Integrand size = 13, antiderivative size = 205 \[ \int \frac {x^{23}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {5 a x^2}{b^{11}}+\frac {x^4}{4 b^{10}}+\frac {a^{11}}{18 b^{12} \left (a+b x^2\right )^9}-\frac {11 a^{10}}{16 b^{12} \left (a+b x^2\right )^8}+\frac {55 a^9}{14 b^{12} \left (a+b x^2\right )^7}-\frac {55 a^8}{4 b^{12} \left (a+b x^2\right )^6}+\frac {33 a^7}{b^{12} \left (a+b x^2\right )^5}-\frac {231 a^6}{4 b^{12} \left (a+b x^2\right )^4}+\frac {77 a^5}{b^{12} \left (a+b x^2\right )^3}-\frac {165 a^4}{2 b^{12} \left (a+b x^2\right )^2}+\frac {165 a^3}{2 b^{12} \left (a+b x^2\right )}+\frac {55 a^2 \log \left (a+b x^2\right )}{2 b^{12}} \]
-5*a*x^2/b^11+1/4*x^4/b^10+1/18*a^11/b^12/(b*x^2+a)^9-11/16*a^10/b^12/(b*x ^2+a)^8+55/14*a^9/b^12/(b*x^2+a)^7-55/4*a^8/b^12/(b*x^2+a)^6+33*a^7/b^12/( b*x^2+a)^5-231/4*a^6/b^12/(b*x^2+a)^4+77*a^5/b^12/(b*x^2+a)^3-165/2*a^4/b^ 12/(b*x^2+a)^2+165/2*a^3/b^12/(b*x^2+a)+55/2*a^2*ln(b*x^2+a)/b^12
Time = 0.02 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.77 \[ \int \frac {x^{23}}{\left (a+b x^2\right )^{10}} \, dx=\frac {42131 a^{11}+351459 a^{10} b x^2+1281096 a^9 b^2 x^4+2656584 a^8 b^3 x^6+3402756 a^7 b^4 x^8+2704212 a^6 b^5 x^{10}+1220688 a^5 b^6 x^{12}+190512 a^4 b^7 x^{14}-77112 a^3 b^8 x^{16}-36288 a^2 b^9 x^{18}-2772 a b^{10} x^{20}+252 b^{11} x^{22}+27720 a^2 \left (a+b x^2\right )^9 \log \left (a+b x^2\right )}{1008 b^{12} \left (a+b x^2\right )^9} \]
(42131*a^11 + 351459*a^10*b*x^2 + 1281096*a^9*b^2*x^4 + 2656584*a^8*b^3*x^ 6 + 3402756*a^7*b^4*x^8 + 2704212*a^6*b^5*x^10 + 1220688*a^5*b^6*x^12 + 19 0512*a^4*b^7*x^14 - 77112*a^3*b^8*x^16 - 36288*a^2*b^9*x^18 - 2772*a*b^10* x^20 + 252*b^11*x^22 + 27720*a^2*(a + b*x^2)^9*Log[a + b*x^2])/(1008*b^12* (a + b*x^2)^9)
Time = 0.37 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.99, 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.
| \(\displaystyle \int \frac {x^{23}}{\left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {x^{22}}{\left (b x^2+a\right )^{10}}dx^2\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {a^{11}}{b^{11} \left (b x^2+a\right )^{10}}+\frac {11 a^{10}}{b^{11} \left (b x^2+a\right )^9}-\frac {55 a^9}{b^{11} \left (b x^2+a\right )^8}+\frac {165 a^8}{b^{11} \left (b x^2+a\right )^7}-\frac {330 a^7}{b^{11} \left (b x^2+a\right )^6}+\frac {462 a^6}{b^{11} \left (b x^2+a\right )^5}-\frac {462 a^5}{b^{11} \left (b x^2+a\right )^4}+\frac {330 a^4}{b^{11} \left (b x^2+a\right )^3}-\frac {165 a^3}{b^{11} \left (b x^2+a\right )^2}+\frac {55 a^2}{b^{11} \left (b x^2+a\right )}-\frac {10 a}{b^{11}}+\frac {x^2}{b^{10}}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {a^{11}}{9 b^{12} \left (a+b x^2\right )^9}-\frac {11 a^{10}}{8 b^{12} \left (a+b x^2\right )^8}+\frac {55 a^9}{7 b^{12} \left (a+b x^2\right )^7}-\frac {55 a^8}{2 b^{12} \left (a+b x^2\right )^6}+\frac {66 a^7}{b^{12} \left (a+b x^2\right )^5}-\frac {231 a^6}{2 b^{12} \left (a+b x^2\right )^4}+\frac {154 a^5}{b^{12} \left (a+b x^2\right )^3}-\frac {165 a^4}{b^{12} \left (a+b x^2\right )^2}+\frac {165 a^3}{b^{12} \left (a+b x^2\right )}+\frac {55 a^2 \log \left (a+b x^2\right )}{b^{12}}-\frac {10 a x^2}{b^{11}}+\frac {x^4}{2 b^{10}}\right )\) |
((-10*a*x^2)/b^11 + x^4/(2*b^10) + a^11/(9*b^12*(a + b*x^2)^9) - (11*a^10) /(8*b^12*(a + b*x^2)^8) + (55*a^9)/(7*b^12*(a + b*x^2)^7) - (55*a^8)/(2*b^ 12*(a + b*x^2)^6) + (66*a^7)/(b^12*(a + b*x^2)^5) - (231*a^6)/(2*b^12*(a + b*x^2)^4) + (154*a^5)/(b^12*(a + b*x^2)^3) - (165*a^4)/(b^12*(a + b*x^2)^ 2) + (165*a^3)/(b^12*(a + b*x^2)) + (55*a^2*Log[a + b*x^2])/b^12)/2
3.2.93.3.1 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 = 142, normalized size of antiderivative = 0.69
| method | result | size |
| norman | \(\frac {\frac {x^{22}}{4 b}-\frac {11 a \,x^{20}}{4 b^{2}}+\frac {78419 a^{11}}{1008 b^{12}}+\frac {495 a^{3} x^{16}}{2 b^{4}}+\frac {1485 a^{4} x^{14}}{b^{5}}+\frac {4235 a^{5} x^{12}}{b^{6}}+\frac {28875 a^{6} x^{10}}{4 b^{7}}+\frac {31647 a^{7} x^{8}}{4 b^{8}}+\frac {11319 a^{8} x^{6}}{2 b^{9}}+\frac {35937 a^{9} x^{4}}{14 b^{10}}+\frac {75339 a^{10} x^{2}}{112 b^{11}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {55 a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{12}}\) | \(142\) |
| risch | \(\frac {x^{4}}{4 b^{10}}-\frac {5 a \,x^{2}}{b^{11}}+\frac {25 a^{2}}{b^{12}}+\frac {\frac {42131 a^{11}}{1008 b}+\frac {39611 a^{10} x^{2}}{112}+\frac {36839 a^{9} b \,x^{4}}{28}+\frac {11253 a^{8} b^{2} x^{6}}{4}+\frac {15147 a^{7} b^{3} x^{8}}{4}+\frac {13167 a^{6} b^{4} x^{10}}{4}+\frac {3619 a^{5} b^{5} x^{12}}{2}+\frac {1155 a^{4} b^{6} x^{14}}{2}+\frac {165 a^{3} b^{7} x^{16}}{2}}{b^{11} \left (b \,x^{2}+a \right )^{9}}+\frac {55 a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{12}}\) | \(148\) |
| default | \(\frac {\left (-b \,x^{2}+10 a \right )^{2}}{4 b^{12}}+\frac {a^{2} \left (\frac {154 a^{3}}{b \left (b \,x^{2}+a \right )^{3}}-\frac {11 a^{8}}{8 b \left (b \,x^{2}+a \right )^{8}}+\frac {66 a^{5}}{b \left (b \,x^{2}+a \right )^{5}}+\frac {a^{9}}{9 b \left (b \,x^{2}+a \right )^{9}}+\frac {55 \ln \left (b \,x^{2}+a \right )}{b}-\frac {55 a^{6}}{2 b \left (b \,x^{2}+a \right )^{6}}-\frac {231 a^{4}}{2 b \left (b \,x^{2}+a \right )^{4}}+\frac {55 a^{7}}{7 b \left (b \,x^{2}+a \right )^{7}}-\frac {165 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}+\frac {165 a}{b \left (b \,x^{2}+a \right )}\right )}{2 b^{11}}\) | \(192\) |
| parallelrisch | \(\frac {78419 a^{11}+3492720 \ln \left (b \,x^{2}+a \right ) x^{10} a^{6} b^{5}+3492720 \ln \left (b \,x^{2}+a \right ) x^{8} a^{7} b^{4}+2328480 \ln \left (b \,x^{2}+a \right ) x^{6} a^{8} b^{3}+997920 \ln \left (b \,x^{2}+a \right ) x^{4} a^{9} b^{2}+249480 \ln \left (b \,x^{2}+a \right ) x^{2} a^{10} b +27720 \ln \left (b \,x^{2}+a \right ) x^{18} a^{2} b^{9}+249480 \ln \left (b \,x^{2}+a \right ) x^{16} a^{3} b^{8}+27720 \ln \left (b \,x^{2}+a \right ) a^{11}+678051 a^{10} b \,x^{2}+252 b^{11} x^{22}+997920 \ln \left (b \,x^{2}+a \right ) x^{14} a^{4} b^{7}+2328480 \ln \left (b \,x^{2}+a \right ) x^{12} a^{5} b^{6}-2772 a \,x^{20} b^{10}+1496880 x^{14} a^{4} b^{7}+7975044 x^{8} a^{7} b^{4}+7276500 x^{10} a^{6} b^{5}+4268880 x^{12} a^{5} b^{6}+5704776 x^{6} a^{8} b^{3}+2587464 x^{4} a^{9} b^{2}+249480 x^{16} a^{3} b^{8}}{1008 b^{12} \left (b \,x^{2}+a \right )^{9}}\) | \(306\) |
(1/4/b*x^22-11/4*a/b^2*x^20+78419/1008*a^11/b^12+495/2*a^3/b^4*x^16+1485*a ^4/b^5*x^14+4235*a^5/b^6*x^12+28875/4*a^6/b^7*x^10+31647/4*a^7/b^8*x^8+113 19/2*a^8/b^9*x^6+35937/14*a^9/b^10*x^4+75339/112*a^10/b^11*x^2)/(b*x^2+a)^ 9+55/2*a^2*ln(b*x^2+a)/b^12
Time = 0.27 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.63 \[ \int \frac {x^{23}}{\left (a+b x^2\right )^{10}} \, dx=\frac {252 \, b^{11} x^{22} - 2772 \, a b^{10} x^{20} - 36288 \, a^{2} b^{9} x^{18} - 77112 \, a^{3} b^{8} x^{16} + 190512 \, a^{4} b^{7} x^{14} + 1220688 \, a^{5} b^{6} x^{12} + 2704212 \, a^{6} b^{5} x^{10} + 3402756 \, a^{7} b^{4} x^{8} + 2656584 \, a^{8} b^{3} x^{6} + 1281096 \, a^{9} b^{2} x^{4} + 351459 \, a^{10} b x^{2} + 42131 \, a^{11} + 27720 \, {\left (a^{2} b^{9} x^{18} + 9 \, a^{3} b^{8} x^{16} + 36 \, a^{4} b^{7} x^{14} + 84 \, a^{5} b^{6} x^{12} + 126 \, a^{6} b^{5} x^{10} + 126 \, a^{7} b^{4} x^{8} + 84 \, a^{8} b^{3} x^{6} + 36 \, a^{9} b^{2} x^{4} + 9 \, a^{10} b x^{2} + a^{11}\right )} \log \left (b x^{2} + a\right )}{1008 \, {\left (b^{21} x^{18} + 9 \, a b^{20} x^{16} + 36 \, a^{2} b^{19} x^{14} + 84 \, a^{3} b^{18} x^{12} + 126 \, a^{4} b^{17} x^{10} + 126 \, a^{5} b^{16} x^{8} + 84 \, a^{6} b^{15} x^{6} + 36 \, a^{7} b^{14} x^{4} + 9 \, a^{8} b^{13} x^{2} + a^{9} b^{12}\right )}} \]
1/1008*(252*b^11*x^22 - 2772*a*b^10*x^20 - 36288*a^2*b^9*x^18 - 77112*a^3* b^8*x^16 + 190512*a^4*b^7*x^14 + 1220688*a^5*b^6*x^12 + 2704212*a^6*b^5*x^ 10 + 3402756*a^7*b^4*x^8 + 2656584*a^8*b^3*x^6 + 1281096*a^9*b^2*x^4 + 351 459*a^10*b*x^2 + 42131*a^11 + 27720*(a^2*b^9*x^18 + 9*a^3*b^8*x^16 + 36*a^ 4*b^7*x^14 + 84*a^5*b^6*x^12 + 126*a^6*b^5*x^10 + 126*a^7*b^4*x^8 + 84*a^8 *b^3*x^6 + 36*a^9*b^2*x^4 + 9*a^10*b*x^2 + a^11)*log(b*x^2 + a))/(b^21*x^1 8 + 9*a*b^20*x^16 + 36*a^2*b^19*x^14 + 84*a^3*b^18*x^12 + 126*a^4*b^17*x^1 0 + 126*a^5*b^16*x^8 + 84*a^6*b^15*x^6 + 36*a^7*b^14*x^4 + 9*a^8*b^13*x^2 + a^9*b^12)
Time = 1.02 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.20 \[ \int \frac {x^{23}}{\left (a+b x^2\right )^{10}} \, dx=\frac {55 a^{2} \log {\left (a + b x^{2} \right )}}{2 b^{12}} - \frac {5 a x^{2}}{b^{11}} + \frac {42131 a^{11} + 356499 a^{10} b x^{2} + 1326204 a^{9} b^{2} x^{4} + 2835756 a^{8} b^{3} x^{6} + 3817044 a^{7} b^{4} x^{8} + 3318084 a^{6} b^{5} x^{10} + 1823976 a^{5} b^{6} x^{12} + 582120 a^{4} b^{7} x^{14} + 83160 a^{3} b^{8} x^{16}}{1008 a^{9} b^{12} + 9072 a^{8} b^{13} x^{2} + 36288 a^{7} b^{14} x^{4} + 84672 a^{6} b^{15} x^{6} + 127008 a^{5} b^{16} x^{8} + 127008 a^{4} b^{17} x^{10} + 84672 a^{3} b^{18} x^{12} + 36288 a^{2} b^{19} x^{14} + 9072 a b^{20} x^{16} + 1008 b^{21} x^{18}} + \frac {x^{4}}{4 b^{10}} \]
55*a**2*log(a + b*x**2)/(2*b**12) - 5*a*x**2/b**11 + (42131*a**11 + 356499 *a**10*b*x**2 + 1326204*a**9*b**2*x**4 + 2835756*a**8*b**3*x**6 + 3817044* a**7*b**4*x**8 + 3318084*a**6*b**5*x**10 + 1823976*a**5*b**6*x**12 + 58212 0*a**4*b**7*x**14 + 83160*a**3*b**8*x**16)/(1008*a**9*b**12 + 9072*a**8*b* *13*x**2 + 36288*a**7*b**14*x**4 + 84672*a**6*b**15*x**6 + 127008*a**5*b** 16*x**8 + 127008*a**4*b**17*x**10 + 84672*a**3*b**18*x**12 + 36288*a**2*b* *19*x**14 + 9072*a*b**20*x**16 + 1008*b**21*x**18) + x**4/(4*b**10)
Time = 0.22 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.13 \[ \int \frac {x^{23}}{\left (a+b x^2\right )^{10}} \, dx=\frac {83160 \, a^{3} b^{8} x^{16} + 582120 \, a^{4} b^{7} x^{14} + 1823976 \, a^{5} b^{6} x^{12} + 3318084 \, a^{6} b^{5} x^{10} + 3817044 \, a^{7} b^{4} x^{8} + 2835756 \, a^{8} b^{3} x^{6} + 1326204 \, a^{9} b^{2} x^{4} + 356499 \, a^{10} b x^{2} + 42131 \, a^{11}}{1008 \, {\left (b^{21} x^{18} + 9 \, a b^{20} x^{16} + 36 \, a^{2} b^{19} x^{14} + 84 \, a^{3} b^{18} x^{12} + 126 \, a^{4} b^{17} x^{10} + 126 \, a^{5} b^{16} x^{8} + 84 \, a^{6} b^{15} x^{6} + 36 \, a^{7} b^{14} x^{4} + 9 \, a^{8} b^{13} x^{2} + a^{9} b^{12}\right )}} + \frac {55 \, a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{12}} + \frac {b x^{4} - 20 \, a x^{2}}{4 \, b^{11}} \]
1/1008*(83160*a^3*b^8*x^16 + 582120*a^4*b^7*x^14 + 1823976*a^5*b^6*x^12 + 3318084*a^6*b^5*x^10 + 3817044*a^7*b^4*x^8 + 2835756*a^8*b^3*x^6 + 1326204 *a^9*b^2*x^4 + 356499*a^10*b*x^2 + 42131*a^11)/(b^21*x^18 + 9*a*b^20*x^16 + 36*a^2*b^19*x^14 + 84*a^3*b^18*x^12 + 126*a^4*b^17*x^10 + 126*a^5*b^16*x ^8 + 84*a^6*b^15*x^6 + 36*a^7*b^14*x^4 + 9*a^8*b^13*x^2 + a^9*b^12) + 55/2 *a^2*log(b*x^2 + a)/b^12 + 1/4*(b*x^4 - 20*a*x^2)/b^11
Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.77 \[ \int \frac {x^{23}}{\left (a+b x^2\right )^{10}} \, dx=\frac {55 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{12}} + \frac {b^{10} x^{4} - 20 \, a b^{9} x^{2}}{4 \, b^{20}} - \frac {78419 \, a^{2} b^{9} x^{18} + 622611 \, a^{3} b^{8} x^{16} + 2240964 \, a^{4} b^{7} x^{14} + 4763220 \, a^{5} b^{6} x^{12} + 6562710 \, a^{6} b^{5} x^{10} + 6063750 \, a^{7} b^{4} x^{8} + 3751440 \, a^{8} b^{3} x^{6} + 1496880 \, a^{9} b^{2} x^{4} + 349272 \, a^{10} b x^{2} + 36288 \, a^{11}}{1008 \, {\left (b x^{2} + a\right )}^{9} b^{12}} \]
55/2*a^2*log(abs(b*x^2 + a))/b^12 + 1/4*(b^10*x^4 - 20*a*b^9*x^2)/b^20 - 1 /1008*(78419*a^2*b^9*x^18 + 622611*a^3*b^8*x^16 + 2240964*a^4*b^7*x^14 + 4 763220*a^5*b^6*x^12 + 6562710*a^6*b^5*x^10 + 6063750*a^7*b^4*x^8 + 3751440 *a^8*b^3*x^6 + 1496880*a^9*b^2*x^4 + 349272*a^10*b*x^2 + 36288*a^11)/((b*x ^2 + a)^9*b^12)
Time = 4.83 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.12 \[ \int \frac {x^{23}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {42131\,a^{11}}{1008\,b}+\frac {39611\,a^{10}\,x^2}{112}+\frac {36839\,a^9\,b\,x^4}{28}+\frac {11253\,a^8\,b^2\,x^6}{4}+\frac {15147\,a^7\,b^3\,x^8}{4}+\frac {13167\,a^6\,b^4\,x^{10}}{4}+\frac {3619\,a^5\,b^5\,x^{12}}{2}+\frac {1155\,a^4\,b^6\,x^{14}}{2}+\frac {165\,a^3\,b^7\,x^{16}}{2}}{a^9\,b^{11}+9\,a^8\,b^{12}\,x^2+36\,a^7\,b^{13}\,x^4+84\,a^6\,b^{14}\,x^6+126\,a^5\,b^{15}\,x^8+126\,a^4\,b^{16}\,x^{10}+84\,a^3\,b^{17}\,x^{12}+36\,a^2\,b^{18}\,x^{14}+9\,a\,b^{19}\,x^{16}+b^{20}\,x^{18}}+\frac {x^4}{4\,b^{10}}-\frac {5\,a\,x^2}{b^{11}}+\frac {55\,a^2\,\ln \left (b\,x^2+a\right )}{2\,b^{12}} \]
((42131*a^11)/(1008*b) + (39611*a^10*x^2)/112 + (36839*a^9*b*x^4)/28 + (11 253*a^8*b^2*x^6)/4 + (15147*a^7*b^3*x^8)/4 + (13167*a^6*b^4*x^10)/4 + (361 9*a^5*b^5*x^12)/2 + (1155*a^4*b^6*x^14)/2 + (165*a^3*b^7*x^16)/2)/(a^9*b^1 1 + b^20*x^18 + 9*a*b^19*x^16 + 9*a^8*b^12*x^2 + 36*a^7*b^13*x^4 + 84*a^6* b^14*x^6 + 126*a^5*b^15*x^8 + 126*a^4*b^16*x^10 + 84*a^3*b^17*x^12 + 36*a^ 2*b^18*x^14) + x^4/(4*b^10) - (5*a*x^2)/b^11 + (55*a^2*log(a + b*x^2))/(2* b^12)