Integrand size = 11, antiderivative size = 159 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=\frac {x}{b^{10}}-\frac {a^{10}}{9 b^{11} (a+b x)^9}+\frac {5 a^9}{4 b^{11} (a+b x)^8}-\frac {45 a^8}{7 b^{11} (a+b x)^7}+\frac {20 a^7}{b^{11} (a+b x)^6}-\frac {42 a^6}{b^{11} (a+b x)^5}+\frac {63 a^5}{b^{11} (a+b x)^4}-\frac {70 a^4}{b^{11} (a+b x)^3}+\frac {60 a^3}{b^{11} (a+b x)^2}-\frac {45 a^2}{b^{11} (a+b x)}-\frac {10 a \log (a+b x)}{b^{11}} \]
x/b^10-1/9*a^10/b^11/(b*x+a)^9+5/4*a^9/b^11/(b*x+a)^8-45/7*a^8/b^11/(b*x+a )^7+20*a^7/b^11/(b*x+a)^6-42*a^6/b^11/(b*x+a)^5+63*a^5/b^11/(b*x+a)^4-70*a ^4/b^11/(b*x+a)^3+60*a^3/b^11/(b*x+a)^2-45*a^2/b^11/(b*x+a)-10*a*ln(b*x+a) /b^11
Time = 0.02 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.86 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=-\frac {4861 a^{10}+41229 a^9 b x+153576 a^8 b^2 x^2+328104 a^7 b^3 x^3+439236 a^6 b^4 x^4+375732 a^5 b^5 x^5+197568 a^4 b^6 x^6+54432 a^3 b^7 x^7+2268 a^2 b^8 x^8-2268 a b^9 x^9-252 b^{10} x^{10}+2520 a (a+b x)^9 \log (a+b x)}{252 b^{11} (a+b x)^9} \]
-1/252*(4861*a^10 + 41229*a^9*b*x + 153576*a^8*b^2*x^2 + 328104*a^7*b^3*x^ 3 + 439236*a^6*b^4*x^4 + 375732*a^5*b^5*x^5 + 197568*a^4*b^6*x^6 + 54432*a ^3*b^7*x^7 + 2268*a^2*b^8*x^8 - 2268*a*b^9*x^9 - 252*b^10*x^10 + 2520*a*(a + b*x)^9*Log[a + b*x])/(b^11*(a + b*x)^9)
Time = 0.33 (sec) , antiderivative size = 159, 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.
| \(\displaystyle \int \frac {x^{10}}{(a+b x)^{10}} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {a^{10}}{b^{10} (a+b x)^{10}}-\frac {10 a^9}{b^{10} (a+b x)^9}+\frac {45 a^8}{b^{10} (a+b x)^8}-\frac {120 a^7}{b^{10} (a+b x)^7}+\frac {210 a^6}{b^{10} (a+b x)^6}-\frac {252 a^5}{b^{10} (a+b x)^5}+\frac {210 a^4}{b^{10} (a+b x)^4}-\frac {120 a^3}{b^{10} (a+b x)^3}+\frac {45 a^2}{b^{10} (a+b x)^2}-\frac {10 a}{b^{10} (a+b x)}+\frac {1}{b^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{10}}{9 b^{11} (a+b x)^9}+\frac {5 a^9}{4 b^{11} (a+b x)^8}-\frac {45 a^8}{7 b^{11} (a+b x)^7}+\frac {20 a^7}{b^{11} (a+b x)^6}-\frac {42 a^6}{b^{11} (a+b x)^5}+\frac {63 a^5}{b^{11} (a+b x)^4}-\frac {70 a^4}{b^{11} (a+b x)^3}+\frac {60 a^3}{b^{11} (a+b x)^2}-\frac {45 a^2}{b^{11} (a+b x)}-\frac {10 a \log (a+b x)}{b^{11}}+\frac {x}{b^{10}}\) |
x/b^10 - a^10/(9*b^11*(a + b*x)^9) + (5*a^9)/(4*b^11*(a + b*x)^8) - (45*a^ 8)/(7*b^11*(a + b*x)^7) + (20*a^7)/(b^11*(a + b*x)^6) - (42*a^6)/(b^11*(a + b*x)^5) + (63*a^5)/(b^11*(a + b*x)^4) - (70*a^4)/(b^11*(a + b*x)^3) + (6 0*a^3)/(b^11*(a + b*x)^2) - (45*a^2)/(b^11*(a + b*x)) - (10*a*Log[a + b*x] )/b^11
3.3.24.3.1 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 = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(\frac {x}{b^{10}}+\frac {-45 a^{2} b^{7} x^{8}-300 a^{3} b^{6} x^{7}-910 a^{4} b^{5} x^{6}-1617 a^{5} b^{4} x^{5}-1827 a^{6} b^{3} x^{4}-1338 a^{7} b^{2} x^{3}-\frac {4329 a^{8} b \,x^{2}}{7}-\frac {4609 a^{9} x}{28}-\frac {4861 a^{10}}{252 b}}{b^{10} \left (b x +a \right )^{9}}-\frac {10 a \ln \left (b x +a \right )}{b^{11}}\) | \(120\) |
| norman | \(\frac {\frac {x^{10}}{b}-\frac {7129 a^{10}}{252 b^{11}}-\frac {90 a^{2} x^{8}}{b^{3}}-\frac {540 a^{3} x^{7}}{b^{4}}-\frac {1540 a^{4} x^{6}}{b^{5}}-\frac {2625 a^{5} x^{5}}{b^{6}}-\frac {2877 a^{6} x^{4}}{b^{7}}-\frac {2058 a^{7} x^{3}}{b^{8}}-\frac {6534 a^{8} x^{2}}{7 b^{9}}-\frac {6849 a^{9} x}{28 b^{10}}}{\left (b x +a \right )^{9}}-\frac {10 a \ln \left (b x +a \right )}{b^{11}}\) | \(124\) |
| default | \(\frac {x}{b^{10}}-\frac {a^{10}}{9 b^{11} \left (b x +a \right )^{9}}+\frac {5 a^{9}}{4 b^{11} \left (b x +a \right )^{8}}-\frac {45 a^{8}}{7 b^{11} \left (b x +a \right )^{7}}+\frac {20 a^{7}}{b^{11} \left (b x +a \right )^{6}}-\frac {42 a^{6}}{b^{11} \left (b x +a \right )^{5}}+\frac {63 a^{5}}{b^{11} \left (b x +a \right )^{4}}-\frac {70 a^{4}}{b^{11} \left (b x +a \right )^{3}}+\frac {60 a^{3}}{b^{11} \left (b x +a \right )^{2}}-\frac {45 a^{2}}{b^{11} \left (b x +a \right )}-\frac {10 a \ln \left (b x +a \right )}{b^{11}}\) | \(154\) |
| parallelrisch | \(-\frac {7129 a^{10}-252 b^{10} x^{10}+2520 \ln \left (b x +a \right ) a^{10}+2520 \ln \left (b x +a \right ) x^{9} a \,b^{9}+22680 \ln \left (b x +a \right ) x^{8} a^{2} b^{8}+90720 \ln \left (b x +a \right ) x^{7} a^{3} b^{7}+211680 \ln \left (b x +a \right ) x^{3} a^{7} b^{3}+90720 \ln \left (b x +a \right ) x^{2} a^{8} b^{2}+22680 \ln \left (b x +a \right ) x \,a^{9} b +211680 \ln \left (b x +a \right ) x^{6} a^{4} b^{6}+317520 \ln \left (b x +a \right ) x^{5} a^{5} b^{5}+317520 \ln \left (b x +a \right ) x^{4} a^{6} b^{4}+61641 a^{9} b x +235224 a^{8} b^{2} x^{2}+518616 a^{7} b^{3} x^{3}+725004 a^{6} b^{4} x^{4}+661500 a^{5} b^{5} x^{5}+388080 a^{4} b^{6} x^{6}+136080 a^{3} b^{7} x^{7}+22680 a^{2} b^{8} x^{8}}{252 b^{11} \left (b x +a \right )^{9}}\) | \(269\) |
x/b^10+(-45*a^2*b^7*x^8-300*a^3*b^6*x^7-910*a^4*b^5*x^6-1617*a^5*b^4*x^5-1 827*a^6*b^3*x^4-1338*a^7*b^2*x^3-4329/7*a^8*b*x^2-4609/28*a^9*x-4861/252*a ^10/b)/b^10/(b*x+a)^9-10*a*ln(b*x+a)/b^11
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (153) = 306\).
Time = 0.22 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.97 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=\frac {252 \, b^{10} x^{10} + 2268 \, a b^{9} x^{9} - 2268 \, a^{2} b^{8} x^{8} - 54432 \, a^{3} b^{7} x^{7} - 197568 \, a^{4} b^{6} x^{6} - 375732 \, a^{5} b^{5} x^{5} - 439236 \, a^{6} b^{4} x^{4} - 328104 \, a^{7} b^{3} x^{3} - 153576 \, a^{8} b^{2} x^{2} - 41229 \, a^{9} b x - 4861 \, a^{10} - 2520 \, {\left (a b^{9} x^{9} + 9 \, a^{2} b^{8} x^{8} + 36 \, a^{3} b^{7} x^{7} + 84 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 126 \, a^{6} b^{4} x^{4} + 84 \, a^{7} b^{3} x^{3} + 36 \, a^{8} b^{2} x^{2} + 9 \, a^{9} b x + a^{10}\right )} \log \left (b x + a\right )}{252 \, {\left (b^{20} x^{9} + 9 \, a b^{19} x^{8} + 36 \, a^{2} b^{18} x^{7} + 84 \, a^{3} b^{17} x^{6} + 126 \, a^{4} b^{16} x^{5} + 126 \, a^{5} b^{15} x^{4} + 84 \, a^{6} b^{14} x^{3} + 36 \, a^{7} b^{13} x^{2} + 9 \, a^{8} b^{12} x + a^{9} b^{11}\right )}} \]
1/252*(252*b^10*x^10 + 2268*a*b^9*x^9 - 2268*a^2*b^8*x^8 - 54432*a^3*b^7*x ^7 - 197568*a^4*b^6*x^6 - 375732*a^5*b^5*x^5 - 439236*a^6*b^4*x^4 - 328104 *a^7*b^3*x^3 - 153576*a^8*b^2*x^2 - 41229*a^9*b*x - 4861*a^10 - 2520*(a*b^ 9*x^9 + 9*a^2*b^8*x^8 + 36*a^3*b^7*x^7 + 84*a^4*b^6*x^6 + 126*a^5*b^5*x^5 + 126*a^6*b^4*x^4 + 84*a^7*b^3*x^3 + 36*a^8*b^2*x^2 + 9*a^9*b*x + a^10)*lo g(b*x + a))/(b^20*x^9 + 9*a*b^19*x^8 + 36*a^2*b^18*x^7 + 84*a^3*b^17*x^6 + 126*a^4*b^16*x^5 + 126*a^5*b^15*x^4 + 84*a^6*b^14*x^3 + 36*a^7*b^13*x^2 + 9*a^8*b^12*x + a^9*b^11)
Time = 0.90 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.41 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=- \frac {10 a \log {\left (a + b x \right )}}{b^{11}} + \frac {- 4861 a^{10} - 41481 a^{9} b x - 155844 a^{8} b^{2} x^{2} - 337176 a^{7} b^{3} x^{3} - 460404 a^{6} b^{4} x^{4} - 407484 a^{5} b^{5} x^{5} - 229320 a^{4} b^{6} x^{6} - 75600 a^{3} b^{7} x^{7} - 11340 a^{2} b^{8} x^{8}}{252 a^{9} b^{11} + 2268 a^{8} b^{12} x + 9072 a^{7} b^{13} x^{2} + 21168 a^{6} b^{14} x^{3} + 31752 a^{5} b^{15} x^{4} + 31752 a^{4} b^{16} x^{5} + 21168 a^{3} b^{17} x^{6} + 9072 a^{2} b^{18} x^{7} + 2268 a b^{19} x^{8} + 252 b^{20} x^{9}} + \frac {x}{b^{10}} \]
-10*a*log(a + b*x)/b**11 + (-4861*a**10 - 41481*a**9*b*x - 155844*a**8*b** 2*x**2 - 337176*a**7*b**3*x**3 - 460404*a**6*b**4*x**4 - 407484*a**5*b**5* x**5 - 229320*a**4*b**6*x**6 - 75600*a**3*b**7*x**7 - 11340*a**2*b**8*x**8 )/(252*a**9*b**11 + 2268*a**8*b**12*x + 9072*a**7*b**13*x**2 + 21168*a**6* b**14*x**3 + 31752*a**5*b**15*x**4 + 31752*a**4*b**16*x**5 + 21168*a**3*b* *17*x**6 + 9072*a**2*b**18*x**7 + 2268*a*b**19*x**8 + 252*b**20*x**9) + x/ b**10
Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.33 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=-\frac {11340 \, a^{2} b^{8} x^{8} + 75600 \, a^{3} b^{7} x^{7} + 229320 \, a^{4} b^{6} x^{6} + 407484 \, a^{5} b^{5} x^{5} + 460404 \, a^{6} b^{4} x^{4} + 337176 \, a^{7} b^{3} x^{3} + 155844 \, a^{8} b^{2} x^{2} + 41481 \, a^{9} b x + 4861 \, a^{10}}{252 \, {\left (b^{20} x^{9} + 9 \, a b^{19} x^{8} + 36 \, a^{2} b^{18} x^{7} + 84 \, a^{3} b^{17} x^{6} + 126 \, a^{4} b^{16} x^{5} + 126 \, a^{5} b^{15} x^{4} + 84 \, a^{6} b^{14} x^{3} + 36 \, a^{7} b^{13} x^{2} + 9 \, a^{8} b^{12} x + a^{9} b^{11}\right )}} + \frac {x}{b^{10}} - \frac {10 \, a \log \left (b x + a\right )}{b^{11}} \]
-1/252*(11340*a^2*b^8*x^8 + 75600*a^3*b^7*x^7 + 229320*a^4*b^6*x^6 + 40748 4*a^5*b^5*x^5 + 460404*a^6*b^4*x^4 + 337176*a^7*b^3*x^3 + 155844*a^8*b^2*x ^2 + 41481*a^9*b*x + 4861*a^10)/(b^20*x^9 + 9*a*b^19*x^8 + 36*a^2*b^18*x^7 + 84*a^3*b^17*x^6 + 126*a^4*b^16*x^5 + 126*a^5*b^15*x^4 + 84*a^6*b^14*x^3 + 36*a^7*b^13*x^2 + 9*a^8*b^12*x + a^9*b^11) + x/b^10 - 10*a*log(b*x + a) /b^11
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.76 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=\frac {x}{b^{10}} - \frac {10 \, a \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {11340 \, a^{2} b^{8} x^{8} + 75600 \, a^{3} b^{7} x^{7} + 229320 \, a^{4} b^{6} x^{6} + 407484 \, a^{5} b^{5} x^{5} + 460404 \, a^{6} b^{4} x^{4} + 337176 \, a^{7} b^{3} x^{3} + 155844 \, a^{8} b^{2} x^{2} + 41481 \, a^{9} b x + 4861 \, a^{10}}{252 \, {\left (b x + a\right )}^{9} b^{11}} \]
x/b^10 - 10*a*log(abs(b*x + a))/b^11 - 1/252*(11340*a^2*b^8*x^8 + 75600*a^ 3*b^7*x^7 + 229320*a^4*b^6*x^6 + 407484*a^5*b^5*x^5 + 460404*a^6*b^4*x^4 + 337176*a^7*b^3*x^3 + 155844*a^8*b^2*x^2 + 41481*a^9*b*x + 4861*a^10)/((b* x + a)^9*b^11)
Time = 1.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.80 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=-\frac {10\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {45\,a^2}{a+b\,x}-\frac {60\,a^3}{{\left (a+b\,x\right )}^2}+\frac {70\,a^4}{{\left (a+b\,x\right )}^3}-\frac {63\,a^5}{{\left (a+b\,x\right )}^4}+\frac {42\,a^6}{{\left (a+b\,x\right )}^5}-\frac {20\,a^7}{{\left (a+b\,x\right )}^6}+\frac {45\,a^8}{7\,{\left (a+b\,x\right )}^7}-\frac {5\,a^9}{4\,{\left (a+b\,x\right )}^8}+\frac {a^{10}}{9\,{\left (a+b\,x\right )}^9}}{b^{11}} \]
-(10*a*log(a + b*x) - b*x + (45*a^2)/(a + b*x) - (60*a^3)/(a + b*x)^2 + (7 0*a^4)/(a + b*x)^3 - (63*a^5)/(a + b*x)^4 + (42*a^6)/(a + b*x)^5 - (20*a^7 )/(a + b*x)^6 + (45*a^8)/(7*(a + b*x)^7) - (5*a^9)/(4*(a + b*x)^8) + a^10/ (9*(a + b*x)^9))/b^11
Time = 0.00 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.23 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=\frac {-2520 \,\mathrm {log}\left (b x +a \right ) a^{10}-22680 \,\mathrm {log}\left (b x +a \right ) a^{9} b x -90720 \,\mathrm {log}\left (b x +a \right ) a^{8} b^{2} x^{2}-211680 \,\mathrm {log}\left (b x +a \right ) a^{7} b^{3} x^{3}-317520 \,\mathrm {log}\left (b x +a \right ) a^{6} b^{4} x^{4}-317520 \,\mathrm {log}\left (b x +a \right ) a^{5} b^{5} x^{5}-211680 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{6} x^{6}-90720 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{7} x^{7}-22680 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{8} x^{8}-2520 \,\mathrm {log}\left (b x +a \right ) a \,b^{9} x^{9}-4609 a^{10}-38961 a^{9} b x -144504 a^{8} b^{2} x^{2}-306936 a^{7} b^{3} x^{3}-407484 a^{6} b^{4} x^{4}-343980 a^{5} b^{5} x^{5}-176400 a^{4} b^{6} x^{6}-45360 a^{3} b^{7} x^{7}+2520 a \,b^{9} x^{9}+252 b^{10} x^{10}}{252 b^{11} \left (b^{9} x^{9}+9 a \,b^{8} x^{8}+36 a^{2} b^{7} x^{7}+84 a^{3} b^{6} x^{6}+126 a^{4} b^{5} x^{5}+126 a^{5} b^{4} x^{4}+84 a^{6} b^{3} x^{3}+36 a^{7} b^{2} x^{2}+9 a^{8} b x +a^{9}\right )} \]
int(x**10/(a**10 + 10*a**9*b*x + 45*a**8*b**2*x**2 + 120*a**7*b**3*x**3 + 210*a**6*b**4*x**4 + 252*a**5*b**5*x**5 + 210*a**4*b**6*x**6 + 120*a**3*b* *7*x**7 + 45*a**2*b**8*x**8 + 10*a*b**9*x**9 + b**10*x**10),x)
( - 2520*log(a + b*x)*a**10 - 22680*log(a + b*x)*a**9*b*x - 90720*log(a + b*x)*a**8*b**2*x**2 - 211680*log(a + b*x)*a**7*b**3*x**3 - 317520*log(a + b*x)*a**6*b**4*x**4 - 317520*log(a + b*x)*a**5*b**5*x**5 - 211680*log(a + b*x)*a**4*b**6*x**6 - 90720*log(a + b*x)*a**3*b**7*x**7 - 22680*log(a + b* x)*a**2*b**8*x**8 - 2520*log(a + b*x)*a*b**9*x**9 - 4609*a**10 - 38961*a** 9*b*x - 144504*a**8*b**2*x**2 - 306936*a**7*b**3*x**3 - 407484*a**6*b**4*x **4 - 343980*a**5*b**5*x**5 - 176400*a**4*b**6*x**6 - 45360*a**3*b**7*x**7 + 2520*a*b**9*x**9 + 252*b**10*x**10)/(252*b**11*(a**9 + 9*a**8*b*x + 36* a**7*b**2*x**2 + 84*a**6*b**3*x**3 + 126*a**5*b**4*x**4 + 126*a**4*b**5*x* *5 + 84*a**3*b**6*x**6 + 36*a**2*b**7*x**7 + 9*a*b**8*x**8 + b**9*x**9))