\(\int \frac {3+8 x^3-3 x^6+x^9}{(-1+x^3)^{2/3} (-27+x^3-2 x^6+x^9)} \, dx\) [15]

Optimal result

Integrand size = 40, antiderivative size = 194 \[ \int \frac {3+8 x^3-3 x^6+x^9}{\left (-1+x^3\right )^{2/3} \left (-27+x^3-2 x^6+x^9\right )} \, dx=\frac {\arctan \left (\frac {258 \sqrt {3} \left (-1+x^3\right )^{2/3} \left (-59 x^2+2 x^5\right )-258 \sqrt {3} \sqrt [3]{-1+x^3} \left (-24 x+56 x^4+13 x^7\right )-\sqrt {3} \left (-10368+7135 x^3+7789 x^6+169 x^9\right )}{13824-25667 x^3+13021 x^6+2197 x^9}\right )}{3 \sqrt {3}}+\frac {1}{18} \log \left (\frac {-27+82 x^3-83 x^6+x^9+18 \left (-1+x^3\right )^{2/3} \left (-5 x^2+2 x^5\right )-9 \sqrt [3]{-1+x^3} \left (3 x-13 x^4+x^7\right )}{-27+x^3-2 x^6+x^9}\right ) \] Output:

1/9*3^(1/2)*arctan((258*3^(1/2)*(2*x^5-59*x^2)*(x^3-1)^(2/3)-258*3^(1/2)*( 
13*x^7+56*x^4-24*x)*(x^3-1)^(1/3)-3^(1/2)*(169*x^9+7789*x^6+7135*x^3-10368 
))/(2197*x^9+13021*x^6-25667*x^3+13824))+1/18*ln((x^9-83*x^6+82*x^3+18*(2* 
x^5-5*x^2)*(x^3-1)^(2/3)-9*(x^7-13*x^4+3*x)*(x^3-1)^(1/3)-27)/(x^9-2*x^6+x 
^3-27))
 

Mathematica [F]

\[ \int \frac {3+8 x^3-3 x^6+x^9}{\left (-1+x^3\right )^{2/3} \left (-27+x^3-2 x^6+x^9\right )} \, dx=\int \frac {3+8 x^3-3 x^6+x^9}{\left (-1+x^3\right )^{2/3} \left (-27+x^3-2 x^6+x^9\right )} \, dx \] Input:

Integrate[(3 + 8*x^3 - 3*x^6 + x^9)/((-1 + x^3)^(2/3)*(-27 + x^3 - 2*x^6 + 
 x^9)),x]
 

Output:

Integrate[(3 + 8*x^3 - 3*x^6 + x^9)/((-1 + x^3)^(2/3)*(-27 + x^3 - 2*x^6 + 
 x^9)), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {x^9-3 x^6+8 x^3+3}{\left (x^3-1\right )^{2/3} \left (x^9-2 x^6+x^3-27\right )} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {\left (-x^2+2 x+5\right ) \left (x^9-3 x^6+8 x^3+3\right )}{135 \left (x^3-1\right )^{2/3} \left (x^3+x^2-2 x-3\right )}+\frac {\left (x^5-4 x^4+6 x^3-16 x^2+16 x-30\right ) \left (x^9-3 x^6+8 x^3+3\right )}{135 \left (x^3-1\right )^{2/3} \left (x^6-x^5+3 x^4-4 x^3+7 x^2-6 x+9\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {10}{9} \int \frac {1}{\left (x^3-1\right )^{2/3} \left (x^3+x^2-2 x-3\right )}dx+\frac {7}{9} \int \frac {x}{\left (x^3-1\right )^{2/3} \left (x^3+x^2-2 x-3\right )}dx-\frac {2}{9} \int \frac {x^2}{\left (x^3-1\right )^{2/3} \left (x^3+x^2-2 x-3\right )}dx-\frac {20}{3} \int \frac {1}{\left (x^3-1\right )^{2/3} \left (x^6-x^5+3 x^4-4 x^3+7 x^2-6 x+9\right )}dx+\frac {41}{9} \int \frac {x}{\left (x^3-1\right )^{2/3} \left (x^6-x^5+3 x^4-4 x^3+7 x^2-6 x+9\right )}dx-\frac {44}{9} \int \frac {x^2}{\left (x^3-1\right )^{2/3} \left (x^6-x^5+3 x^4-4 x^3+7 x^2-6 x+9\right )}dx+\int \frac {x^3}{\left (x^3-1\right )^{2/3} \left (x^6-x^5+3 x^4-4 x^3+7 x^2-6 x+9\right )}dx-\frac {11}{9} \int \frac {x^4}{\left (x^3-1\right )^{2/3} \left (x^6-x^5+3 x^4-4 x^3+7 x^2-6 x+9\right )}dx+\frac {2}{9} \int \frac {x^5}{\left (x^3-1\right )^{2/3} \left (x^6-x^5+3 x^4-4 x^3+7 x^2-6 x+9\right )}dx+\frac {x \left (1-x^3\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )}{\left (x^3-1\right )^{2/3}}\)

Input:

Int[(3 + 8*x^3 - 3*x^6 + x^9)/((-1 + x^3)^(2/3)*(-27 + x^3 - 2*x^6 + x^9)) 
,x]
 

Output:

$Aborted
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.43 (sec) , antiderivative size = 845, normalized size of antiderivative = 4.36

method	result	size
trager	\(\text {Expression too large to display}\)	\(845\)

Input:

int((x^9-3*x^6+8*x^3+3)/(x^3-1)^(2/3)/(x^9-2*x^6+x^3-27),x,method=_RETURNV 
ERBOSE)
 

Output:

1/9*ln((-206955*RootOf(81*_Z^2+9*_Z+1)^2*x^9-1011123*RootOf(81*_Z^2+9*_Z+1 
)^2*(x^3-1)^(1/3)*x^7-106434*RootOf(81*_Z^2+9*_Z+1)*x^9+4516155*(x^3-1)^(2 
/3)*RootOf(81*_Z^2+9*_Z+1)^2*x^5+206955*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(1/ 
3)*x^7-9271*x^9+407187*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(2/3)*x^5+35478*(x^3 
-1)^(1/3)*x^7-5173875*RootOf(81*_Z^2+9*_Z+1)^2*x^6-63162*(x^3-1)^(2/3)*x^5 
-8088984*RootOf(81*_Z^2+9*_Z+1)^2*(x^3-1)^(1/3)*x^4-2039985*RootOf(81*_Z^2 
+9*_Z+1)*x^6+2329722*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(1/3)*x^4+18542*x^6-24 
42069*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(2/3)*x^2+70956*(x^3-1)^(1/3)*x^4-206 
955*RootOf(81*_Z^2+9*_Z+1)^2*x^3-419166*(x^3-1)^(2/3)*x^2+514431*RootOf(81 
*_Z^2+9*_Z+1)*x^3+337041*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(1/3)*x+241046*x^3 
-106434*(x^3-1)^(1/3)*x-620865*RootOf(81*_Z^2+9*_Z+1)-250317)/(x^3+x^2-2*x 
-3)/(x^6-x^5+3*x^4-4*x^3+7*x^2-6*x+9))+RootOf(81*_Z^2+9*_Z+1)*ln((-60444*R 
ootOf(81*_Z^2+9*_Z+1)^2*x^9-319302*RootOf(81*_Z^2+9*_Z+1)^2*(x^3-1)^(1/3)* 
x^7-9271*RootOf(81*_Z^2+9*_Z+1)*x^9+501795*(x^3-1)^(2/3)*RootOf(81*_Z^2+9* 
_Z+1)^2*x^5+12483*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(1/3)*x^7+66267*RootOf(81 
*_Z^2+9*_Z+1)*(x^3-1)^(2/3)*x^5-1511100*RootOf(81*_Z^2+9*_Z+1)^2*x^6-5850* 
(x^3-1)^(2/3)*x^5-2554416*RootOf(81*_Z^2+9*_Z+1)^2*(x^3-1)^(1/3)*x^4-41310 
7*RootOf(81*_Z^2+9*_Z+1)*x^6-113004*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(1/3)*x 
^4-27813*x^6+271341*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(2/3)*x^2+8322*(x^3-1)^ 
(1/3)*x^4-60444*RootOf(81*_Z^2+9*_Z+1)^2*x^3-16425*(x^3-1)^(2/3)*x^2-19...
 

Fricas [A] (verification not implemented)

Time = 3.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.90 \[ \int \frac {3+8 x^3-3 x^6+x^9}{\left (-1+x^3\right )^{2/3} \left (-27+x^3-2 x^6+x^9\right )} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {258 \, \sqrt {3} {\left (2 \, x^{5} - 59 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 258 \, \sqrt {3} {\left (13 \, x^{7} + 56 \, x^{4} - 24 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (169 \, x^{9} + 7789 \, x^{6} + 7135 \, x^{3} - 10368\right )}}{2197 \, x^{9} + 13021 \, x^{6} - 25667 \, x^{3} + 13824}\right ) + \frac {1}{18} \, \log \left (\frac {x^{9} - 83 \, x^{6} + 82 \, x^{3} + 18 \, {\left (2 \, x^{5} - 5 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 9 \, {\left (x^{7} - 13 \, x^{4} + 3 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 27}{x^{9} - 2 \, x^{6} + x^{3} - 27}\right ) \] Input:

integrate((x^9-3*x^6+8*x^3+3)/(x^3-1)^(2/3)/(x^9-2*x^6+x^3-27),x, algorith 
m="fricas")
 

Output:

1/9*sqrt(3)*arctan((258*sqrt(3)*(2*x^5 - 59*x^2)*(x^3 - 1)^(2/3) - 258*sqr 
t(3)*(13*x^7 + 56*x^4 - 24*x)*(x^3 - 1)^(1/3) - sqrt(3)*(169*x^9 + 7789*x^ 
6 + 7135*x^3 - 10368))/(2197*x^9 + 13021*x^6 - 25667*x^3 + 13824)) + 1/18* 
log((x^9 - 83*x^6 + 82*x^3 + 18*(2*x^5 - 5*x^2)*(x^3 - 1)^(2/3) - 9*(x^7 - 
 13*x^4 + 3*x)*(x^3 - 1)^(1/3) - 27)/(x^9 - 2*x^6 + x^3 - 27))
 

Sympy [F]

\[ \int \frac {3+8 x^3-3 x^6+x^9}{\left (-1+x^3\right )^{2/3} \left (-27+x^3-2 x^6+x^9\right )} \, dx=\int \frac {x^{9} - 3 x^{6} + 8 x^{3} + 3}{\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} + x^{2} - 2 x - 3\right ) \left (x^{6} - x^{5} + 3 x^{4} - 4 x^{3} + 7 x^{2} - 6 x + 9\right )}\, dx \] Input:

integrate((x**9-3*x**6+8*x**3+3)/(x**3-1)**(2/3)/(x**9-2*x**6+x**3-27),x)
 

Output:

Integral((x**9 - 3*x**6 + 8*x**3 + 3)/(((x - 1)*(x**2 + x + 1))**(2/3)*(x* 
*3 + x**2 - 2*x - 3)*(x**6 - x**5 + 3*x**4 - 4*x**3 + 7*x**2 - 6*x + 9)), 
x)
 

Maxima [F]

\[ \int \frac {3+8 x^3-3 x^6+x^9}{\left (-1+x^3\right )^{2/3} \left (-27+x^3-2 x^6+x^9\right )} \, dx=\int { \frac {x^{9} - 3 \, x^{6} + 8 \, x^{3} + 3}{{\left (x^{9} - 2 \, x^{6} + x^{3} - 27\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate((x^9-3*x^6+8*x^3+3)/(x^3-1)^(2/3)/(x^9-2*x^6+x^3-27),x, algorith 
m="maxima")
 

Output:

integrate((x^9 - 3*x^6 + 8*x^3 + 3)/((x^9 - 2*x^6 + x^3 - 27)*(x^3 - 1)^(2 
/3)), x)
 

Giac [F]

\[ \int \frac {3+8 x^3-3 x^6+x^9}{\left (-1+x^3\right )^{2/3} \left (-27+x^3-2 x^6+x^9\right )} \, dx=\int { \frac {x^{9} - 3 \, x^{6} + 8 \, x^{3} + 3}{{\left (x^{9} - 2 \, x^{6} + x^{3} - 27\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate((x^9-3*x^6+8*x^3+3)/(x^3-1)^(2/3)/(x^9-2*x^6+x^3-27),x, algorith 
m="giac")
 

Output:

integrate((x^9 - 3*x^6 + 8*x^3 + 3)/((x^9 - 2*x^6 + x^3 - 27)*(x^3 - 1)^(2 
/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {3+8 x^3-3 x^6+x^9}{\left (-1+x^3\right )^{2/3} \left (-27+x^3-2 x^6+x^9\right )} \, dx=\int \frac {x^9-3\,x^6+8\,x^3+3}{{\left (x^3-1\right )}^{2/3}\,\left (x^9-2\,x^6+x^3-27\right )} \,d x \] Input:

int((8*x^3 - 3*x^6 + x^9 + 3)/((x^3 - 1)^(2/3)*(x^3 - 2*x^6 + x^9 - 27)),x 
)
 

Output:

int((8*x^3 - 3*x^6 + x^9 + 3)/((x^3 - 1)^(2/3)*(x^3 - 2*x^6 + x^9 - 27)), 
x)
 

Reduce [F]

\[ \int \frac {3+8 x^3-3 x^6+x^9}{\left (-1+x^3\right )^{2/3} \left (-27+x^3-2 x^6+x^9\right )} \, dx=\int \frac {x^{9}}{\left (x^{3}-1\right )^{\frac {2}{3}} x^{9}-2 \left (x^{3}-1\right )^{\frac {2}{3}} x^{6}+\left (x^{3}-1\right )^{\frac {2}{3}} x^{3}-27 \left (x^{3}-1\right )^{\frac {2}{3}}}d x -3 \left (\int \frac {x^{6}}{\left (x^{3}-1\right )^{\frac {2}{3}} x^{9}-2 \left (x^{3}-1\right )^{\frac {2}{3}} x^{6}+\left (x^{3}-1\right )^{\frac {2}{3}} x^{3}-27 \left (x^{3}-1\right )^{\frac {2}{3}}}d x \right )+8 \left (\int \frac {x^{3}}{\left (x^{3}-1\right )^{\frac {2}{3}} x^{9}-2 \left (x^{3}-1\right )^{\frac {2}{3}} x^{6}+\left (x^{3}-1\right )^{\frac {2}{3}} x^{3}-27 \left (x^{3}-1\right )^{\frac {2}{3}}}d x \right )+3 \left (\int \frac {1}{\left (x^{3}-1\right )^{\frac {2}{3}} x^{9}-2 \left (x^{3}-1\right )^{\frac {2}{3}} x^{6}+\left (x^{3}-1\right )^{\frac {2}{3}} x^{3}-27 \left (x^{3}-1\right )^{\frac {2}{3}}}d x \right ) \] Input:

int((x^9-3*x^6+8*x^3+3)/(x^3-1)^(2/3)/(x^9-2*x^6+x^3-27),x)
 

Output:

int(x**9/((x**3 - 1)**(2/3)*x**9 - 2*(x**3 - 1)**(2/3)*x**6 + (x**3 - 1)** 
(2/3)*x**3 - 27*(x**3 - 1)**(2/3)),x) - 3*int(x**6/((x**3 - 1)**(2/3)*x**9 
 - 2*(x**3 - 1)**(2/3)*x**6 + (x**3 - 1)**(2/3)*x**3 - 27*(x**3 - 1)**(2/3 
)),x) + 8*int(x**3/((x**3 - 1)**(2/3)*x**9 - 2*(x**3 - 1)**(2/3)*x**6 + (x 
**3 - 1)**(2/3)*x**3 - 27*(x**3 - 1)**(2/3)),x) + 3*int(1/((x**3 - 1)**(2/ 
3)*x**9 - 2*(x**3 - 1)**(2/3)*x**6 + (x**3 - 1)**(2/3)*x**3 - 27*(x**3 - 1 
)**(2/3)),x)