\(\int \frac {(1+x^3)^{2/3} (-1+3 x^6)}{x^9 (1+2 x^3)} \, dx\) [1497]

Optimal result

Integrand size = 29, antiderivative size = 104 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (5-14 x^3+x^6\right )}{40 x^8}+\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{1+x^3}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]

[Out]

1/40*(x^3+1)^(2/3)*(x^6-14*x^3+5)/x^8+1/3*3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^3+1)^(1/3)))+1/3*ln(x+(x^3+1)^(1/3
))-1/6*ln(x^2-x*(x^3+1)^(1/3)+(x^3+1)^(2/3))

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {6857, 277, 270, 283, 245, 399, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (2 x^3+1\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^3+1}-x\right )+\frac {\left (x^3+1\right )^{5/3}}{8 x^8}-\frac {19 \left (x^3+1\right )^{5/3}}{40 x^5}+\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]

[In]

Int[((1 + x^3)^(2/3)*(-1 + 3*x^6))/(x^9*(1 + 2*x^3)),x]

[Out]

(1 + x^3)^(2/3)/(2*x^2) + (1 + x^3)^(5/3)/(8*x^8) - (19*(1 + x^3)^(5/3))/(40*x^5) - ArcTan[(1 - (2*x)/(1 + x^3
)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[1 + 2*x^3]/6 + Log[-x - (1 + x^3)^(1/3)]/2

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 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(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 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 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\left (1+x^3\right )^{2/3}}{x^9}+\frac {2 \left (1+x^3\right )^{2/3}}{x^6}-\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {2 \left (1+x^3\right )^{2/3}}{1+2 x^3}\right ) \, dx \\ & = 2 \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+2 \int \frac {\left (1+x^3\right )^{2/3}}{1+2 x^3} \, dx-\int \frac {\left (1+x^3\right )^{2/3}}{x^9} \, dx-\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx \\ & = \frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{8 x^8}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {1}{\sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx \\ & = \frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{8 x^8}-\frac {19 \left (1+x^3\right )^{5/3}}{40 x^5}-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (1+2 x^3\right )+\frac {1}{2} \log \left (-x-\sqrt [3]{1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (5-14 x^3+x^6\right )}{40 x^8}+\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{1+x^3}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]

[In]

Integrate[((1 + x^3)^(2/3)*(-1 + 3*x^6))/(x^9*(1 + 2*x^3)),x]

[Out]

((1 + x^3)^(2/3)*(5 - 14*x^3 + x^6))/(40*x^8) + ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^3)^(1/3))]/Sqrt[3] + Log[x +
(1 + x^3)^(1/3)]/3 - Log[x^2 - x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6

Maple [A] (verified)

Time = 2.54 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {40 \ln \left (\frac {x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{8}+3 \left (x^{6}-14 x^{3}+5\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+20 x^{8} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )-\ln \left (\frac {x^{2}-x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{120 x^{8}}\)	\(104\)
risch	\(\frac {x^{9}-13 x^{6}-9 x^{3}+5}{40 x^{8} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {\ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 x \left (x^{3}+1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+x^{3}+1}{2 x^{3}+1}\right )}{3}-\frac {\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{2 x^{3}+1}\right )}{3}-\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{2 x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\)	\(393\)
trager	\(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (x^{6}-14 x^{3}+5\right )}{40 x^{8}}-\frac {\ln \left (\frac {63 \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 +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}+21 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+10 x^{3}-63 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+5}{2 x^{3}+1}\right )}{3}-\ln \left (\frac {63 \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 +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}+21 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+10 x^{3}-63 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+5}{2 x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {63 \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 -18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-15 x \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-2 x^{3}-63 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-48 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4}{2 x^{3}+1}\right )\)	\(489\)

[In]

int((x^3+1)^(2/3)*(3*x^6-1)/x^9/(2*x^3+1),x,method=_RETURNVERBOSE)

[Out]

1/120*(40*ln((x+(x^3+1)^(1/3))/x)*x^8+3*(x^6-14*x^3+5)*(x^3+1)^(2/3)+20*x^8*(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^8

Fricas [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.22 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=-\frac {40 \, \sqrt {3} x^{8} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} + 1\right )}}{7 \, x^{3} - 1}\right ) - 20 \, x^{8} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{3} + 1}\right ) - 3 \, {\left (x^{6} - 14 \, x^{3} + 5\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{120 \, x^{8}} \]

[In]

integrate((x^3+1)^(2/3)*(3*x^6-1)/x^9/(2*x^3+1),x, algorithm="fricas")

[Out]

-1/120*(40*sqrt(3)*x^8*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)) - 20*x^8*log((2*x^3 + 3*(x^3 + 1)^(1/3)*x^2 + 3*(x^3 + 1)^(2/3)*x + 1)/(2*x^3 + 1)) - 3*(x^6 -
14*x^3 + 5)*(x^3 + 1)^(2/3))/x^8

Sympy [F]

\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \cdot \left (3 x^{6} - 1\right )}{x^{9} \cdot \left (2 x^{3} + 1\right )}\, dx \]

[In]

integrate((x**3+1)**(2/3)*(3*x**6-1)/x**9/(2*x**3+1),x)

[Out]

Integral(((x + 1)*(x**2 - x + 1))**(2/3)*(3*x**6 - 1)/(x**9*(2*x**3 + 1)), x)

Maxima [F]

\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} + 1\right )} x^{9}} \,d x } \]

[In]

integrate((x^3+1)^(2/3)*(3*x^6-1)/x^9/(2*x^3+1),x, algorithm="maxima")

[Out]

integrate((3*x^6 - 1)*(x^3 + 1)^(2/3)/((2*x^3 + 1)*x^9), x)

Giac [F]

\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} + 1\right )} x^{9}} \,d x } \]

[In]

integrate((x^3+1)^(2/3)*(3*x^6-1)/x^9/(2*x^3+1),x, algorithm="giac")

[Out]

integrate((3*x^6 - 1)*(x^3 + 1)^(2/3)/((2*x^3 + 1)*x^9), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (3\,x^6-1\right )}{x^9\,\left (2\,x^3+1\right )} \,d x \]

[In]

int(((x^3 + 1)^(2/3)*(3*x^6 - 1))/(x^9*(2*x^3 + 1)),x)

[Out]

int(((x^3 + 1)^(2/3)*(3*x^6 - 1))/(x^9*(2*x^3 + 1)), x)