21 \left (x^3-2\right )^{2/3}}{10 x^2} \]

[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*} \text {integral}& = \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}-\frac {5 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{-2+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5}{6} \log \left (-1+x^3\right )+\frac {5}{2} \log \left (-x-\sqrt [3]{-2+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-1+x^3\right )} \, dx=\frac {\left (-2+x^3\right )^{2/3} \left (8+21 x^3\right )}{10 x^5}+\frac {5 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-2+x^3}}\right )}{\sqrt {3}}+\frac {5}{3} \log \left (x+\sqrt [3]{-2+x^3}\right )-\frac {5}{6} \log \left (x^2-x \sqrt [3]{-2+x^3}+\left (-2+x^3\right )^{2/3}\right ) \]

[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

Maple [A] (verified)

Time = 3.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(\frac {50 \ln \left (\frac {x +\left (x^{3}-2\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (63 x^{3}+24\right ) \left (x^{3}-2\right )^{\frac {2}{3}}+25 x^{5} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{3}-2\right )^{\frac {1}{3}}\right )}{3 x}\right )-\ln \left (\frac {x^{2}-x \left (x^{3}-2\right )^{\frac {1}{3}}+\left (x^{3}-2\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{30 x^{5}}\)	\(100\)
risch	\(\frac {21 x^{6}-34 x^{3}-16}{10 x^{5} \left (x^{3}-2\right )^{\frac {1}{3}}}+\frac {5 \ln \left (-\frac {-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{3}-2\right )^{\frac {2}{3}} x -6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}+5 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 \left (x^{3}-2\right )^{\frac {2}{3}} x +\left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )\)	\(268\)
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 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+1075968 \operatorname {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}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-2991072 \operatorname {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 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-3250368 \operatorname {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 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+1075968 \operatorname {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}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-2991072 \operatorname {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 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-3250368 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-3020}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+160 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \ln \left (-\frac {-100353024 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+1075968 \operatorname {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}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-5081760 \operatorname {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] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+56274}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )\)	\(500\)

[In]

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

[Out]

1/30*(50*ln((x+(x^3-2)^(1/3))/x)*x^5+(63*x^3+24)*(x^3-2)^(2/3)+25*x^5*(2*3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^3-
2)^(1/3))/x)-ln((x^2-x*(x^3-2)^(1/3)+(x^3-2)^(2/3))/x^2)))/x^5

Fricas [A] (verification not implemented)

none

Time = 0.62 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.20 \[ \int \frac {\left (-2+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-1+x^3\right )} \, dx=-\frac {50 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} - 2\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} - 2\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} - 2\right )}}{7 \, x^{3} + 2}\right ) - 25 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} - 2\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 2\right )}^{\frac {2}{3}} x - 2}{x^{3} - 1}\right ) - 3 \, {\left (21 \, x^{3} + 8\right )} {\left (x^{3} - 2\right )}^{\frac {2}{3}}}{30 \, x^{5}} \]

[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

Sympy [F]

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

[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)

Maxima [F]

\[ \int \frac {\left (-2+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-1+x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 4\right )} {\left (x^{3} - 2\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )} x^{6}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (-2+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-1+x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 4\right )} {\left (x^{3} - 2\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )} x^{6}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-1+x^3\right )} \, dx=\int \frac {{\left (x^3-2\right )}^{2/3}\,\left (x^3+4\right )}{x^6\,\left (x^3-1\right )} \,d x \]

[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)