Integrand size = 13, antiderivative size = 188 \[ \int \frac {x^{21}}{\left (a+b x^2\right )^{10}} \, dx=\frac {x^2}{2 b^{10}}-\frac {a^{10}}{18 b^{11} \left (a+b x^2\right )^9}+\frac {5 a^9}{8 b^{11} \left (a+b x^2\right )^8}-\frac {45 a^8}{14 b^{11} \left (a+b x^2\right )^7}+\frac {10 a^7}{b^{11} \left (a+b x^2\right )^6}-\frac {21 a^6}{b^{11} \left (a+b x^2\right )^5}+\frac {63 a^5}{2 b^{11} \left (a+b x^2\right )^4}-\frac {35 a^4}{b^{11} \left (a+b x^2\right )^3}+\frac {30 a^3}{b^{11} \left (a+b x^2\right )^2}-\frac {45 a^2}{2 b^{11} \left (a+b x^2\right )}-\frac {5 a \log \left (a+b x^2\right )}{b^{11}} \]
1/2*x^2/b^10-1/18*a^10/b^11/(b*x^2+a)^9+5/8*a^9/b^11/(b*x^2+a)^8-45/14*a^8 /b^11/(b*x^2+a)^7+10*a^7/b^11/(b*x^2+a)^6-21*a^6/b^11/(b*x^2+a)^5+63/2*a^5 /b^11/(b*x^2+a)^4-35*a^4/b^11/(b*x^2+a)^3+30*a^3/b^11/(b*x^2+a)^2-45/2*a^2 /b^11/(b*x^2+a)-5*a*ln(b*x^2+a)/b^11
Time = 0.02 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.77 \[ \int \frac {x^{21}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {4861 a^{10}+41229 a^9 b x^2+153576 a^8 b^2 x^4+328104 a^7 b^3 x^6+439236 a^6 b^4 x^8+375732 a^5 b^5 x^{10}+197568 a^4 b^6 x^{12}+54432 a^3 b^7 x^{14}+2268 a^2 b^8 x^{16}-2268 a b^9 x^{18}-252 b^{10} x^{20}+2520 a \left (a+b x^2\right )^9 \log \left (a+b x^2\right )}{504 b^{11} \left (a+b x^2\right )^9} \]
-1/504*(4861*a^10 + 41229*a^9*b*x^2 + 153576*a^8*b^2*x^4 + 328104*a^7*b^3* x^6 + 439236*a^6*b^4*x^8 + 375732*a^5*b^5*x^10 + 197568*a^4*b^6*x^12 + 544 32*a^3*b^7*x^14 + 2268*a^2*b^8*x^16 - 2268*a*b^9*x^18 - 252*b^10*x^20 + 25 20*a*(a + b*x^2)^9*Log[a + b*x^2])/(b^11*(a + b*x^2)^9)
Time = 0.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {243, 49, 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.
| \(\displaystyle \int \frac {x^{21}}{\left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {x^{20}}{\left (b x^2+a\right )^{10}}dx^2\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {a^{10}}{b^{10} \left (b x^2+a\right )^{10}}-\frac {10 a^9}{b^{10} \left (b x^2+a\right )^9}+\frac {45 a^8}{b^{10} \left (b x^2+a\right )^8}-\frac {120 a^7}{b^{10} \left (b x^2+a\right )^7}+\frac {210 a^6}{b^{10} \left (b x^2+a\right )^6}-\frac {252 a^5}{b^{10} \left (b x^2+a\right )^5}+\frac {210 a^4}{b^{10} \left (b x^2+a\right )^4}-\frac {120 a^3}{b^{10} \left (b x^2+a\right )^3}+\frac {45 a^2}{b^{10} \left (b x^2+a\right )^2}-\frac {10 a}{b^{10} \left (b x^2+a\right )}+\frac {1}{b^{10}}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {a^{10}}{9 b^{11} \left (a+b x^2\right )^9}+\frac {5 a^9}{4 b^{11} \left (a+b x^2\right )^8}-\frac {45 a^8}{7 b^{11} \left (a+b x^2\right )^7}+\frac {20 a^7}{b^{11} \left (a+b x^2\right )^6}-\frac {42 a^6}{b^{11} \left (a+b x^2\right )^5}+\frac {63 a^5}{b^{11} \left (a+b x^2\right )^4}-\frac {70 a^4}{b^{11} \left (a+b x^2\right )^3}+\frac {60 a^3}{b^{11} \left (a+b x^2\right )^2}-\frac {45 a^2}{b^{11} \left (a+b x^2\right )}-\frac {10 a \log \left (a+b x^2\right )}{b^{11}}+\frac {x^2}{b^{10}}\right )\) |
(x^2/b^10 - a^10/(9*b^11*(a + b*x^2)^9) + (5*a^9)/(4*b^11*(a + b*x^2)^8) - (45*a^8)/(7*b^11*(a + b*x^2)^7) + (20*a^7)/(b^11*(a + b*x^2)^6) - (42*a^6 )/(b^11*(a + b*x^2)^5) + (63*a^5)/(b^11*(a + b*x^2)^4) - (70*a^4)/(b^11*(a + b*x^2)^3) + (60*a^3)/(b^11*(a + b*x^2)^2) - (45*a^2)/(b^11*(a + b*x^2)) - (10*a*Log[a + b*x^2])/b^11)/2
3.2.94.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 1.77 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(\frac {x^{2}}{2 b^{10}}+\frac {-\frac {4861 a^{10}}{504 b}-\frac {4609 a^{9} x^{2}}{56}-\frac {4329 a^{8} b \,x^{4}}{14}-669 a^{7} b^{2} x^{6}-\frac {1827 a^{6} b^{3} x^{8}}{2}-\frac {1617 a^{5} b^{4} x^{10}}{2}-455 a^{4} b^{5} x^{12}-150 a^{3} b^{6} x^{14}-\frac {45 a^{2} b^{7} x^{16}}{2}}{b^{10} \left (b \,x^{2}+a \right )^{9}}-\frac {5 a \ln \left (b \,x^{2}+a \right )}{b^{11}}\) | \(129\) |
| norman | \(\frac {\frac {x^{20}}{2 b}-\frac {7129 a^{10}}{504 b^{11}}-\frac {45 a^{2} x^{16}}{b^{3}}-\frac {270 a^{3} x^{14}}{b^{4}}-\frac {770 a^{4} x^{12}}{b^{5}}-\frac {2625 a^{5} x^{10}}{2 b^{6}}-\frac {2877 a^{6} x^{8}}{2 b^{7}}-\frac {1029 a^{7} x^{6}}{b^{8}}-\frac {3267 a^{8} x^{4}}{7 b^{9}}-\frac {6849 a^{9} x^{2}}{56 b^{10}}}{\left (b \,x^{2}+a \right )^{9}}-\frac {5 a \ln \left (b \,x^{2}+a \right )}{b^{11}}\) | \(131\) |
| default | \(\frac {x^{2}}{2 b^{10}}-\frac {a \left (\frac {70 a^{3}}{b \left (b \,x^{2}+a \right )^{3}}+\frac {42 a^{5}}{b \left (b \,x^{2}+a \right )^{5}}+\frac {a^{9}}{9 b \left (b \,x^{2}+a \right )^{9}}+\frac {10 \ln \left (b \,x^{2}+a \right )}{b}-\frac {20 a^{6}}{b \left (b \,x^{2}+a \right )^{6}}-\frac {63 a^{4}}{b \left (b \,x^{2}+a \right )^{4}}-\frac {5 a^{8}}{4 b \left (b \,x^{2}+a \right )^{8}}+\frac {45 a^{7}}{7 b \left (b \,x^{2}+a \right )^{7}}-\frac {60 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}+\frac {45 a}{b \left (b \,x^{2}+a \right )}\right )}{2 b^{10}}\) | \(181\) |
| parallelrisch | \(-\frac {7129 a^{10}+61641 a^{9} b \,x^{2}-252 b^{10} x^{20}+235224 a^{8} b^{2} x^{4}+518616 a^{7} b^{3} x^{6}+661500 a^{5} b^{5} x^{10}+725004 a^{6} b^{4} x^{8}+388080 a^{4} b^{6} x^{12}+22680 a^{2} b^{8} x^{16}+136080 a^{3} b^{7} x^{14}+2520 \ln \left (b \,x^{2}+a \right ) x^{18} a \,b^{9}+22680 \ln \left (b \,x^{2}+a \right ) x^{16} a^{2} b^{8}+90720 \ln \left (b \,x^{2}+a \right ) x^{14} a^{3} b^{7}+211680 \ln \left (b \,x^{2}+a \right ) x^{12} a^{4} b^{6}+317520 \ln \left (b \,x^{2}+a \right ) x^{10} a^{5} b^{5}+317520 \ln \left (b \,x^{2}+a \right ) x^{8} a^{6} b^{4}+211680 \ln \left (b \,x^{2}+a \right ) x^{6} a^{7} b^{3}+90720 \ln \left (b \,x^{2}+a \right ) x^{4} a^{8} b^{2}+22680 \ln \left (b \,x^{2}+a \right ) x^{2} a^{9} b +2520 \ln \left (b \,x^{2}+a \right ) a^{10}}{504 b^{11} \left (b \,x^{2}+a \right )^{9}}\) | \(295\) |
1/2*x^2/b^10+(-4861/504*a^10/b-4609/56*a^9*x^2-4329/14*a^8*b*x^4-669*a^7*b ^2*x^6-1827/2*a^6*b^3*x^8-1617/2*a^5*b^4*x^10-455*a^4*b^5*x^12-150*a^3*b^6 *x^14-45/2*a^2*b^7*x^16)/b^10/(b*x^2+a)^9-5*a*ln(b*x^2+a)/b^11
Time = 0.24 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.71 \[ \int \frac {x^{21}}{\left (a+b x^2\right )^{10}} \, dx=\frac {252 \, b^{10} x^{20} + 2268 \, a b^{9} x^{18} - 2268 \, a^{2} b^{8} x^{16} - 54432 \, a^{3} b^{7} x^{14} - 197568 \, a^{4} b^{6} x^{12} - 375732 \, a^{5} b^{5} x^{10} - 439236 \, a^{6} b^{4} x^{8} - 328104 \, a^{7} b^{3} x^{6} - 153576 \, a^{8} b^{2} x^{4} - 41229 \, a^{9} b x^{2} - 4861 \, a^{10} - 2520 \, {\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \log \left (b x^{2} + a\right )}{504 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}} \]
1/504*(252*b^10*x^20 + 2268*a*b^9*x^18 - 2268*a^2*b^8*x^16 - 54432*a^3*b^7 *x^14 - 197568*a^4*b^6*x^12 - 375732*a^5*b^5*x^10 - 439236*a^6*b^4*x^8 - 3 28104*a^7*b^3*x^6 - 153576*a^8*b^2*x^4 - 41229*a^9*b*x^2 - 4861*a^10 - 252 0*(a*b^9*x^18 + 9*a^2*b^8*x^16 + 36*a^3*b^7*x^14 + 84*a^4*b^6*x^12 + 126*a ^5*b^5*x^10 + 126*a^6*b^4*x^8 + 84*a^7*b^3*x^6 + 36*a^8*b^2*x^4 + 9*a^9*b* x^2 + a^10)*log(b*x^2 + a))/(b^20*x^18 + 9*a*b^19*x^16 + 36*a^2*b^18*x^14 + 84*a^3*b^17*x^12 + 126*a^4*b^16*x^10 + 126*a^5*b^15*x^8 + 84*a^6*b^14*x^ 6 + 36*a^7*b^13*x^4 + 9*a^8*b^12*x^2 + a^9*b^11)
Time = 0.97 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.24 \[ \int \frac {x^{21}}{\left (a+b x^2\right )^{10}} \, dx=- \frac {5 a \log {\left (a + b x^{2} \right )}}{b^{11}} + \frac {- 4861 a^{10} - 41481 a^{9} b x^{2} - 155844 a^{8} b^{2} x^{4} - 337176 a^{7} b^{3} x^{6} - 460404 a^{6} b^{4} x^{8} - 407484 a^{5} b^{5} x^{10} - 229320 a^{4} b^{6} x^{12} - 75600 a^{3} b^{7} x^{14} - 11340 a^{2} b^{8} x^{16}}{504 a^{9} b^{11} + 4536 a^{8} b^{12} x^{2} + 18144 a^{7} b^{13} x^{4} + 42336 a^{6} b^{14} x^{6} + 63504 a^{5} b^{15} x^{8} + 63504 a^{4} b^{16} x^{10} + 42336 a^{3} b^{17} x^{12} + 18144 a^{2} b^{18} x^{14} + 4536 a b^{19} x^{16} + 504 b^{20} x^{18}} + \frac {x^{2}}{2 b^{10}} \]
-5*a*log(a + b*x**2)/b**11 + (-4861*a**10 - 41481*a**9*b*x**2 - 155844*a** 8*b**2*x**4 - 337176*a**7*b**3*x**6 - 460404*a**6*b**4*x**8 - 407484*a**5* b**5*x**10 - 229320*a**4*b**6*x**12 - 75600*a**3*b**7*x**14 - 11340*a**2*b **8*x**16)/(504*a**9*b**11 + 4536*a**8*b**12*x**2 + 18144*a**7*b**13*x**4 + 42336*a**6*b**14*x**6 + 63504*a**5*b**15*x**8 + 63504*a**4*b**16*x**10 + 42336*a**3*b**17*x**12 + 18144*a**2*b**18*x**14 + 4536*a*b**19*x**16 + 50 4*b**20*x**18) + x**2/(2*b**10)
Time = 0.22 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.17 \[ \int \frac {x^{21}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {11340 \, a^{2} b^{8} x^{16} + 75600 \, a^{3} b^{7} x^{14} + 229320 \, a^{4} b^{6} x^{12} + 407484 \, a^{5} b^{5} x^{10} + 460404 \, a^{6} b^{4} x^{8} + 337176 \, a^{7} b^{3} x^{6} + 155844 \, a^{8} b^{2} x^{4} + 41481 \, a^{9} b x^{2} + 4861 \, a^{10}}{504 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}} + \frac {x^{2}}{2 \, b^{10}} - \frac {5 \, a \log \left (b x^{2} + a\right )}{b^{11}} \]
-1/504*(11340*a^2*b^8*x^16 + 75600*a^3*b^7*x^14 + 229320*a^4*b^6*x^12 + 40 7484*a^5*b^5*x^10 + 460404*a^6*b^4*x^8 + 337176*a^7*b^3*x^6 + 155844*a^8*b ^2*x^4 + 41481*a^9*b*x^2 + 4861*a^10)/(b^20*x^18 + 9*a*b^19*x^16 + 36*a^2* b^18*x^14 + 84*a^3*b^17*x^12 + 126*a^4*b^16*x^10 + 126*a^5*b^15*x^8 + 84*a ^6*b^14*x^6 + 36*a^7*b^13*x^4 + 9*a^8*b^12*x^2 + a^9*b^11) + 1/2*x^2/b^10 - 5*a*log(b*x^2 + a)/b^11
Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.74 \[ \int \frac {x^{21}}{\left (a+b x^2\right )^{10}} \, dx=\frac {x^{2}}{2 \, b^{10}} - \frac {5 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{b^{11}} + \frac {7129 \, a b^{9} x^{18} + 52821 \, a^{2} b^{8} x^{16} + 181044 \, a^{3} b^{7} x^{14} + 369516 \, a^{4} b^{6} x^{12} + 490770 \, a^{5} b^{5} x^{10} + 437850 \, a^{6} b^{4} x^{8} + 261660 \, a^{7} b^{3} x^{6} + 100800 \, a^{8} b^{2} x^{4} + 22680 \, a^{9} b x^{2} + 2268 \, a^{10}}{504 \, {\left (b x^{2} + a\right )}^{9} b^{11}} \]
1/2*x^2/b^10 - 5*a*log(abs(b*x^2 + a))/b^11 + 1/504*(7129*a*b^9*x^18 + 528 21*a^2*b^8*x^16 + 181044*a^3*b^7*x^14 + 369516*a^4*b^6*x^12 + 490770*a^5*b ^5*x^10 + 437850*a^6*b^4*x^8 + 261660*a^7*b^3*x^6 + 100800*a^8*b^2*x^4 + 2 2680*a^9*b*x^2 + 2268*a^10)/((b*x^2 + a)^9*b^11)
Time = 0.44 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.17 \[ \int \frac {x^{21}}{\left (a+b x^2\right )^{10}} \, dx=\frac {x^2}{2\,b^{10}}-\frac {\frac {4861\,a^{10}}{504\,b}+\frac {4609\,a^9\,x^2}{56}+\frac {4329\,a^8\,b\,x^4}{14}+669\,a^7\,b^2\,x^6+\frac {1827\,a^6\,b^3\,x^8}{2}+\frac {1617\,a^5\,b^4\,x^{10}}{2}+455\,a^4\,b^5\,x^{12}+150\,a^3\,b^6\,x^{14}+\frac {45\,a^2\,b^7\,x^{16}}{2}}{a^9\,b^{10}+9\,a^8\,b^{11}\,x^2+36\,a^7\,b^{12}\,x^4+84\,a^6\,b^{13}\,x^6+126\,a^5\,b^{14}\,x^8+126\,a^4\,b^{15}\,x^{10}+84\,a^3\,b^{16}\,x^{12}+36\,a^2\,b^{17}\,x^{14}+9\,a\,b^{18}\,x^{16}+b^{19}\,x^{18}}-\frac {5\,a\,\ln \left (b\,x^2+a\right )}{b^{11}} \]
x^2/(2*b^10) - ((4861*a^10)/(504*b) + (4609*a^9*x^2)/56 + (4329*a^8*b*x^4) /14 + 669*a^7*b^2*x^6 + (1827*a^6*b^3*x^8)/2 + (1617*a^5*b^4*x^10)/2 + 455 *a^4*b^5*x^12 + 150*a^3*b^6*x^14 + (45*a^2*b^7*x^16)/2)/(a^9*b^10 + b^19*x ^18 + 9*a*b^18*x^16 + 9*a^8*b^11*x^2 + 36*a^7*b^12*x^4 + 84*a^6*b^13*x^6 + 126*a^5*b^14*x^8 + 126*a^4*b^15*x^10 + 84*a^3*b^16*x^12 + 36*a^2*b^17*x^1 4) - (5*a*log(a + b*x^2))/b^11