3.21.18 \(\int \frac {(-1+x^3)^{2/3} (4+4 x^3+x^6)}{x^9 (1+x^3)} \, dx\) [2018]

3.21.18.1 Optimal result

Integrand size = 30, antiderivative size = 143 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-5+2 x^3-2 x^6\right )}{10 x^8}+\frac {2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]

output

1/10*(x^3-1)^(2/3)*(-2*x^6+2*x^3-5)/x^8+1/3*arctan(3^(1/2)*x/(x+2^(2/3)*(x 
^3-1)^(1/3)))*2^(2/3)*3^(1/2)-1/3*ln(-2*x+2^(2/3)*(x^3-1)^(1/3))*2^(2/3)+1 
/6*ln(2*x^2+2^(2/3)*x*(x^3-1)^(1/3)+2^(1/3)*(x^3-1)^(2/3))*2^(2/3)
 

3.21.18.2 Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-5+2 x^3-2 x^6\right )}{10 x^8}+\frac {2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]

input

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

output

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

3.21.18.3 Rubi [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.90, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1380, 1056, 1050, 27, 1053, 25, 27, 901, 1053, 27, 901}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1380

\(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (x^3+2\right )^2}{x^9 \left (x^3+1\right )}dx\)

\(\Big \downarrow \) 1056

\(\displaystyle 2 \int \frac {\left (x^3-1\right )^{2/3} \left (x^3+2\right )}{x^9 \left (x^3+1\right )}dx+\frac {1}{8} \int \frac {\left (x^3-1\right )^{2/3} \left (x^3+2\right )}{x^6 \left (x^3+1\right )}dx\)

\(\Big \downarrow \) 1050

\(\displaystyle \frac {1}{8} \left (\frac {1}{5} \int \frac {9-x^3}{x^3 \sqrt [3]{x^3-1} \left (x^3+1\right )}dx-\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )+2 \left (\frac {1}{8} \int \frac {4 \left (3-x^3\right )}{x^6 \sqrt [3]{x^3-1} \left (x^3+1\right )}dx-\frac {\left (x^3-1\right )^{2/3}}{4 x^8}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{5} \int \frac {9-x^3}{x^3 \sqrt [3]{x^3-1} \left (x^3+1\right )}dx-\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )+2 \left (\frac {1}{2} \int \frac {3-x^3}{x^6 \sqrt [3]{x^3-1} \left (x^3+1\right )}dx-\frac {\left (x^3-1\right )^{2/3}}{4 x^8}\right )\)

\(\Big \downarrow \) 1053

\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {1}{5} \int -\frac {11-9 x^3}{x^3 \sqrt [3]{x^3-1} \left (x^3+1\right )}dx+\frac {3 \left (x^3-1\right )^{2/3}}{5 x^5}\right )-\frac {\left (x^3-1\right )^{2/3}}{4 x^8}\right )+\frac {1}{8} \left (\frac {1}{5} \left (\frac {1}{2} \int -\frac {20}{\sqrt [3]{x^3-1} \left (x^3+1\right )}dx+\frac {9 \left (x^3-1\right )^{2/3}}{2 x^2}\right )-\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {3 \left (x^3-1\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {11-9 x^3}{x^3 \sqrt [3]{x^3-1} \left (x^3+1\right )}dx\right )-\frac {\left (x^3-1\right )^{2/3}}{4 x^8}\right )+\frac {1}{8} \left (\frac {1}{5} \left (\frac {1}{2} \int -\frac {20}{\sqrt [3]{x^3-1} \left (x^3+1\right )}dx+\frac {9 \left (x^3-1\right )^{2/3}}{2 x^2}\right )-\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {3 \left (x^3-1\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {11-9 x^3}{x^3 \sqrt [3]{x^3-1} \left (x^3+1\right )}dx\right )-\frac {\left (x^3-1\right )^{2/3}}{4 x^8}\right )+\frac {1}{8} \left (\frac {1}{5} \left (\frac {9 \left (x^3-1\right )^{2/3}}{2 x^2}-10 \int \frac {1}{\sqrt [3]{x^3-1} \left (x^3+1\right )}dx\right )-\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )\)

\(\Big \downarrow \) 901

\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {3 \left (x^3-1\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {11-9 x^3}{x^3 \sqrt [3]{x^3-1} \left (x^3+1\right )}dx\right )-\frac {\left (x^3-1\right )^{2/3}}{4 x^8}\right )+\frac {1}{8} \left (\frac {1}{5} \left (\frac {9 \left (x^3-1\right )^{2/3}}{2 x^2}-10 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{2 \sqrt [3]{2}}\right )\right )-\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )\)

\(\Big \downarrow \) 1053

\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {1}{5} \left (-\frac {1}{2} \int -\frac {40}{\sqrt [3]{x^3-1} \left (x^3+1\right )}dx-\frac {11 \left (x^3-1\right )^{2/3}}{2 x^2}\right )+\frac {3 \left (x^3-1\right )^{2/3}}{5 x^5}\right )-\frac {\left (x^3-1\right )^{2/3}}{4 x^8}\right )+\frac {1}{8} \left (\frac {1}{5} \left (\frac {9 \left (x^3-1\right )^{2/3}}{2 x^2}-10 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{2 \sqrt [3]{2}}\right )\right )-\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {1}{5} \left (20 \int \frac {1}{\sqrt [3]{x^3-1} \left (x^3+1\right )}dx-\frac {11 \left (x^3-1\right )^{2/3}}{2 x^2}\right )+\frac {3 \left (x^3-1\right )^{2/3}}{5 x^5}\right )-\frac {\left (x^3-1\right )^{2/3}}{4 x^8}\right )+\frac {1}{8} \left (\frac {1}{5} \left (\frac {9 \left (x^3-1\right )^{2/3}}{2 x^2}-10 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{2 \sqrt [3]{2}}\right )\right )-\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )\)

\(\Big \downarrow \) 901

\(\displaystyle \frac {1}{8} \left (\frac {1}{5} \left (\frac {9 \left (x^3-1\right )^{2/3}}{2 x^2}-10 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{2 \sqrt [3]{2}}\right )\right )-\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )+2 \left (\frac {1}{2} \left (\frac {1}{5} \left (20 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{2 \sqrt [3]{2}}\right )-\frac {11 \left (x^3-1\right )^{2/3}}{2 x^2}\right )+\frac {3 \left (x^3-1\right )^{2/3}}{5 x^5}\right )-\frac {\left (x^3-1\right )^{2/3}}{4 x^8}\right )\)

input

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

output

((-2*(-1 + x^3)^(2/3))/(5*x^5) + ((9*(-1 + x^3)^(2/3))/(2*x^2) - 10*(ArcTa 
n[(1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3]) + Log[1 
+ x^3]/(6*2^(1/3)) - Log[2^(1/3)*x - (-1 + x^3)^(1/3)]/(2*2^(1/3))))/5)/8 
+ 2*(-1/4*(-1 + x^3)^(2/3)/x^8 + ((3*(-1 + x^3)^(2/3))/(5*x^5) + ((-11*(-1 
 + x^3)^(2/3))/(2*x^2) + 20*(ArcTan[(1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/S 
qrt[3]]/(2^(1/3)*Sqrt[3]) + Log[1 + x^3]/(6*2^(1/3)) - Log[2^(1/3)*x - (-1 
 + x^3)^(1/3)]/(2*2^(1/3))))/5)/2)
 

3.21.18.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 901

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit 
h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S 
qrt[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 1050

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] - Simp[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] && G 
tQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
 

rule 1053

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] + Simp[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]
 

rule 1056

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Simp[e   Int[(g*x)^m 
*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^(r - 1), x], x] + Simp[f/e^n   Int 
[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^(r - 1), x], x] /; F 
reeQ[{a, b, c, d, e, f, g, m, p, q}, x] && IGtQ[n, 0] && IGtQ[r, 0]
 

rule 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

3.21.18.4 Maple [A] (verified)

Time = 13.60 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {-5 x^{8} \left (2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {2}{3}}-3 \left (x^{3}-1\right )^{\frac {2}{3}} \left (2 x^{6}-2 x^{3}+5\right )}{30 x^{8}}\)	\(119\)
risch	\(\text {Expression too large to display}\)	\(913\)
trager	\(\text {Expression too large to display}\)	\(1125\)

input

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

output

1/30*(-5*x^8*(2*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(x^3-1)^(1/3)))*3^(1/2)+2* 
ln((-2^(1/3)*x+(x^3-1)^(1/3))/x)-ln((2^(2/3)*x^2+2^(1/3)*x*(x^3-1)^(1/3)+( 
x^3-1)^(2/3))/x^2))*2^(2/3)-3*(x^3-1)^(2/3)*(2*x^6-2*x^3+5))/x^8
 

3.21.18.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (110) = 220\).

Time = 1.74 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.93 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=-\frac {10 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{8} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \, \left (-4\right )^{\frac {1}{3}} x^{8} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 5 \, \left (-4\right )^{\frac {1}{3}} x^{8} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 9 \, {\left (2 \, x^{6} - 2 \, x^{3} + 5\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{8}} \]

input

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

output

-1/90*(10*sqrt(3)*(-4)^(1/3)*x^8*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(5*x^7 + 
 4*x^4 - x)*(x^3 - 1)^(2/3) + 6*sqrt(3)*(-4)^(1/3)*(19*x^8 - 16*x^5 + x^2) 
*(x^3 - 1)^(1/3) - sqrt(3)*(71*x^9 - 111*x^6 + 33*x^3 - 1))/(109*x^9 - 105 
*x^6 + 3*x^3 + 1)) - 10*(-4)^(1/3)*x^8*log(-(3*(-4)^(2/3)*(x^3 - 1)^(1/3)* 
x^2 - 6*(x^3 - 1)^(2/3)*x + (-4)^(1/3)*(x^3 + 1))/(x^3 + 1)) + 5*(-4)^(1/3 
)*x^8*log(-(6*(-4)^(1/3)*(5*x^4 - x)*(x^3 - 1)^(2/3) - (-4)^(2/3)*(19*x^6 
- 16*x^3 + 1) - 24*(2*x^5 - x^2)*(x^3 - 1)^(1/3))/(x^6 + 2*x^3 + 1)) + 9*( 
2*x^6 - 2*x^3 + 5)*(x^3 - 1)^(2/3))/x^8
 

3.21.18.6 Sympy [F]

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

input

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

output

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

3.21.18.7 Maxima [F]

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

input

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

output

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

3.21.18.8 Giac [F]

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

input

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

output

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

3.21.18.9 Mupad [F(-1)]

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

input

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

output

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