\(\int \frac {(-1+x^3)^{2/3} (4+x^3+x^6)}{x^9 (-2+x^3)} \, dx\) [2035]

Optimal result

Integrand size = 28, antiderivative size = 145 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3+x^6\right )}{x^9 \left (-2+x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (10+8 x^3+7 x^6\right )}{40 x^8}-\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3} \sqrt {3}}+\frac {5 \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{12\ 2^{2/3}}-\frac {5 \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{24\ 2^{2/3}} \]

[Out]

1/40*(x^3-1)^(2/3)*(7*x^6+8*x^3+10)/x^8-5/24*3^(1/2)*arctan(3^(1/2)*x/(x+2*2^(1/3)*(x^3-1)^(1/3)))*2^(1/3)+5/2
4*ln(-x+2^(1/3)*(x^3-1)^(1/3))*2^(1/3)-5/48*ln(x^2+2^(1/3)*x*(x^3-1)^(1/3)+2^(2/3)*(x^3-1)^(2/3))*2^(1/3)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6857, 277, 270, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3+x^6\right )}{x^9 \left (-2+x^3\right )} \, dx=-\frac {5 \arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3}}-\frac {5 \log \left (x^3-2\right )}{24\ 2^{2/3}}+\frac {5 \log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{x^3-1}\right )}{8\ 2^{2/3}}-\frac {\left (x^3-1\right )^{5/3}}{4 x^8}-\frac {9 \left (x^3-1\right )^{5/3}}{20 x^5}+\frac {5 \left (x^3-1\right )^{2/3}}{8 x^2} \]

[In]

Int[((-1 + x^3)^(2/3)*(4 + x^3 + x^6))/(x^9*(-2 + x^3)),x]

[Out]

(5*(-1 + x^3)^(2/3))/(8*x^2) - (-1 + x^3)^(5/3)/(4*x^8) - (9*(-1 + x^3)^(5/3))/(20*x^5) - (5*ArcTan[(1 + (2^(2
/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(4*2^(2/3)*Sqrt[3]) - (5*Log[-2 + x^3])/(24*2^(2/3)) + (5*Log[x/2^(1/3) - (
-1 + x^3)^(1/3)])/(8*2^(2/3))

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 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 {2 \left (-1+x^3\right )^{2/3}}{x^9}-\frac {3 \left (-1+x^3\right )^{2/3}}{2 x^6}-\frac {5 \left (-1+x^3\right )^{2/3}}{4 x^3}+\frac {5 \left (-1+x^3\right )^{2/3}}{4 \left (-2+x^3\right )}\right ) \, dx \\ & = -\left (\frac {5}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )+\frac {5}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx-\frac {3}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx-2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^9} \, dx \\ & = \frac {5 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}-\frac {3 \left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {3}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx+\frac {5}{4} \int \frac {1}{\left (-2+x^3\right ) \sqrt [3]{-1+x^3}} \, dx \\ & = \frac {5 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}-\frac {9 \left (-1+x^3\right )^{5/3}}{20 x^5}-\frac {5 \arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3}}-\frac {5 \log \left (-2+x^3\right )}{24\ 2^{2/3}}+\frac {5 \log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{-1+x^3}\right )}{8\ 2^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3+x^6\right )}{x^9 \left (-2+x^3\right )} \, dx=\frac {1}{240} \left (\frac {6 \left (-1+x^3\right )^{2/3} \left (10+8 x^3+7 x^6\right )}{x^8}-50 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+50 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )-25 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )\right ) \]

[In]

Integrate[((-1 + x^3)^(2/3)*(4 + x^3 + x^6))/(x^9*(-2 + x^3)),x]

[Out]

((6*(-1 + x^3)^(2/3)*(10 + 8*x^3 + 7*x^6))/x^8 - 50*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^
3)^(1/3))] + 50*2^(1/3)*Log[-x + 2^(1/3)*(-1 + x^3)^(1/3)] - 25*2^(1/3)*Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) +
2^(2/3)*(-1 + x^3)^(2/3)])/240

Maple [A] (verified)

Time = 13.80 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(\frac {\left (42 x^{6}+48 x^{3}+60\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+50 \,2^{\frac {1}{3}} x^{8} \left (\arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}-\frac {\ln \left (\frac {2^{\frac {2}{3}} x {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}+\ln \left (\frac {-2^{\frac {2}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (2\right )}{2}\right )}{240 x^{8}}\)	\(139\)
risch	\(\text {Expression too large to display}\)	\(612\)
trager	\(\text {Expression too large to display}\)	\(749\)

[In]

int((x^3-1)^(2/3)*(x^6+x^3+4)/x^9/(x^3-2),x,method=_RETURNVERBOSE)

[Out]

1/240*((42*x^6+48*x^3+60)*(x^3-1)^(2/3)+50*2^(1/3)*x^8*(arctan(1/3*3^(1/2)/x*(x+2*2^(1/3)*(x^3-1)^(1/3)))*3^(1
/2)-1/2*ln((2^(2/3)*x*((-1+x)*(x^2+x+1))^(1/3)+2^(1/3)*x^2+2*((-1+x)*(x^2+x+1))^(2/3))/x^2)+ln((-2^(2/3)*x+2*(
(-1+x)*(x^2+x+1))^(1/3))/x)-1/2*ln(2)))/x^8

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (109) = 218\).

Time = 2.06 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.91 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3+x^6\right )}{x^9 \left (-2+x^3\right )} \, dx=\frac {100 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{8} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, \sqrt {3} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 50 \cdot 4^{\frac {2}{3}} x^{8} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 25 \cdot 4^{\frac {2}{3}} x^{8} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) + 36 \, {\left (7 \, x^{6} + 8 \, x^{3} + 10\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{1440 \, x^{8}} \]

[In]

integrate((x^3-1)^(2/3)*(x^6+x^3+4)/x^9/(x^3-2),x, algorithm="fricas")

[Out]

1/1440*(100*4^(1/6)*sqrt(3)*x^8*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(2*x^7 - 5*x^4 + 2*x)*(x^3 - 1)^(2/3) +
4^(1/3)*sqrt(3)*(91*x^9 - 168*x^6 + 84*x^3 - 8) + 12*sqrt(3)*(19*x^8 - 22*x^5 + 4*x^2)*(x^3 - 1)^(1/3))/(53*x
^9 - 48*x^6 - 12*x^3 + 8)) + 50*4^(2/3)*x^8*log((6*4^(1/3)*(x^3 - 1)^(1/3)*x^2 + 4^(2/3)*(x^3 - 2) - 12*(x^3 -
1)^(2/3)*x)/(x^3 - 2)) - 25*4^(2/3)*x^8*log((6*4^(2/3)*(2*x^4 - x)*(x^3 - 1)^(2/3) + 4^(1/3)*(19*x^6 - 22*x^3
+ 4) + 6*(5*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 - 4*x^3 + 4)) + 36*(7*x^6 + 8*x^3 + 10)*(x^3 - 1)^(2/3))/x^8

Sympy [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3+x^6\right )}{x^9 \left (-2+x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + x^{3} + 4\right )}{x^{9} \left (x^{3} - 2\right )}\, dx \]

[In]

integrate((x**3-1)**(2/3)*(x**6+x**3+4)/x**9/(x**3-2),x)

[Out]

Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x**6 + x**3 + 4)/(x**9*(x**3 - 2)), x)

Maxima [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3+x^6\right )}{x^9 \left (-2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} + x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{9}} \,d x } \]

[In]

integrate((x^3-1)^(2/3)*(x^6+x^3+4)/x^9/(x^3-2),x, algorithm="maxima")

[Out]

integrate((x^6 + x^3 + 4)*(x^3 - 1)^(2/3)/((x^3 - 2)*x^9), x)

Giac [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3+x^6\right )}{x^9 \left (-2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} + x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{9}} \,d x } \]

[In]

integrate((x^3-1)^(2/3)*(x^6+x^3+4)/x^9/(x^3-2),x, algorithm="giac")

[Out]

integrate((x^6 + x^3 + 4)*(x^3 - 1)^(2/3)/((x^3 - 2)*x^9), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3+x^6\right )}{x^9 \left (-2+x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+x^3+4\right )}{x^9\,\left (x^3-2\right )} \,d x \]

[In]

int(((x^3 - 1)^(2/3)*(x^3 + x^6 + 4))/(x^9*(x^3 - 2)),x)

[Out]

int(((x^3 - 1)^(2/3)*(x^3 + x^6 + 4))/(x^9*(x^3 - 2)), x)