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

Optimal result

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

[Out] 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+8 
2*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))
 

Rubi [F]

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

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

[Out] (x*(1 - x^3)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, x^3])/(-1 + x^3)^( 
2/3) + (10*Defer[Int][1/((-1 + x^3)^(2/3)*(-3 - 2*x + x^2 + x^3)), x]) 
/9 + (7*Defer[Int][x/((-1 + x^3)^(2/3)*(-3 - 2*x + x^2 + x^3)), x])/9 
- (2*Defer[Int][x^2/((-1 + x^3)^(2/3)*(-3 - 2*x + x^2 + x^3)), x])/9 - 
 (20*Defer[Int][1/((-1 + x^3)^(2/3)*(9 - 6*x + 7*x^2 - 4*x^3 + 3*x^4 - 
 x^5 + x^6)), x])/3 + (41*Defer[Int][x/((-1 + x^3)^(2/3)*(9 - 6*x + 7* 
x^2 - 4*x^3 + 3*x^4 - x^5 + x^6)), x])/9 - (44*Defer[Int][x^2/((-1 + x 
^3)^(2/3)*(9 - 6*x + 7*x^2 - 4*x^3 + 3*x^4 - x^5 + x^6)), x])/9 + Defe 
r[Int][x^3/((-1 + x^3)^(2/3)*(9 - 6*x + 7*x^2 - 4*x^3 + 3*x^4 - x^5 + 
x^6)), x] - (11*Defer[Int][x^4/((-1 + x^3)^(2/3)*(9 - 6*x + 7*x^2 - 4* 
x^3 + 3*x^4 - x^5 + x^6)), x])/9 + (2*Defer[Int][x^5/((-1 + x^3)^(2/3) 
*(9 - 6*x + 7*x^2 - 4*x^3 + 3*x^4 - x^5 + x^6)), x])/9
 

Rubi steps \begin{align*} \text {integral}= \int \left (\frac {\left (5+2 x-x^2\right ) \left (3+8 x^3-3 x^6+x^9\right )}{135 \left (-1+x^3\right )^{2/3} \left (-3-2 x+x^2+x^3\right )}+\frac {\left (-30+16 x-16 x^2+6 x^3-4 x^4+x^5\right ) \left (3+8 x^3-3 x^6+x^9\right )}{135 \left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )}\right ) \, dx \\ = \frac {1}{135} \int \frac {\left (5+2 x-x^2\right ) \left (3+8 x^3-3 x^6+x^9\right )}{\left (-1+x^3\right )^{2/3} \left (-3-2 x+x^2+x^3\right )} \, dx+\frac {1}{135} \int \frac {\left (-30+16 x-16 x^2+6 x^3-4 x^4+x^5\right ) \left (3+8 x^3-3 x^6+x^9\right )}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )} \, dx \\ = \frac {1}{135} \int \left (\frac {45}{\left (-1+x^3\right )^{2/3}}+\frac {3 x}{\left (-1+x^3\right )^{2/3}}+\frac {4 x^2}{\left (-1+x^3\right )^{2/3}}-\frac {3 x^4}{\left (-1+x^3\right )^{2/3}}+\frac {6 x^5}{\left (-1+x^3\right )^{2/3}}+\frac {3 x^7}{\left (-1+x^3\right )^{2/3}}-\frac {x^8}{\left (-1+x^3\right )^{2/3}}+\frac {15 \left (10+7 x-2 x^2\right )}{\left (-1+x^3\right )^{2/3} \left (-3-2 x+x^2+x^3\right )}\right ) \, dx+\frac {1}{135} \int \left (\frac {90}{\left (-1+x^3\right )^{2/3}}-\frac {3 x}{\left (-1+x^3\right )^{2/3}}-\frac {4 x^2}{\left (-1+x^3\right )^{2/3}}+\frac {3 x^4}{\left (-1+x^3\right )^{2/3}}-\frac {6 x^5}{\left (-1+x^3\right )^{2/3}}-\frac {3 x^7}{\left (-1+x^3\right )^{2/3}}+\frac {x^8}{\left (-1+x^3\right )^{2/3}}-\frac {15 \left (60-41 x+44 x^2-9 x^3+11 x^4-2 x^5\right )}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )}\right ) \, dx \\ = \frac {1}{9} \int \frac {10+7 x-2 x^2}{\left (-1+x^3\right )^{2/3} \left (-3-2 x+x^2+x^3\right )} \, dx-\frac {1}{9} \int \frac {60-41 x+44 x^2-9 x^3+11 x^4-2 x^5}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )} \, dx+\frac {1}{3} \int \frac {1}{\left (-1+x^3\right )^{2/3}} \, dx+\frac {2}{3} \int \frac {1}{\left (-1+x^3\right )^{2/3}} \, dx \\ = \frac {1}{9} \int \left (\frac {10}{\left (-1+x^3\right )^{2/3} \left (-3-2 x+x^2+x^3\right )}+\frac {7 x}{\left (-1+x^3\right )^{2/3} \left (-3-2 x+x^2+x^3\right )}-\frac {2 x^2}{\left (-1+x^3\right )^{2/3} \left (-3-2 x+x^2+x^3\right )}\right ) \, dx-\frac {1}{9} \int \left (\frac {60}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )}-\frac {41 x}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )}+\frac {44 x^2}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )}-\frac {9 x^3}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )}+\frac {11 x^4}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )}-\frac {2 x^5}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )}\right ) \, dx+\frac {\left (1-x^3\right )^{2/3}}{3 \left (-1+x^3\right )^{2/3}} \int \frac {1}{\left (1-x^3\right )^{2/3}} \, dx+\frac {2 \left (1-x^3\right )^{2/3}}{3 \left (-1+x^3\right )^{2/3}} \int \frac {1}{\left (1-x^3\right )^{2/3}} \, dx \\ = \frac {x \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};x^3\right )}{\left (-1+x^3\right )^{2/3}}-\frac {2}{9} \int \frac {x^2}{\left (-1+x^3\right )^{2/3} \left (-3-2 x+x^2+x^3\right )} \, dx+\frac {2}{9} \int \frac {x^5}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )} \, dx+\frac {7}{9} \int \frac {x}{\left (-1+x^3\right )^{2/3} \left (-3-2 x+x^2+x^3\right )} \, dx+\frac {10}{9} \int \frac {1}{\left (-1+x^3\right )^{2/3} \left (-3-2 x+x^2+x^3\right )} \, dx-\frac {11}{9} \int \frac {x^4}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )} \, dx+\frac {41}{9} \int \frac {x}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )} \, dx-\frac {44}{9} \int \frac {x^2}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )} \, dx-\frac {20}{3} \int \frac {1}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )} \, dx+\int \frac {x^3}{\left (-1+x^3\right )^{2/3} \left (9-6 x+7 x^2-4 x^3+3 x^4-x^5+x^6\right )} \, dx \\ \end{align*}

Mathematica [F]

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

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

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

Maple [C] (verified)

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

Time = 1.28 (sec) , antiderivative size = 845, normalized size of antiderivative = 4.36

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

[In] 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=_RET 
URNVERBOSE)
 

[Out] 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*Ro 
otOf(81*_Z^2+9*_Z+1)^2*(x^3-1)^(2/3)*x^5+206955*RootOf(81*_Z^2+9*_Z+1) 
*(x^3-1)^(1/3)*x^7-9271*x^9-5173875*RootOf(81*_Z^2+9*_Z+1)^2*x^6+40718 
7*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(2/3)*x^5+35478*(x^3-1)^(1/3)*x^7-808 
8984*RootOf(81*_Z^2+9*_Z+1)^2*(x^3-1)^(1/3)*x^4-2039985*RootOf(81*_Z^2 
+9*_Z+1)*x^6-63162*(x^3-1)^(2/3)*x^5+2329722*RootOf(81*_Z^2+9*_Z+1)*(x 
^3-1)^(1/3)*x^4+18542*x^6-206955*RootOf(81*_Z^2+9*_Z+1)^2*x^3-2442069* 
RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(2/3)*x^2+70956*(x^3-1)^(1/3)*x^4+51443 
1*RootOf(81*_Z^2+9*_Z+1)*x^3-419166*(x^3-1)^(2/3)*x^2+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*RootOf(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*RootOf(81*_Z^2+9*_Z+1)^2*(x^3-1)^(2/3)*x^5+12 
483*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(1/3)*x^7-1511100*RootOf(81*_Z^2+9* 
_Z+1)^2*x^6+66267*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(2/3)*x^5-2554416*Roo 
tOf(81*_Z^2+9*_Z+1)^2*(x^3-1)^(1/3)*x^4-413107*RootOf(81*_Z^2+9*_Z+1)* 
x^6-5850*(x^3-1)^(2/3)*x^5-113004*RootOf(81*_Z^2+9*_Z+1)*(x^3-1)^(1/3) 
*x^4-27813*x^6-60444*RootOf(81*_Z^2+9*_Z+1)^2*x^3+271341*RootOf(81*_Z^ 
2+9*_Z+1)*(x^3-1)^(2/3)*x^2+8322*(x^3-1)^(1/3)*x^4-190603*RootOf(81*_Z 
^2+9*_Z+1)*x^3-16425*(x^3-1)^(2/3)*x^2-106434*RootOf(81*_Z^2+9*_Z+1)*( 
x^3-1)^(1/3)*x-27813*x^3+4161*(x^3-1)^(1/3)*x+181332*RootOf(81*_Z^2+9* 
_Z+1)+27813)/(x^3+x^2-2*x-3)/(x^6-x^5+3*x^4-4*x^3+7*x^2-6*x+9))
 

Fricas [A] (verification not implemented)

none

Time = 3.21 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.90 \[ \int \frac {3+8 x^3-3 x^6+x^9}{\left (-1+x^3\right )^{2/3} \left (-27+x^3-2 x^6+x^9\right )} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {258 \, \sqrt {3} {\left (2 \, x^{5} - 59 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 258 \, \sqrt {3} {\left (13 \, x^{7} + 56 \, x^{4} - 24 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (169 \, x^{9} + 7789 \, x^{6} + 7135 \, x^{3} - 10368\right )}}{2197 \, x^{9} + 13021 \, x^{6} - 25667 \, x^{3} + 13824}\right ) + \frac {1}{18} \, \log \left (\frac {x^{9} - 83 \, x^{6} + 82 \, x^{3} + 18 \, {\left (2 \, x^{5} - 5 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 9 \, {\left (x^{7} - 13 \, x^{4} + 3 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 27}{x^{9} - 2 \, x^{6} + x^{3} - 27}\right ) \]

[In] integrate((x^9-3*x^6+8*x^3+3)/((x^3-1)^(2/3)*(x^9-2*x^6+x^3-27)),x, al 
gorithm="fricas")
 

[Out] 1/9*sqrt(3)*arctan((258*sqrt(3)*(2*x^5 - 59*x^2)*(x^3 - 1)^(2/3) - 258 
*sqrt(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 + 138 
24)) + 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]

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

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

[Out] 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]

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

[In] integrate((x^9-3*x^6+8*x^3+3)/((x^3-1)^(2/3)*(x^9-2*x^6+x^3-27)),x, al 
gorithm="maxima")
 

[Out] 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]

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

[In] integrate((x^9-3*x^6+8*x^3+3)/((x^3-1)^(2/3)*(x^9-2*x^6+x^3-27)),x, al 
gorithm="giac")
 

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

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

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