Integrand size = 22, antiderivative size = 121 \[ \int x^5 \sqrt {1-x^3} \left (1+x^9\right )^2 \, dx=-\frac {8}{9} \left (1-x^3\right )^{3/2}+\frac {32}{15} \left (1-x^3\right )^{5/2}-\frac {22}{7} \left (1-x^3\right )^{7/2}+\frac {86}{27} \left (1-x^3\right )^{9/2}-\frac {74}{33} \left (1-x^3\right )^{11/2}+\frac {14}{13} \left (1-x^3\right )^{13/2}-\frac {14}{45} \left (1-x^3\right )^{15/2}+\frac {2}{51} \left (1-x^3\right )^{17/2} \]
-8/9*(-x^3+1)^(3/2)+32/15*(-x^3+1)^(5/2)-22/7*(-x^3+1)^(7/2)+86/27*(-x^3+1 )^(9/2)-74/33*(-x^3+1)^(11/2)+14/13*(-x^3+1)^(13/2)-14/45*(-x^3+1)^(15/2)+ 2/51*(-x^3+1)^(17/2)
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.47 \[ \int x^5 \sqrt {1-x^3} \left (1+x^9\right )^2 \, dx=\frac {2 \sqrt {1-x^3} \left (-173014-86507 x^3+126561 x^6-22160 x^9-19390 x^{12}+135702 x^{15}-3234 x^{18}-3003 x^{21}+45045 x^{24}\right )}{2297295} \]
(2*Sqrt[1 - x^3]*(-173014 - 86507*x^3 + 126561*x^6 - 22160*x^9 - 19390*x^1 2 + 135702*x^15 - 3234*x^18 - 3003*x^21 + 45045*x^24))/2297295
Time = 0.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2361, 2123, 2009}
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.
\(\displaystyle \int x^5 \sqrt {1-x^3} \left (x^9+1\right )^2 \, dx\) |
\(\Big \downarrow \) 2361 |
\(\displaystyle \frac {1}{3} \int x^3 \sqrt {1-x^3} \left (x^9+1\right )^2dx^3\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \frac {1}{3} \int \left (-\left (1-x^3\right )^{15/2}+7 \left (1-x^3\right )^{13/2}-21 \left (1-x^3\right )^{11/2}+37 \left (1-x^3\right )^{9/2}-43 \left (1-x^3\right )^{7/2}+33 \left (1-x^3\right )^{5/2}-16 \left (1-x^3\right )^{3/2}+4 \sqrt {1-x^3}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {2}{17} \left (1-x^3\right )^{17/2}-\frac {14}{15} \left (1-x^3\right )^{15/2}+\frac {42}{13} \left (1-x^3\right )^{13/2}-\frac {74}{11} \left (1-x^3\right )^{11/2}+\frac {86}{9} \left (1-x^3\right )^{9/2}-\frac {66}{7} \left (1-x^3\right )^{7/2}+\frac {32}{5} \left (1-x^3\right )^{5/2}-\frac {8}{3} \left (1-x^3\right )^{3/2}\right )\) |
((-8*(1 - x^3)^(3/2))/3 + (32*(1 - x^3)^(5/2))/5 - (66*(1 - x^3)^(7/2))/7 + (86*(1 - x^3)^(9/2))/9 - (74*(1 - x^3)^(11/2))/11 + (42*(1 - x^3)^(13/2) )/13 - (14*(1 - x^3)^(15/2))/15 + (2*(1 - x^3)^(17/2))/17)/3
3.7.86.3.1 Defintions of rubi rules used
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*SubstFor[x^n, Pq, x]*(a + b*x)^p, x ], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[S implify[(m + 1)/n]]
Time = 1.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.40
method | result | size |
pseudoelliptic | \(-\frac {2 \left (-x^{3}+1\right )^{\frac {3}{2}} \left (45045 x^{21}+42042 x^{18}+38808 x^{15}+174510 x^{12}+155120 x^{9}+132960 x^{6}+259521 x^{3}+173014\right )}{2297295}\) | \(49\) |
trager | \(\left (\frac {2}{51} x^{24}-\frac {2}{765} x^{21}-\frac {28}{9945} x^{18}+\frac {1436}{12155} x^{15}-\frac {1108}{65637} x^{12}-\frac {8864}{459459} x^{9}+\frac {84374}{765765} x^{6}-\frac {173014}{2297295} x^{3}-\frac {346028}{2297295}\right ) \sqrt {-x^{3}+1}\) | \(53\) |
gosper | \(\frac {2 \sqrt {-x^{3}+1}\, \left (45045 x^{21}+42042 x^{18}+38808 x^{15}+174510 x^{12}+155120 x^{9}+132960 x^{6}+259521 x^{3}+173014\right ) \left (x -1\right ) \left (x^{2}+x +1\right )}{2297295}\) | \(58\) |
risch | \(-\frac {2 \left (45045 x^{24}-3003 x^{21}-3234 x^{18}+135702 x^{15}-19390 x^{12}-22160 x^{9}+126561 x^{6}-86507 x^{3}-173014\right ) \left (x^{3}-1\right )}{2297295 \sqrt {-x^{3}+1}}\) | \(59\) |
default | \(\frac {84374 x^{6} \sqrt {-x^{3}+1}}{765765}-\frac {173014 x^{3} \sqrt {-x^{3}+1}}{2297295}-\frac {346028 \sqrt {-x^{3}+1}}{2297295}+\frac {2 x^{24} \sqrt {-x^{3}+1}}{51}-\frac {2 x^{21} \sqrt {-x^{3}+1}}{765}-\frac {28 x^{18} \sqrt {-x^{3}+1}}{9945}+\frac {1436 x^{15} \sqrt {-x^{3}+1}}{12155}-\frac {1108 x^{12} \sqrt {-x^{3}+1}}{65637}-\frac {8864 x^{9} \sqrt {-x^{3}+1}}{459459}\) | \(125\) |
elliptic | \(\frac {84374 x^{6} \sqrt {-x^{3}+1}}{765765}-\frac {173014 x^{3} \sqrt {-x^{3}+1}}{2297295}-\frac {346028 \sqrt {-x^{3}+1}}{2297295}+\frac {2 x^{24} \sqrt {-x^{3}+1}}{51}-\frac {2 x^{21} \sqrt {-x^{3}+1}}{765}-\frac {28 x^{18} \sqrt {-x^{3}+1}}{9945}+\frac {1436 x^{15} \sqrt {-x^{3}+1}}{12155}-\frac {1108 x^{12} \sqrt {-x^{3}+1}}{65637}-\frac {8864 x^{9} \sqrt {-x^{3}+1}}{459459}\) | \(125\) |
meijerg | \(-\frac {-\frac {8192 \sqrt {\pi }}{109395}+\frac {4 \sqrt {\pi }\, \left (-x^{3}+1\right )^{\frac {3}{2}} \left (6435 x^{21}+6006 x^{18}+5544 x^{15}+5040 x^{12}+4480 x^{9}+3840 x^{6}+3072 x^{3}+2048\right )}{109395}}{6 \sqrt {\pi }}+\frac {\frac {512 \sqrt {\pi }}{3465}-\frac {4 \sqrt {\pi }\, \left (-x^{3}+1\right )^{\frac {3}{2}} \left (315 x^{12}+280 x^{9}+240 x^{6}+192 x^{3}+128\right )}{3465}}{3 \sqrt {\pi }}-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (-x^{3}+1\right )^{\frac {3}{2}} \left (3 x^{3}+2\right )}{15}}{6 \sqrt {\pi }}\) | \(143\) |
-2/2297295*(-x^3+1)^(3/2)*(45045*x^21+42042*x^18+38808*x^15+174510*x^12+15 5120*x^9+132960*x^6+259521*x^3+173014)
Time = 0.33 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.44 \[ \int x^5 \sqrt {1-x^3} \left (1+x^9\right )^2 \, dx=\frac {2}{2297295} \, {\left (45045 \, x^{24} - 3003 \, x^{21} - 3234 \, x^{18} + 135702 \, x^{15} - 19390 \, x^{12} - 22160 \, x^{9} + 126561 \, x^{6} - 86507 \, x^{3} - 173014\right )} \sqrt {-x^{3} + 1} \]
2/2297295*(45045*x^24 - 3003*x^21 - 3234*x^18 + 135702*x^15 - 19390*x^12 - 22160*x^9 + 126561*x^6 - 86507*x^3 - 173014)*sqrt(-x^3 + 1)
Time = 0.91 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int x^5 \sqrt {1-x^3} \left (1+x^9\right )^2 \, dx=\frac {2 x^{24} \sqrt {1 - x^{3}}}{51} - \frac {2 x^{21} \sqrt {1 - x^{3}}}{765} - \frac {28 x^{18} \sqrt {1 - x^{3}}}{9945} + \frac {1436 x^{15} \sqrt {1 - x^{3}}}{12155} - \frac {1108 x^{12} \sqrt {1 - x^{3}}}{65637} - \frac {8864 x^{9} \sqrt {1 - x^{3}}}{459459} + \frac {84374 x^{6} \sqrt {1 - x^{3}}}{765765} - \frac {173014 x^{3} \sqrt {1 - x^{3}}}{2297295} - \frac {346028 \sqrt {1 - x^{3}}}{2297295} \]
2*x**24*sqrt(1 - x**3)/51 - 2*x**21*sqrt(1 - x**3)/765 - 28*x**18*sqrt(1 - x**3)/9945 + 1436*x**15*sqrt(1 - x**3)/12155 - 1108*x**12*sqrt(1 - x**3)/ 65637 - 8864*x**9*sqrt(1 - x**3)/459459 + 84374*x**6*sqrt(1 - x**3)/765765 - 173014*x**3*sqrt(1 - x**3)/2297295 - 346028*sqrt(1 - x**3)/2297295
Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.74 \[ \int x^5 \sqrt {1-x^3} \left (1+x^9\right )^2 \, dx=\frac {2}{51} \, {\left (-x^{3} + 1\right )}^{\frac {17}{2}} - \frac {14}{45} \, {\left (-x^{3} + 1\right )}^{\frac {15}{2}} + \frac {14}{13} \, {\left (-x^{3} + 1\right )}^{\frac {13}{2}} - \frac {74}{33} \, {\left (-x^{3} + 1\right )}^{\frac {11}{2}} + \frac {86}{27} \, {\left (-x^{3} + 1\right )}^{\frac {9}{2}} - \frac {22}{7} \, {\left (-x^{3} + 1\right )}^{\frac {7}{2}} + \frac {32}{15} \, {\left (-x^{3} + 1\right )}^{\frac {5}{2}} - \frac {8}{9} \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} \]
2/51*(-x^3 + 1)^(17/2) - 14/45*(-x^3 + 1)^(15/2) + 14/13*(-x^3 + 1)^(13/2) - 74/33*(-x^3 + 1)^(11/2) + 86/27*(-x^3 + 1)^(9/2) - 22/7*(-x^3 + 1)^(7/2 ) + 32/15*(-x^3 + 1)^(5/2) - 8/9*(-x^3 + 1)^(3/2)
Time = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.14 \[ \int x^5 \sqrt {1-x^3} \left (1+x^9\right )^2 \, dx=\frac {2}{51} \, {\left (x^{3} - 1\right )}^{8} \sqrt {-x^{3} + 1} + \frac {14}{45} \, {\left (x^{3} - 1\right )}^{7} \sqrt {-x^{3} + 1} + \frac {14}{13} \, {\left (x^{3} - 1\right )}^{6} \sqrt {-x^{3} + 1} + \frac {74}{33} \, {\left (x^{3} - 1\right )}^{5} \sqrt {-x^{3} + 1} + \frac {86}{27} \, {\left (x^{3} - 1\right )}^{4} \sqrt {-x^{3} + 1} + \frac {22}{7} \, {\left (x^{3} - 1\right )}^{3} \sqrt {-x^{3} + 1} + \frac {32}{15} \, {\left (x^{3} - 1\right )}^{2} \sqrt {-x^{3} + 1} - \frac {8}{9} \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} \]
2/51*(x^3 - 1)^8*sqrt(-x^3 + 1) + 14/45*(x^3 - 1)^7*sqrt(-x^3 + 1) + 14/13 *(x^3 - 1)^6*sqrt(-x^3 + 1) + 74/33*(x^3 - 1)^5*sqrt(-x^3 + 1) + 86/27*(x^ 3 - 1)^4*sqrt(-x^3 + 1) + 22/7*(x^3 - 1)^3*sqrt(-x^3 + 1) + 32/15*(x^3 - 1 )^2*sqrt(-x^3 + 1) - 8/9*(-x^3 + 1)^(3/2)
Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int x^5 \sqrt {1-x^3} \left (1+x^9\right )^2 \, dx=\frac {84374\,x^6\,\sqrt {1-x^3}}{765765}-\frac {173014\,x^3\,\sqrt {1-x^3}}{2297295}-\frac {8864\,x^9\,\sqrt {1-x^3}}{459459}-\frac {1108\,x^{12}\,\sqrt {1-x^3}}{65637}+\frac {1436\,x^{15}\,\sqrt {1-x^3}}{12155}-\frac {28\,x^{18}\,\sqrt {1-x^3}}{9945}-\frac {2\,x^{21}\,\sqrt {1-x^3}}{765}+\frac {2\,x^{24}\,\sqrt {1-x^3}}{51}-\frac {346028\,\sqrt {1-x^3}}{2297295} \]
(84374*x^6*(1 - x^3)^(1/2))/765765 - (173014*x^3*(1 - x^3)^(1/2))/2297295 - (8864*x^9*(1 - x^3)^(1/2))/459459 - (1108*x^12*(1 - x^3)^(1/2))/65637 + (1436*x^15*(1 - x^3)^(1/2))/12155 - (28*x^18*(1 - x^3)^(1/2))/9945 - (2*x^ 21*(1 - x^3)^(1/2))/765 + (2*x^24*(1 - x^3)^(1/2))/51 - (346028*(1 - x^3)^ (1/2))/2297295