Integrand size = 11, antiderivative size = 154 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {a^9}{9 b^{10} (a+b x)^9}-\frac {9 a^8}{8 b^{10} (a+b x)^8}+\frac {36 a^7}{7 b^{10} (a+b x)^7}-\frac {14 a^6}{b^{10} (a+b x)^6}+\frac {126 a^5}{5 b^{10} (a+b x)^5}-\frac {63 a^4}{2 b^{10} (a+b x)^4}+\frac {28 a^3}{b^{10} (a+b x)^3}-\frac {18 a^2}{b^{10} (a+b x)^2}+\frac {9 a}{b^{10} (a+b x)}+\frac {\log (a+b x)}{b^{10}} \]
1/9*a^9/b^10/(b*x+a)^9-9/8*a^8/b^10/(b*x+a)^8+36/7*a^7/b^10/(b*x+a)^7-14*a ^6/b^10/(b*x+a)^6+126/5*a^5/b^10/(b*x+a)^5-63/2*a^4/b^10/(b*x+a)^4+28*a^3/ b^10/(b*x+a)^3-18*a^2/b^10/(b*x+a)^2+9*a/b^10/(b*x+a)+ln(b*x+a)/b^10
Time = 0.02 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.72 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {a \left (7129 a^8+61641 a^7 b x+235224 a^6 b^2 x^2+518616 a^5 b^3 x^3+725004 a^4 b^4 x^4+661500 a^3 b^5 x^5+388080 a^2 b^6 x^6+136080 a b^7 x^7+22680 b^8 x^8\right )}{2520 b^{10} (a+b x)^9}+\frac {\log (a+b x)}{b^{10}} \]
(a*(7129*a^8 + 61641*a^7*b*x + 235224*a^6*b^2*x^2 + 518616*a^5*b^3*x^3 + 7 25004*a^4*b^4*x^4 + 661500*a^3*b^5*x^5 + 388080*a^2*b^6*x^6 + 136080*a*b^7 *x^7 + 22680*b^8*x^8))/(2520*b^10*(a + b*x)^9) + Log[a + b*x]/b^10
Time = 0.31 (sec) , antiderivative size = 154, 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.
\(\displaystyle \int \frac {x^9}{(a+b x)^{10}} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (-\frac {a^9}{b^9 (a+b x)^{10}}+\frac {9 a^8}{b^9 (a+b x)^9}-\frac {36 a^7}{b^9 (a+b x)^8}+\frac {84 a^6}{b^9 (a+b x)^7}-\frac {126 a^5}{b^9 (a+b x)^6}+\frac {126 a^4}{b^9 (a+b x)^5}-\frac {84 a^3}{b^9 (a+b x)^4}+\frac {36 a^2}{b^9 (a+b x)^3}-\frac {9 a}{b^9 (a+b x)^2}+\frac {1}{b^9 (a+b x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^9}{9 b^{10} (a+b x)^9}-\frac {9 a^8}{8 b^{10} (a+b x)^8}+\frac {36 a^7}{7 b^{10} (a+b x)^7}-\frac {14 a^6}{b^{10} (a+b x)^6}+\frac {126 a^5}{5 b^{10} (a+b x)^5}-\frac {63 a^4}{2 b^{10} (a+b x)^4}+\frac {28 a^3}{b^{10} (a+b x)^3}-\frac {18 a^2}{b^{10} (a+b x)^2}+\frac {9 a}{b^{10} (a+b x)}+\frac {\log (a+b x)}{b^{10}}\) |
a^9/(9*b^10*(a + b*x)^9) - (9*a^8)/(8*b^10*(a + b*x)^8) + (36*a^7)/(7*b^10 *(a + b*x)^7) - (14*a^6)/(b^10*(a + b*x)^6) + (126*a^5)/(5*b^10*(a + b*x)^ 5) - (63*a^4)/(2*b^10*(a + b*x)^4) + (28*a^3)/(b^10*(a + b*x)^3) - (18*a^2 )/(b^10*(a + b*x)^2) + (9*a)/(b^10*(a + b*x)) + Log[a + b*x]/b^10
3.3.25.3.1 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 = 113, normalized size of antiderivative = 0.73
method | result | size |
norman | \(\frac {\frac {7129 a^{9}}{2520 b^{10}}+\frac {9 a \,x^{8}}{b^{2}}+\frac {54 a^{2} x^{7}}{b^{3}}+\frac {154 a^{3} x^{6}}{b^{4}}+\frac {525 a^{4} x^{5}}{2 b^{5}}+\frac {2877 a^{5} x^{4}}{10 b^{6}}+\frac {1029 a^{6} x^{3}}{5 b^{7}}+\frac {3267 a^{7} x^{2}}{35 b^{8}}+\frac {6849 a^{8} x}{280 b^{9}}}{\left (b x +a \right )^{9}}+\frac {\ln \left (b x +a \right )}{b^{10}}\) | \(113\) |
risch | \(\frac {\frac {7129 a^{9}}{2520 b^{10}}+\frac {9 a \,x^{8}}{b^{2}}+\frac {54 a^{2} x^{7}}{b^{3}}+\frac {154 a^{3} x^{6}}{b^{4}}+\frac {525 a^{4} x^{5}}{2 b^{5}}+\frac {2877 a^{5} x^{4}}{10 b^{6}}+\frac {1029 a^{6} x^{3}}{5 b^{7}}+\frac {3267 a^{7} x^{2}}{35 b^{8}}+\frac {6849 a^{8} x}{280 b^{9}}}{\left (b x +a \right )^{9}}+\frac {\ln \left (b x +a \right )}{b^{10}}\) | \(113\) |
default | \(\frac {a^{9}}{9 b^{10} \left (b x +a \right )^{9}}-\frac {9 a^{8}}{8 b^{10} \left (b x +a \right )^{8}}+\frac {36 a^{7}}{7 b^{10} \left (b x +a \right )^{7}}-\frac {14 a^{6}}{b^{10} \left (b x +a \right )^{6}}+\frac {126 a^{5}}{5 b^{10} \left (b x +a \right )^{5}}-\frac {63 a^{4}}{2 b^{10} \left (b x +a \right )^{4}}+\frac {28 a^{3}}{b^{10} \left (b x +a \right )^{3}}-\frac {18 a^{2}}{b^{10} \left (b x +a \right )^{2}}+\frac {9 a}{b^{10} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{10}}\) | \(145\) |
parallelrisch | \(\frac {7129 a^{9}+2520 \ln \left (b x +a \right ) a^{9}+22680 \ln \left (b x +a \right ) x^{8} a \,b^{8}+90720 \ln \left (b x +a \right ) x^{7} a^{2} b^{7}+22680 \ln \left (b x +a \right ) x \,a^{8} b +211680 \ln \left (b x +a \right ) x^{6} a^{3} b^{6}+317520 \ln \left (b x +a \right ) x^{5} a^{4} b^{5}+317520 \ln \left (b x +a \right ) x^{4} a^{5} b^{4}+211680 \ln \left (b x +a \right ) x^{3} a^{6} b^{3}+90720 \ln \left (b x +a \right ) x^{2} a^{7} b^{2}+661500 a^{4} x^{5} b^{5}+2520 \ln \left (b x +a \right ) x^{9} b^{9}+388080 x^{6} a^{3} b^{6}+22680 a \,x^{8} b^{8}+725004 a^{5} b^{4} x^{4}+518616 a^{6} b^{3} x^{3}+235224 a^{7} b^{2} x^{2}+61641 a^{8} b x +136080 a^{2} x^{7} b^{7}}{2520 b^{10} \left (b x +a \right )^{9}}\) | \(256\) |
(7129/2520*a^9/b^10+9*a/b^2*x^8+54*a^2/b^3*x^7+154*a^3/b^4*x^6+525/2*a^4/b ^5*x^5+2877/10*a^5/b^6*x^4+1029/5*a^6/b^7*x^3+3267/35*a^7/b^8*x^2+6849/280 *a^8/b^9*x)/(b*x+a)^9+ln(b*x+a)/b^10
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (144) = 288\).
Time = 0.22 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.90 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {22680 \, a b^{8} x^{8} + 136080 \, a^{2} b^{7} x^{7} + 388080 \, a^{3} b^{6} x^{6} + 661500 \, a^{4} b^{5} x^{5} + 725004 \, a^{5} b^{4} x^{4} + 518616 \, a^{6} b^{3} x^{3} + 235224 \, a^{7} b^{2} x^{2} + 61641 \, a^{8} b x + 7129 \, a^{9} + 2520 \, {\left (b^{9} x^{9} + 9 \, a b^{8} x^{8} + 36 \, a^{2} b^{7} x^{7} + 84 \, a^{3} b^{6} x^{6} + 126 \, a^{4} b^{5} x^{5} + 126 \, a^{5} b^{4} x^{4} + 84 \, a^{6} b^{3} x^{3} + 36 \, a^{7} b^{2} x^{2} + 9 \, a^{8} b x + a^{9}\right )} \log \left (b x + a\right )}{2520 \, {\left (b^{19} x^{9} + 9 \, a b^{18} x^{8} + 36 \, a^{2} b^{17} x^{7} + 84 \, a^{3} b^{16} x^{6} + 126 \, a^{4} b^{15} x^{5} + 126 \, a^{5} b^{14} x^{4} + 84 \, a^{6} b^{13} x^{3} + 36 \, a^{7} b^{12} x^{2} + 9 \, a^{8} b^{11} x + a^{9} b^{10}\right )}} \]
1/2520*(22680*a*b^8*x^8 + 136080*a^2*b^7*x^7 + 388080*a^3*b^6*x^6 + 661500 *a^4*b^5*x^5 + 725004*a^5*b^4*x^4 + 518616*a^6*b^3*x^3 + 235224*a^7*b^2*x^ 2 + 61641*a^8*b*x + 7129*a^9 + 2520*(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)*log(b*x + a))/(b^19*x^9 + 9*a*b^18*x^8 + 36*a^2*b^17*x^7 + 84*a^3*b^16*x^6 + 126*a^4*b^15*x^5 + 126*a^5*b^14*x^4 + 84*a^6*b^13*x^3 + 36*a^7*b^12*x^2 + 9*a^8*b^11*x + a^9*b^10)
Time = 0.57 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.38 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {7129 a^{9} + 61641 a^{8} b x + 235224 a^{7} b^{2} x^{2} + 518616 a^{6} b^{3} x^{3} + 725004 a^{5} b^{4} x^{4} + 661500 a^{4} b^{5} x^{5} + 388080 a^{3} b^{6} x^{6} + 136080 a^{2} b^{7} x^{7} + 22680 a b^{8} x^{8}}{2520 a^{9} b^{10} + 22680 a^{8} b^{11} x + 90720 a^{7} b^{12} x^{2} + 211680 a^{6} b^{13} x^{3} + 317520 a^{5} b^{14} x^{4} + 317520 a^{4} b^{15} x^{5} + 211680 a^{3} b^{16} x^{6} + 90720 a^{2} b^{17} x^{7} + 22680 a b^{18} x^{8} + 2520 b^{19} x^{9}} + \frac {\log {\left (a + b x \right )}}{b^{10}} \]
(7129*a**9 + 61641*a**8*b*x + 235224*a**7*b**2*x**2 + 518616*a**6*b**3*x** 3 + 725004*a**5*b**4*x**4 + 661500*a**4*b**5*x**5 + 388080*a**3*b**6*x**6 + 136080*a**2*b**7*x**7 + 22680*a*b**8*x**8)/(2520*a**9*b**10 + 22680*a**8 *b**11*x + 90720*a**7*b**12*x**2 + 211680*a**6*b**13*x**3 + 317520*a**5*b* *14*x**4 + 317520*a**4*b**15*x**5 + 211680*a**3*b**16*x**6 + 90720*a**2*b* *17*x**7 + 22680*a*b**18*x**8 + 2520*b**19*x**9) + log(a + b*x)/b**10
Time = 0.23 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.31 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {22680 \, a b^{8} x^{8} + 136080 \, a^{2} b^{7} x^{7} + 388080 \, a^{3} b^{6} x^{6} + 661500 \, a^{4} b^{5} x^{5} + 725004 \, a^{5} b^{4} x^{4} + 518616 \, a^{6} b^{3} x^{3} + 235224 \, a^{7} b^{2} x^{2} + 61641 \, a^{8} b x + 7129 \, a^{9}}{2520 \, {\left (b^{19} x^{9} + 9 \, a b^{18} x^{8} + 36 \, a^{2} b^{17} x^{7} + 84 \, a^{3} b^{16} x^{6} + 126 \, a^{4} b^{15} x^{5} + 126 \, a^{5} b^{14} x^{4} + 84 \, a^{6} b^{13} x^{3} + 36 \, a^{7} b^{12} x^{2} + 9 \, a^{8} b^{11} x + a^{9} b^{10}\right )}} + \frac {\log \left (b x + a\right )}{b^{10}} \]
1/2520*(22680*a*b^8*x^8 + 136080*a^2*b^7*x^7 + 388080*a^3*b^6*x^6 + 661500 *a^4*b^5*x^5 + 725004*a^5*b^4*x^4 + 518616*a^6*b^3*x^3 + 235224*a^7*b^2*x^ 2 + 61641*a^8*b*x + 7129*a^9)/(b^19*x^9 + 9*a*b^18*x^8 + 36*a^2*b^17*x^7 + 84*a^3*b^16*x^6 + 126*a^4*b^15*x^5 + 126*a^5*b^14*x^4 + 84*a^6*b^13*x^3 + 36*a^7*b^12*x^2 + 9*a^8*b^11*x + a^9*b^10) + log(b*x + a)/b^10
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.73 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {\log \left ({\left | b x + a \right |}\right )}{b^{10}} + \frac {22680 \, a b^{7} x^{8} + 136080 \, a^{2} b^{6} x^{7} + 388080 \, a^{3} b^{5} x^{6} + 661500 \, a^{4} b^{4} x^{5} + 725004 \, a^{5} b^{3} x^{4} + 518616 \, a^{6} b^{2} x^{3} + 235224 \, a^{7} b x^{2} + 61641 \, a^{8} x + \frac {7129 \, a^{9}}{b}}{2520 \, {\left (b x + a\right )}^{9} b^{9}} \]
log(abs(b*x + a))/b^10 + 1/2520*(22680*a*b^7*x^8 + 136080*a^2*b^6*x^7 + 38 8080*a^3*b^5*x^6 + 661500*a^4*b^4*x^5 + 725004*a^5*b^3*x^4 + 518616*a^6*b^ 2*x^3 + 235224*a^7*b*x^2 + 61641*a^8*x + 7129*a^9/b)/((b*x + a)^9*b^9)
Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.76 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {\ln \left (a+b\,x\right )+\frac {9\,a}{a+b\,x}-\frac {18\,a^2}{{\left (a+b\,x\right )}^2}+\frac {28\,a^3}{{\left (a+b\,x\right )}^3}-\frac {63\,a^4}{2\,{\left (a+b\,x\right )}^4}+\frac {126\,a^5}{5\,{\left (a+b\,x\right )}^5}-\frac {14\,a^6}{{\left (a+b\,x\right )}^6}+\frac {36\,a^7}{7\,{\left (a+b\,x\right )}^7}-\frac {9\,a^8}{8\,{\left (a+b\,x\right )}^8}+\frac {a^9}{9\,{\left (a+b\,x\right )}^9}}{b^{10}} \]
(log(a + b*x) + (9*a)/(a + b*x) - (18*a^2)/(a + b*x)^2 + (28*a^3)/(a + b*x )^3 - (63*a^4)/(2*(a + b*x)^4) + (126*a^5)/(5*(a + b*x)^5) - (14*a^6)/(a + b*x)^6 + (36*a^7)/(7*(a + b*x)^7) - (9*a^8)/(8*(a + b*x)^8) + a^9/(9*(a + b*x)^9))/b^10
Time = 0.00 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.22 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {2520 \,\mathrm {log}\left (b x +a \right ) a^{9}+22680 \,\mathrm {log}\left (b x +a \right ) a^{8} b x +90720 \,\mathrm {log}\left (b x +a \right ) a^{7} b^{2} x^{2}+211680 \,\mathrm {log}\left (b x +a \right ) a^{6} b^{3} x^{3}+317520 \,\mathrm {log}\left (b x +a \right ) a^{5} b^{4} x^{4}+317520 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{5} x^{5}+211680 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{6} x^{6}+90720 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{7} x^{7}+22680 \,\mathrm {log}\left (b x +a \right ) a \,b^{8} x^{8}+2520 \,\mathrm {log}\left (b x +a \right ) b^{9} x^{9}+4609 a^{9}+38961 a^{8} b x +144504 a^{7} b^{2} x^{2}+306936 a^{6} b^{3} x^{3}+407484 a^{5} b^{4} x^{4}+343980 a^{4} b^{5} x^{5}+176400 a^{3} b^{6} x^{6}+45360 a^{2} b^{7} x^{7}-2520 b^{9} x^{9}}{2520 b^{10} \left (b^{9} x^{9}+9 a \,b^{8} x^{8}+36 a^{2} b^{7} x^{7}+84 a^{3} b^{6} x^{6}+126 a^{4} b^{5} x^{5}+126 a^{5} b^{4} x^{4}+84 a^{6} b^{3} x^{3}+36 a^{7} b^{2} x^{2}+9 a^{8} b x +a^{9}\right )} \]
int(x**9/(a**10 + 10*a**9*b*x + 45*a**8*b**2*x**2 + 120*a**7*b**3*x**3 + 2 10*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**9 + 22680*log(a + b*x)*a**8*b*x + 90720*log(a + b*x) *a**7*b**2*x**2 + 211680*log(a + b*x)*a**6*b**3*x**3 + 317520*log(a + b*x) *a**5*b**4*x**4 + 317520*log(a + b*x)*a**4*b**5*x**5 + 211680*log(a + b*x) *a**3*b**6*x**6 + 90720*log(a + b*x)*a**2*b**7*x**7 + 22680*log(a + b*x)*a *b**8*x**8 + 2520*log(a + b*x)*b**9*x**9 + 4609*a**9 + 38961*a**8*b*x + 14 4504*a**7*b**2*x**2 + 306936*a**6*b**3*x**3 + 407484*a**5*b**4*x**4 + 3439 80*a**4*b**5*x**5 + 176400*a**3*b**6*x**6 + 45360*a**2*b**7*x**7 - 2520*b* *9*x**9)/(2520*b**10*(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))