Integrand size = 11, antiderivative size = 177 \[ \int \frac {x^{11}}{(a+b x)^{10}} \, dx=-\frac {10 a x}{b^{11}}+\frac {x^2}{2 b^{10}}+\frac {a^{11}}{9 b^{12} (a+b x)^9}-\frac {11 a^{10}}{8 b^{12} (a+b x)^8}+\frac {55 a^9}{7 b^{12} (a+b x)^7}-\frac {55 a^8}{2 b^{12} (a+b x)^6}+\frac {66 a^7}{b^{12} (a+b x)^5}-\frac {231 a^6}{2 b^{12} (a+b x)^4}+\frac {154 a^5}{b^{12} (a+b x)^3}-\frac {165 a^4}{b^{12} (a+b x)^2}+\frac {165 a^3}{b^{12} (a+b x)}+\frac {55 a^2 \log (a+b x)}{b^{12}} \]
-10*a*x/b^11+1/2*x^2/b^10+1/9*a^11/b^12/(b*x+a)^9-11/8*a^10/b^12/(b*x+a)^8 +55/7*a^9/b^12/(b*x+a)^7-55/2*a^8/b^12/(b*x+a)^6+66*a^7/b^12/(b*x+a)^5-231 /2*a^6/b^12/(b*x+a)^4+154*a^5/b^12/(b*x+a)^3-165*a^4/b^12/(b*x+a)^2+165*a^ 3/b^12/(b*x+a)+55*a^2*ln(b*x+a)/b^12
Time = 0.02 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.85 \[ \int \frac {x^{11}}{(a+b x)^{10}} \, dx=\frac {42131 a^{11}+351459 a^{10} b x+1281096 a^9 b^2 x^2+2656584 a^8 b^3 x^3+3402756 a^7 b^4 x^4+2704212 a^6 b^5 x^5+1220688 a^5 b^6 x^6+190512 a^4 b^7 x^7-77112 a^3 b^8 x^8-36288 a^2 b^9 x^9-2772 a b^{10} x^{10}+252 b^{11} x^{11}+27720 a^2 (a+b x)^9 \log (a+b x)}{504 b^{12} (a+b x)^9} \]
(42131*a^11 + 351459*a^10*b*x + 1281096*a^9*b^2*x^2 + 2656584*a^8*b^3*x^3 + 3402756*a^7*b^4*x^4 + 2704212*a^6*b^5*x^5 + 1220688*a^5*b^6*x^6 + 190512 *a^4*b^7*x^7 - 77112*a^3*b^8*x^8 - 36288*a^2*b^9*x^9 - 2772*a*b^10*x^10 + 252*b^11*x^11 + 27720*a^2*(a + b*x)^9*Log[a + b*x])/(504*b^12*(a + b*x)^9)
Time = 0.37 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {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.
| \(\displaystyle \int \frac {x^{11}}{(a+b x)^{10}} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {a^{11}}{b^{11} (a+b x)^{10}}+\frac {11 a^{10}}{b^{11} (a+b x)^9}-\frac {55 a^9}{b^{11} (a+b x)^8}+\frac {165 a^8}{b^{11} (a+b x)^7}-\frac {330 a^7}{b^{11} (a+b x)^6}+\frac {462 a^6}{b^{11} (a+b x)^5}-\frac {462 a^5}{b^{11} (a+b x)^4}+\frac {330 a^4}{b^{11} (a+b x)^3}-\frac {165 a^3}{b^{11} (a+b x)^2}+\frac {55 a^2}{b^{11} (a+b x)}-\frac {10 a}{b^{11}}+\frac {x}{b^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{11}}{9 b^{12} (a+b x)^9}-\frac {11 a^{10}}{8 b^{12} (a+b x)^8}+\frac {55 a^9}{7 b^{12} (a+b x)^7}-\frac {55 a^8}{2 b^{12} (a+b x)^6}+\frac {66 a^7}{b^{12} (a+b x)^5}-\frac {231 a^6}{2 b^{12} (a+b x)^4}+\frac {154 a^5}{b^{12} (a+b x)^3}-\frac {165 a^4}{b^{12} (a+b x)^2}+\frac {165 a^3}{b^{12} (a+b x)}+\frac {55 a^2 \log (a+b x)}{b^{12}}-\frac {10 a x}{b^{11}}+\frac {x^2}{2 b^{10}}\) |
(-10*a*x)/b^11 + x^2/(2*b^10) + a^11/(9*b^12*(a + b*x)^9) - (11*a^10)/(8*b ^12*(a + b*x)^8) + (55*a^9)/(7*b^12*(a + b*x)^7) - (55*a^8)/(2*b^12*(a + b *x)^6) + (66*a^7)/(b^12*(a + b*x)^5) - (231*a^6)/(2*b^12*(a + b*x)^4) + (1 54*a^5)/(b^12*(a + b*x)^3) - (165*a^4)/(b^12*(a + b*x)^2) + (165*a^3)/(b^1 2*(a + b*x)) + (55*a^2*Log[a + b*x])/b^12
3.3.23.3.1 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]
Time = 0.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(\frac {x^{2}}{2 b^{10}}-\frac {10 a x}{b^{11}}+\frac {165 a^{3} b^{7} x^{8}+1155 a^{4} b^{6} x^{7}+3619 a^{5} b^{5} x^{6}+\frac {13167 a^{6} b^{4} x^{5}}{2}+\frac {15147 a^{7} b^{3} x^{4}}{2}+\frac {11253 a^{8} b^{2} x^{3}}{2}+\frac {36839 a^{9} b \,x^{2}}{14}+\frac {39611 a^{10} x}{56}+\frac {42131 a^{11}}{504 b}}{b^{11} \left (b x +a \right )^{9}}+\frac {55 a^{2} \ln \left (b x +a \right )}{b^{12}}\) | \(132\) |
| norman | \(\frac {\frac {x^{11}}{2 b}-\frac {11 a \,x^{10}}{2 b^{2}}+\frac {78419 a^{11}}{504 b^{12}}+\frac {495 a^{3} x^{8}}{b^{4}}+\frac {2970 a^{4} x^{7}}{b^{5}}+\frac {8470 a^{5} x^{6}}{b^{6}}+\frac {28875 a^{6} x^{5}}{2 b^{7}}+\frac {31647 a^{7} x^{4}}{2 b^{8}}+\frac {11319 a^{8} x^{3}}{b^{9}}+\frac {35937 a^{9} x^{2}}{7 b^{10}}+\frac {75339 a^{10} x}{56 b^{11}}}{\left (b x +a \right )^{9}}+\frac {55 a^{2} \ln \left (b x +a \right )}{b^{12}}\) | \(136\) |
| default | \(-\frac {-\frac {1}{2} b \,x^{2}+10 a x}{b^{11}}+\frac {a^{11}}{9 b^{12} \left (b x +a \right )^{9}}+\frac {55 a^{2} \ln \left (b x +a \right )}{b^{12}}-\frac {55 a^{8}}{2 b^{12} \left (b x +a \right )^{6}}+\frac {55 a^{9}}{7 b^{12} \left (b x +a \right )^{7}}-\frac {11 a^{10}}{8 b^{12} \left (b x +a \right )^{8}}-\frac {231 a^{6}}{2 b^{12} \left (b x +a \right )^{4}}+\frac {154 a^{5}}{b^{12} \left (b x +a \right )^{3}}+\frac {66 a^{7}}{b^{12} \left (b x +a \right )^{5}}-\frac {165 a^{4}}{b^{12} \left (b x +a \right )^{2}}+\frac {165 a^{3}}{b^{12} \left (b x +a \right )}\) | \(167\) |
| parallelrisch | \(\frac {78419 a^{11}+27720 \ln \left (b x +a \right ) a^{11}+27720 \ln \left (b x +a \right ) x^{9} a^{2} b^{9}+249480 \ln \left (b x +a \right ) x^{8} a^{3} b^{8}+997920 \ln \left (b x +a \right ) x^{7} a^{4} b^{7}+2328480 \ln \left (b x +a \right ) x^{6} a^{5} b^{6}+3492720 \ln \left (b x +a \right ) x^{5} a^{6} b^{5}+3492720 \ln \left (b x +a \right ) x^{4} a^{7} b^{4}+2328480 \ln \left (b x +a \right ) x^{3} a^{8} b^{3}+997920 \ln \left (b x +a \right ) x^{2} a^{9} b^{2}+249480 \ln \left (b x +a \right ) x \,a^{10} b +252 b^{11} x^{11}+249480 a^{3} x^{8} b^{8}+1496880 a^{4} b^{7} x^{7}+4268880 a^{5} b^{6} x^{6}+7276500 a^{6} b^{5} x^{5}+7975044 a^{7} b^{4} x^{4}+5704776 b^{3} a^{8} x^{3}+2587464 b^{2} a^{9} x^{2}+678051 a^{10} b x -2772 a \,x^{10} b^{10}}{504 b^{12} \left (b x +a \right )^{9}}\) | \(280\) |
1/2*x^2/b^10-10*a*x/b^11+(165*a^3*b^7*x^8+1155*a^4*b^6*x^7+3619*a^5*b^5*x^ 6+13167/2*a^6*b^4*x^5+15147/2*a^7*b^3*x^4+11253/2*a^8*b^2*x^3+36839/14*a^9 *b*x^2+39611/56*a^10*x+42131/504/b*a^11)/b^11/(b*x+a)^9+55*a^2*ln(b*x+a)/b ^12
Time = 0.22 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.85 \[ \int \frac {x^{11}}{(a+b x)^{10}} \, dx=\frac {252 \, b^{11} x^{11} - 2772 \, a b^{10} x^{10} - 36288 \, a^{2} b^{9} x^{9} - 77112 \, a^{3} b^{8} x^{8} + 190512 \, a^{4} b^{7} x^{7} + 1220688 \, a^{5} b^{6} x^{6} + 2704212 \, a^{6} b^{5} x^{5} + 3402756 \, a^{7} b^{4} x^{4} + 2656584 \, a^{8} b^{3} x^{3} + 1281096 \, a^{9} b^{2} x^{2} + 351459 \, a^{10} b x + 42131 \, a^{11} + 27720 \, {\left (a^{2} b^{9} x^{9} + 9 \, a^{3} b^{8} x^{8} + 36 \, a^{4} b^{7} x^{7} + 84 \, a^{5} b^{6} x^{6} + 126 \, a^{6} b^{5} x^{5} + 126 \, a^{7} b^{4} x^{4} + 84 \, a^{8} b^{3} x^{3} + 36 \, a^{9} b^{2} x^{2} + 9 \, a^{10} b x + a^{11}\right )} \log \left (b x + a\right )}{504 \, {\left (b^{21} x^{9} + 9 \, a b^{20} x^{8} + 36 \, a^{2} b^{19} x^{7} + 84 \, a^{3} b^{18} x^{6} + 126 \, a^{4} b^{17} x^{5} + 126 \, a^{5} b^{16} x^{4} + 84 \, a^{6} b^{15} x^{3} + 36 \, a^{7} b^{14} x^{2} + 9 \, a^{8} b^{13} x + a^{9} b^{12}\right )}} \]
1/504*(252*b^11*x^11 - 2772*a*b^10*x^10 - 36288*a^2*b^9*x^9 - 77112*a^3*b^ 8*x^8 + 190512*a^4*b^7*x^7 + 1220688*a^5*b^6*x^6 + 2704212*a^6*b^5*x^5 + 3 402756*a^7*b^4*x^4 + 2656584*a^8*b^3*x^3 + 1281096*a^9*b^2*x^2 + 351459*a^ 10*b*x + 42131*a^11 + 27720*(a^2*b^9*x^9 + 9*a^3*b^8*x^8 + 36*a^4*b^7*x^7 + 84*a^5*b^6*x^6 + 126*a^6*b^5*x^5 + 126*a^7*b^4*x^4 + 84*a^8*b^3*x^3 + 36 *a^9*b^2*x^2 + 9*a^10*b*x + a^11)*log(b*x + a))/(b^21*x^9 + 9*a*b^20*x^8 + 36*a^2*b^19*x^7 + 84*a^3*b^18*x^6 + 126*a^4*b^17*x^5 + 126*a^5*b^16*x^4 + 84*a^6*b^15*x^3 + 36*a^7*b^14*x^2 + 9*a^8*b^13*x + a^9*b^12)
Time = 0.78 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.33 \[ \int \frac {x^{11}}{(a+b x)^{10}} \, dx=\frac {55 a^{2} \log {\left (a + b x \right )}}{b^{12}} - \frac {10 a x}{b^{11}} + \frac {42131 a^{11} + 356499 a^{10} b x + 1326204 a^{9} b^{2} x^{2} + 2835756 a^{8} b^{3} x^{3} + 3817044 a^{7} b^{4} x^{4} + 3318084 a^{6} b^{5} x^{5} + 1823976 a^{5} b^{6} x^{6} + 582120 a^{4} b^{7} x^{7} + 83160 a^{3} b^{8} x^{8}}{504 a^{9} b^{12} + 4536 a^{8} b^{13} x + 18144 a^{7} b^{14} x^{2} + 42336 a^{6} b^{15} x^{3} + 63504 a^{5} b^{16} x^{4} + 63504 a^{4} b^{17} x^{5} + 42336 a^{3} b^{18} x^{6} + 18144 a^{2} b^{19} x^{7} + 4536 a b^{20} x^{8} + 504 b^{21} x^{9}} + \frac {x^{2}}{2 b^{10}} \]
55*a**2*log(a + b*x)/b**12 - 10*a*x/b**11 + (42131*a**11 + 356499*a**10*b* x + 1326204*a**9*b**2*x**2 + 2835756*a**8*b**3*x**3 + 3817044*a**7*b**4*x* *4 + 3318084*a**6*b**5*x**5 + 1823976*a**5*b**6*x**6 + 582120*a**4*b**7*x* *7 + 83160*a**3*b**8*x**8)/(504*a**9*b**12 + 4536*a**8*b**13*x + 18144*a** 7*b**14*x**2 + 42336*a**6*b**15*x**3 + 63504*a**5*b**16*x**4 + 63504*a**4* b**17*x**5 + 42336*a**3*b**18*x**6 + 18144*a**2*b**19*x**7 + 4536*a*b**20* x**8 + 504*b**21*x**9) + x**2/(2*b**10)
Time = 0.22 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.26 \[ \int \frac {x^{11}}{(a+b x)^{10}} \, dx=\frac {83160 \, a^{3} b^{8} x^{8} + 582120 \, a^{4} b^{7} x^{7} + 1823976 \, a^{5} b^{6} x^{6} + 3318084 \, a^{6} b^{5} x^{5} + 3817044 \, a^{7} b^{4} x^{4} + 2835756 \, a^{8} b^{3} x^{3} + 1326204 \, a^{9} b^{2} x^{2} + 356499 \, a^{10} b x + 42131 \, a^{11}}{504 \, {\left (b^{21} x^{9} + 9 \, a b^{20} x^{8} + 36 \, a^{2} b^{19} x^{7} + 84 \, a^{3} b^{18} x^{6} + 126 \, a^{4} b^{17} x^{5} + 126 \, a^{5} b^{16} x^{4} + 84 \, a^{6} b^{15} x^{3} + 36 \, a^{7} b^{14} x^{2} + 9 \, a^{8} b^{13} x + a^{9} b^{12}\right )}} + \frac {55 \, a^{2} \log \left (b x + a\right )}{b^{12}} + \frac {b x^{2} - 20 \, a x}{2 \, b^{11}} \]
1/504*(83160*a^3*b^8*x^8 + 582120*a^4*b^7*x^7 + 1823976*a^5*b^6*x^6 + 3318 084*a^6*b^5*x^5 + 3817044*a^7*b^4*x^4 + 2835756*a^8*b^3*x^3 + 1326204*a^9* b^2*x^2 + 356499*a^10*b*x + 42131*a^11)/(b^21*x^9 + 9*a*b^20*x^8 + 36*a^2* b^19*x^7 + 84*a^3*b^18*x^6 + 126*a^4*b^17*x^5 + 126*a^5*b^16*x^4 + 84*a^6* b^15*x^3 + 36*a^7*b^14*x^2 + 9*a^8*b^13*x + a^9*b^12) + 55*a^2*log(b*x + a )/b^12 + 1/2*(b*x^2 - 20*a*x)/b^11
Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.78 \[ \int \frac {x^{11}}{(a+b x)^{10}} \, dx=\frac {55 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{12}} + \frac {b^{10} x^{2} - 20 \, a b^{9} x}{2 \, b^{20}} + \frac {83160 \, a^{3} b^{8} x^{8} + 582120 \, a^{4} b^{7} x^{7} + 1823976 \, a^{5} b^{6} x^{6} + 3318084 \, a^{6} b^{5} x^{5} + 3817044 \, a^{7} b^{4} x^{4} + 2835756 \, a^{8} b^{3} x^{3} + 1326204 \, a^{9} b^{2} x^{2} + 356499 \, a^{10} b x + 42131 \, a^{11}}{504 \, {\left (b x + a\right )}^{9} b^{12}} \]
55*a^2*log(abs(b*x + a))/b^12 + 1/2*(b^10*x^2 - 20*a*b^9*x)/b^20 + 1/504*( 83160*a^3*b^8*x^8 + 582120*a^4*b^7*x^7 + 1823976*a^5*b^6*x^6 + 3318084*a^6 *b^5*x^5 + 3817044*a^7*b^4*x^4 + 2835756*a^8*b^3*x^3 + 1326204*a^9*b^2*x^2 + 356499*a^10*b*x + 42131*a^11)/((b*x + a)^9*b^12)
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.78 \[ \int \frac {x^{11}}{(a+b x)^{10}} \, dx=\frac {\frac {{\left (a+b\,x\right )}^2}{2}+\frac {165\,a^3}{a+b\,x}-\frac {165\,a^4}{{\left (a+b\,x\right )}^2}+\frac {154\,a^5}{{\left (a+b\,x\right )}^3}-\frac {231\,a^6}{2\,{\left (a+b\,x\right )}^4}+\frac {66\,a^7}{{\left (a+b\,x\right )}^5}-\frac {55\,a^8}{2\,{\left (a+b\,x\right )}^6}+\frac {55\,a^9}{7\,{\left (a+b\,x\right )}^7}-\frac {11\,a^{10}}{8\,{\left (a+b\,x\right )}^8}+\frac {a^{11}}{9\,{\left (a+b\,x\right )}^9}+55\,a^2\,\ln \left (a+b\,x\right )-11\,a\,b\,x}{b^{12}} \]