\(\int \frac {(1+x^3)^{2/3} (2+x^3)}{x^6 (1+2 x^3)} \, dx\) [1348]

Optimal result

Integrand size = 27, antiderivative size = 97 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (-4+11 x^3\right )}{10 x^5}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )+\log \left (x+\sqrt [3]{1+x^3}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]

[Out]

1/10*(x^3+1)^(2/3)*(11*x^3-4)/x^5+3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^3+1)^(1/3)))+ln(x+(x^3+1)^(1/3))-1/2*ln(x^
2-x*(x^3+1)^(1/3)+(x^3+1)^(2/3))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {594, 597, 12, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (2 x^3+1\right )+\frac {3}{2} \log \left (-\sqrt [3]{x^3+1}-x\right )-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}+\frac {11 \left (x^3+1\right )^{2/3}}{10 x^2} \]

[In]

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

[Out]

(-2*(1 + x^3)^(2/3))/(5*x^5) + (11*(1 + x^3)^(2/3))/(10*x^2) - Sqrt[3]*ArcTan[(1 - (2*x)/(1 + x^3)^(1/3))/Sqrt
[3]] - Log[1 + 2*x^3]/2 + (3*Log[-x - (1 + 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 {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {1}{5} \int \frac {-11-7 x^3}{x^3 \sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {1}{10} \int -\frac {30}{\sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}+3 \int \frac {1}{\sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (1+2 x^3\right )+\frac {3}{2} \log \left (-x-\sqrt [3]{1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (-4+11 x^3\right )}{10 x^5}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )+\log \left (x+\sqrt [3]{1+x^3}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]

[In]

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

[Out]

((1 + x^3)^(2/3)*(-4 + 11*x^3))/(10*x^5) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^3)^(1/3))] + Log[x + (1 +
x^3)^(1/3)] - Log[x^2 - x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/2

Maple [A] (verified)

Time = 6.42 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(\frac {10 \ln \left (\frac {x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (11 x^{3}-4\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+5 x^{5} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )-\ln \left (\frac {x^{2}-x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{10 x^{5}}\)	\(100\)
risch	\(\frac {11 x^{6}+7 x^{3}-4}{10 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}}+\ln \left (-\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +6 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-x \left (x^{3}+1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-1}{2 x^{3}+1}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 x \left (x^{3}+1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2 x^{3}+1}\right )\)	\(258\)
trager	\(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (11 x^{3}-4\right )}{10 x^{5}}+\ln \left (\frac {-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +63 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+21 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-15 x \left (x^{3}+1\right )^{\frac {2}{3}}+6 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-5 x^{3}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-10}{2 x^{3}+1}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -45 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+24 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+21 x \left (x^{3}+1\right )^{\frac {2}{3}}+6 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-4 x^{3}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+21 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{2 x^{3}+1}\right )\)	\(327\)

[In]

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

[Out]

1/10*(10*ln((x+(x^3+1)^(1/3))/x)*x^5+(11*x^3-4)*(x^3+1)^(2/3)+5*x^5*(2*3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^3+1)
^(1/3))/x)-ln((x^2-x*(x^3+1)^(1/3)+(x^3+1)^(2/3))/x^2)))/x^5

Fricas [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.28 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} + 1\right )}}{7 \, x^{3} - 1}\right ) - 5 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{3} + 1}\right ) - {\left (11 \, x^{3} - 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \]

[In]

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

[Out]

-1/10*(10*sqrt(3)*x^5*arctan((4*sqrt(3)*(x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(x^3 + 1)^(2/3)*x + sqrt(3)*(x^3 + 1))
/(7*x^3 - 1)) - 5*x^5*log((2*x^3 + 3*(x^3 + 1)^(1/3)*x^2 + 3*(x^3 + 1)^(2/3)*x + 1)/(2*x^3 + 1)) - (11*x^3 - 4
)*(x^3 + 1)^(2/3))/x^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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