\(\int \frac {(1+x^3)^{2/3} (2+x^3+x^6)}{x^6 (-2+x^3)^2} \, dx\) [2342]

Optimal result

Integrand size = 28, antiderivative size = 185 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (24+102 x^3-97 x^6\right )}{120 x^5 \left (-2+x^3\right )}+\frac {35 \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )}{12\ 2^{2/3} 3^{5/6}}-\frac {35 \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1+x^3}\right )}{36\ 2^{2/3} \sqrt [3]{3}}+\frac {35 \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1+x^3}+2^{2/3} \sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{72\ 2^{2/3} \sqrt [3]{3}} \]

[Out]

1/120*(x^3+1)^(2/3)*(-97*x^6+102*x^3+24)/x^5/(x^3-2)+35/72*arctan(3^(5/6)*x/(3^(1/3)*x+2*2^(1/3)*(x^3+1)^(1/3)
))*2^(1/3)*3^(1/6)-35/216*ln(-3*x+2^(1/3)*3^(2/3)*(x^3+1)^(1/3))*2^(1/3)*3^(2/3)+35/432*ln(3*x^2+2^(1/3)*3^(2/
3)*x*(x^3+1)^(1/3)+2^(2/3)*3^(1/3)*(x^3+1)^(2/3))*2^(1/3)*3^(2/3)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.35, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6874, 270, 283, 245, 386, 384, 399} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\frac {3 \sqrt [6]{3} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3}}+\frac {\sqrt [3]{2} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{3\ 3^{5/6}}+\frac {x \left (x^3+1\right )^{2/3}}{3 \left (2-x^3\right )}+\frac {\log \left (x^3-2\right )}{9\ 2^{2/3} \sqrt [3]{3}}+\frac {1}{8} \left (\frac {3}{2}\right )^{2/3} \log \left (x^3-2\right )-\frac {\log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3+1}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {3}{8} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3+1}\right )-\frac {\left (x^3+1\right )^{5/3}}{10 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{8 x^2} \]

[In]

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

[Out]

(-3*(1 + x^3)^(2/3))/(8*x^2) + (x*(1 + x^3)^(2/3))/(3*(2 - x^3)) - (1 + x^3)^(5/3)/(10*x^5) + (2^(1/3)*ArcTan[
(1 + (2^(2/3)*3^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(3*3^(5/6)) + (3*3^(1/6)*ArcTan[(1 + (2^(2/3)*3^(1/3)*x)/(
1 + x^3)^(1/3))/Sqrt[3]])/(4*2^(2/3)) + ((3/2)^(2/3)*Log[-2 + x^3])/8 + Log[-2 + x^3]/(9*2^(2/3)*3^(1/3)) - (3
*(3/2)^(2/3)*Log[(3/2)^(1/3)*x - (1 + x^3)^(1/3)])/8 - Log[(3/2)^(1/3)*x - (1 + x^3)^(1/3)]/(3*2^(2/3)*3^(1/3)
)

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, 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 386

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[(-x)*(a + b*x^n)^(p + 1)*(
(c + d*x^n)^q/(a*n*(p + 1))), x] - Dist[c*(q/(a*(p + 1))), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1+x^3\right )^{2/3}}{2 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{4 x^3}+\frac {2 \left (1+x^3\right )^{2/3}}{\left (-2+x^3\right )^2}-\frac {3 \left (1+x^3\right )^{2/3}}{4 \left (-2+x^3\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+\frac {3}{4} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx-\frac {3}{4} \int \frac {\left (1+x^3\right )^{2/3}}{-2+x^3} \, dx+2 \int \frac {\left (1+x^3\right )^{2/3}}{\left (-2+x^3\right )^2} \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{8 x^2}+\frac {x \left (1+x^3\right )^{2/3}}{3 \left (2-x^3\right )}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}-\frac {2}{3} \int \frac {1}{\left (-2+x^3\right ) \sqrt [3]{1+x^3}} \, dx-\frac {9}{4} \int \frac {1}{\left (-2+x^3\right ) \sqrt [3]{1+x^3}} \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{8 x^2}+\frac {x \left (1+x^3\right )^{2/3}}{3 \left (2-x^3\right )}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {\sqrt [3]{2} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3\ 3^{5/6}}+\frac {3 \sqrt [6]{3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3}}+\frac {1}{8} \left (\frac {3}{2}\right )^{2/3} \log \left (-2+x^3\right )+\frac {\log \left (-2+x^3\right )}{9\ 2^{2/3} \sqrt [3]{3}}-\frac {3}{8} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{1+x^3}\right )-\frac {\log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{1+x^3}\right )}{3\ 2^{2/3} \sqrt [3]{3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.98 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\frac {\frac {18 \left (1+x^3\right )^{2/3} \left (24+102 x^3-97 x^6\right )}{x^5 \left (-2+x^3\right )}+1050 \sqrt [3]{2} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )-350 \sqrt [3]{2} 3^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1+x^3}\right )+175 \sqrt [3]{2} 3^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1+x^3}+2^{2/3} \sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{2160} \]

[In]

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

[Out]

((18*(1 + x^3)^(2/3)*(24 + 102*x^3 - 97*x^6))/(x^5*(-2 + x^3)) + 1050*2^(1/3)*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1
/3)*x + 2*2^(1/3)*(1 + x^3)^(1/3))] - 350*2^(1/3)*3^(2/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(1 + x^3)^(1/3)] + 175*2^
(1/3)*3^(2/3)*Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(1 + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(1 + x^3)^(2/3)])/2160

Maple [A] (verified)

Time = 15.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(\frac {\left (-1746 x^{6}+1836 x^{3}+432\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+175 \left (x^{3}-2\right ) 2^{\frac {1}{3}} x^{5} \left (\left (\ln \left (\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (1+x \right ) \left (x^{2}-x +1\right )\right )}^{\frac {1}{3}} x +2 {\left (\left (1+x \right ) \left (x^{2}-x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 {\left (\left (1+x \right ) \left (x^{2}-x +1\right )\right )}^{\frac {1}{3}}}{x}\right )+\ln \left (2\right )\right ) 3^{\frac {2}{3}}-6 \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) 3^{\frac {1}{6}}\right )}{2160 \left (x^{3}-2\right ) x^{5}}\)	\(175\)
trager	\(\text {Expression too large to display}\)	\(756\)
risch	\(\text {Expression too large to display}\)	\(920\)

[In]

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

[Out]

1/2160*((-1746*x^6+1836*x^3+432)*(x^3+1)^(2/3)+175*(x^3-2)*2^(1/3)*x^5*((ln((2^(1/3)*3^(2/3)*x^2+2^(2/3)*3^(1/
3)*((1+x)*(x^2-x+1))^(1/3)*x+2*((1+x)*(x^2-x+1))^(2/3))/x^2)-2*ln((-2^(2/3)*3^(1/3)*x+2*((1+x)*(x^2-x+1))^(1/3
))/x)+ln(2))*3^(2/3)-6*arctan(1/9*3^(1/2)*(2*2^(1/3)*3^(2/3)*(x^3+1)^(1/3)+3*x)/x)*3^(1/6)))/(x^3-2)/x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (137) = 274\).

Time = 2.38 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.72 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3+x^6\right )}{x^6 \left (-2+x^3\right )^2} \, dx=\frac {350 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{8} - 2 \, x^{5}\right )} \log \left (\frac {18 \cdot 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - 2\right )} - 36 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 175 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{8} - 2 \, x^{5}\right )} \log \left (-\frac {6 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (4 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (55 \, x^{6} + 50 \, x^{3} + 4\right )} - 18 \, {\left (7 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 2100 \cdot 12^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} {\left (x^{8} - 2 \, x^{5}\right )} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (4 \, x^{7} - 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 36 \, \left (-1\right )^{\frac {1}{3}} {\left (55 \, x^{8} + 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} + 600 \, x^{6} + 204 \, x^{3} + 8\right )}\right )}}{6 \, {\left (487 \, x^{9} + 480 \, x^{6} + 12 \, x^{3} - 8\right )}}\right ) - 108 \, {\left (97 \, x^{6} - 102 \, x^{3} - 24\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{12960 \, {\left (x^{8} - 2 \, x^{5}\right )}} \]

[In]

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

[Out]

1/12960*(350*12^(2/3)*(-1)^(1/3)*(x^8 - 2*x^5)*log((18*12^(1/3)*(-1)^(2/3)*(x^3 + 1)^(1/3)*x^2 + 12^(2/3)*(-1)
^(1/3)*(x^3 - 2) - 36*(x^3 + 1)^(2/3)*x)/(x^3 - 2)) - 175*12^(2/3)*(-1)^(1/3)*(x^8 - 2*x^5)*log(-(6*12^(2/3)*(
-1)^(1/3)*(4*x^4 + x)*(x^3 + 1)^(2/3) - 12^(1/3)*(-1)^(2/3)*(55*x^6 + 50*x^3 + 4) - 18*(7*x^5 + 4*x^2)*(x^3 +
1)^(1/3))/(x^6 - 4*x^3 + 4)) - 2100*12^(1/6)*(-1)^(1/3)*(x^8 - 2*x^5)*arctan(1/6*12^(1/6)*(12*12^(2/3)*(-1)^(2
/3)*(4*x^7 - 7*x^4 - 2*x)*(x^3 + 1)^(2/3) + 36*(-1)^(1/3)*(55*x^8 + 50*x^5 + 4*x^2)*(x^3 + 1)^(1/3) - 12^(1/3)
*(377*x^9 + 600*x^6 + 204*x^3 + 8))/(487*x^9 + 480*x^6 + 12*x^3 - 8)) - 108*(97*x^6 - 102*x^3 - 24)*(x^3 + 1)^
(2/3))/(x^8 - 2*x^5)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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