Integrand size = 11, antiderivative size = 158 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=-\frac {1}{a^{10} x}-\frac {b}{9 a^2 (a+b x)^9}-\frac {b}{4 a^3 (a+b x)^8}-\frac {3 b}{7 a^4 (a+b x)^7}-\frac {2 b}{3 a^5 (a+b x)^6}-\frac {b}{a^6 (a+b x)^5}-\frac {3 b}{2 a^7 (a+b x)^4}-\frac {7 b}{3 a^8 (a+b x)^3}-\frac {4 b}{a^9 (a+b x)^2}-\frac {9 b}{a^{10} (a+b x)}-\frac {10 b \log (x)}{a^{11}}+\frac {10 b \log (a+b x)}{a^{11}} \]
-1/a^10/x-1/9*b/a^2/(b*x+a)^9-1/4*b/a^3/(b*x+a)^8-3/7*b/a^4/(b*x+a)^7-2/3* b/a^5/(b*x+a)^6-b/a^6/(b*x+a)^5-3/2*b/a^7/(b*x+a)^4-7/3*b/a^8/(b*x+a)^3-4* b/a^9/(b*x+a)^2-9*b/a^10/(b*x+a)-10*b*ln(x)/a^11+10*b*ln(b*x+a)/a^11
Time = 0.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=-\frac {\frac {a \left (252 a^9+7129 a^8 b x+41481 a^7 b^2 x^2+120564 a^6 b^3 x^3+210756 a^5 b^4 x^4+236754 a^4 b^5 x^5+173250 a^3 b^6 x^6+80220 a^2 b^7 x^7+21420 a b^8 x^8+2520 b^9 x^9\right )}{x (a+b x)^9}+2520 b \log (x)-2520 b \log (a+b x)}{252 a^{11}} \]
-1/252*((a*(252*a^9 + 7129*a^8*b*x + 41481*a^7*b^2*x^2 + 120564*a^6*b^3*x^ 3 + 210756*a^5*b^4*x^4 + 236754*a^4*b^5*x^5 + 173250*a^3*b^6*x^6 + 80220*a ^2*b^7*x^7 + 21420*a*b^8*x^8 + 2520*b^9*x^9))/(x*(a + b*x)^9) + 2520*b*Log [x] - 2520*b*Log[a + b*x])/a^11
Time = 0.33 (sec) , antiderivative size = 158, 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 = {54, 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.
| \(\displaystyle \int \frac {1}{x^2 (a+b x)^{10}} \, dx\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \int \left (\frac {10 b^2}{a^{11} (a+b x)}-\frac {10 b}{a^{11} x}+\frac {9 b^2}{a^{10} (a+b x)^2}+\frac {1}{a^{10} x^2}+\frac {8 b^2}{a^9 (a+b x)^3}+\frac {7 b^2}{a^8 (a+b x)^4}+\frac {6 b^2}{a^7 (a+b x)^5}+\frac {5 b^2}{a^6 (a+b x)^6}+\frac {4 b^2}{a^5 (a+b x)^7}+\frac {3 b^2}{a^4 (a+b x)^8}+\frac {2 b^2}{a^3 (a+b x)^9}+\frac {b^2}{a^2 (a+b x)^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b \log (x)}{a^{11}}+\frac {10 b \log (a+b x)}{a^{11}}-\frac {9 b}{a^{10} (a+b x)}-\frac {1}{a^{10} x}-\frac {4 b}{a^9 (a+b x)^2}-\frac {7 b}{3 a^8 (a+b x)^3}-\frac {3 b}{2 a^7 (a+b x)^4}-\frac {b}{a^6 (a+b x)^5}-\frac {2 b}{3 a^5 (a+b x)^6}-\frac {3 b}{7 a^4 (a+b x)^7}-\frac {b}{4 a^3 (a+b x)^8}-\frac {b}{9 a^2 (a+b x)^9}\) |
-(1/(a^10*x)) - b/(9*a^2*(a + b*x)^9) - b/(4*a^3*(a + b*x)^8) - (3*b)/(7*a ^4*(a + b*x)^7) - (2*b)/(3*a^5*(a + b*x)^6) - b/(a^6*(a + b*x)^5) - (3*b)/ (2*a^7*(a + b*x)^4) - (7*b)/(3*a^8*(a + b*x)^3) - (4*b)/(a^9*(a + b*x)^2) - (9*b)/(a^10*(a + b*x)) - (10*b*Log[x])/a^11 + (10*b*Log[a + b*x])/a^11
3.3.36.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Time = 0.06 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87
| method | result | size |
| risch | \(\frac {-\frac {10 b^{9} x^{9}}{a^{10}}-\frac {85 b^{8} x^{8}}{a^{9}}-\frac {955 b^{7} x^{7}}{3 a^{8}}-\frac {1375 b^{6} x^{6}}{2 a^{7}}-\frac {1879 b^{5} x^{5}}{2 a^{6}}-\frac {2509 b^{4} x^{4}}{3 a^{5}}-\frac {3349 b^{3} x^{3}}{7 a^{4}}-\frac {4609 b^{2} x^{2}}{28 a^{3}}-\frac {7129 b x}{252 a^{2}}-\frac {1}{a}}{x \left (b x +a \right )^{9}}+\frac {10 b \ln \left (-b x -a \right )}{a^{11}}-\frac {10 b \ln \left (x \right )}{a^{11}}\) | \(137\) |
| norman | \(\frac {-\frac {1}{a}+\frac {90 b^{2} x^{2}}{a^{3}}+\frac {540 b^{3} x^{3}}{a^{4}}+\frac {1540 b^{4} x^{4}}{a^{5}}+\frac {2625 b^{5} x^{5}}{a^{6}}+\frac {2877 b^{6} x^{6}}{a^{7}}+\frac {2058 b^{7} x^{7}}{a^{8}}+\frac {6534 b^{8} x^{8}}{7 a^{9}}+\frac {6849 b^{9} x^{9}}{28 a^{10}}+\frac {7129 b^{10} x^{10}}{252 a^{11}}}{x \left (b x +a \right )^{9}}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {10 b \ln \left (b x +a \right )}{a^{11}}\) | \(138\) |
| default | \(-\frac {1}{a^{10} x}-\frac {b}{9 a^{2} \left (b x +a \right )^{9}}-\frac {b}{4 a^{3} \left (b x +a \right )^{8}}-\frac {3 b}{7 a^{4} \left (b x +a \right )^{7}}-\frac {2 b}{3 a^{5} \left (b x +a \right )^{6}}-\frac {b}{a^{6} \left (b x +a \right )^{5}}-\frac {3 b}{2 a^{7} \left (b x +a \right )^{4}}-\frac {7 b}{3 a^{8} \left (b x +a \right )^{3}}-\frac {4 b}{a^{9} \left (b x +a \right )^{2}}-\frac {9 b}{a^{10} \left (b x +a \right )}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {10 b \ln \left (b x +a \right )}{a^{11}}\) | \(147\) |
| parallelrisch | \(-\frac {252 a^{10}-7129 b^{10} x^{10}-22680 \ln \left (b x +a \right ) x^{9} a \,b^{9}-90720 \ln \left (b x +a \right ) x^{8} a^{2} b^{8}-211680 \ln \left (b x +a \right ) x^{7} a^{3} b^{7}-90720 \ln \left (b x +a \right ) x^{3} a^{7} b^{3}-22680 \ln \left (b x +a \right ) x^{2} a^{8} b^{2}-2520 \ln \left (b x +a \right ) x \,a^{9} b -317520 \ln \left (b x +a \right ) x^{6} a^{4} b^{6}-317520 \ln \left (b x +a \right ) x^{5} a^{5} b^{5}-211680 \ln \left (b x +a \right ) x^{4} a^{6} b^{4}+90720 a^{7} b^{3} \ln \left (x \right ) x^{3}+211680 a^{6} b^{4} \ln \left (x \right ) x^{4}+22680 a \,b^{9} \ln \left (x \right ) x^{9}+211680 a^{3} b^{7} \ln \left (x \right ) x^{7}+90720 a^{2} b^{8} \ln \left (x \right ) x^{8}+317520 a^{4} b^{6} \ln \left (x \right ) x^{6}+317520 a^{5} b^{5} \ln \left (x \right ) x^{5}+22680 a^{8} b^{2} \ln \left (x \right ) x^{2}+2520 a^{9} b \ln \left (x \right ) x -22680 a^{8} b^{2} x^{2}-136080 a^{7} b^{3} x^{3}-388080 a^{6} b^{4} x^{4}-661500 a^{5} b^{5} x^{5}-725004 a^{4} b^{6} x^{6}-518616 a^{3} b^{7} x^{7}-235224 a^{2} b^{8} x^{8}-61641 a \,b^{9} x^{9}+2520 b^{10} \ln \left (x \right ) x^{10}-2520 \ln \left (b x +a \right ) x^{10} b^{10}}{252 a^{11} x \left (b x +a \right )^{9}}\) | \(398\) |
(-10*b^9/a^10*x^9-85*b^8/a^9*x^8-955/3*b^7/a^8*x^7-1375/2*b^6/a^7*x^6-1879 /2*b^5/a^6*x^5-2509/3*b^4/a^5*x^4-3349/7*b^3/a^4*x^3-4609/28*b^2/a^3*x^2-7 129/252*b/a^2*x-1/a)/x/(b*x+a)^9+10/a^11*b*ln(-b*x-a)-10*b*ln(x)/a^11
Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (146) = 292\).
Time = 0.24 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.64 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=-\frac {2520 \, a b^{9} x^{9} + 21420 \, a^{2} b^{8} x^{8} + 80220 \, a^{3} b^{7} x^{7} + 173250 \, a^{4} b^{6} x^{6} + 236754 \, a^{5} b^{5} x^{5} + 210756 \, a^{6} b^{4} x^{4} + 120564 \, a^{7} b^{3} x^{3} + 41481 \, a^{8} b^{2} x^{2} + 7129 \, a^{9} b x + 252 \, a^{10} - 2520 \, {\left (b^{10} x^{10} + 9 \, a b^{9} x^{9} + 36 \, a^{2} b^{8} x^{8} + 84 \, a^{3} b^{7} x^{7} + 126 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 84 \, a^{6} b^{4} x^{4} + 36 \, a^{7} b^{3} x^{3} + 9 \, a^{8} b^{2} x^{2} + a^{9} b x\right )} \log \left (b x + a\right ) + 2520 \, {\left (b^{10} x^{10} + 9 \, a b^{9} x^{9} + 36 \, a^{2} b^{8} x^{8} + 84 \, a^{3} b^{7} x^{7} + 126 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 84 \, a^{6} b^{4} x^{4} + 36 \, a^{7} b^{3} x^{3} + 9 \, a^{8} b^{2} x^{2} + a^{9} b x\right )} \log \left (x\right )}{252 \, {\left (a^{11} b^{9} x^{10} + 9 \, a^{12} b^{8} x^{9} + 36 \, a^{13} b^{7} x^{8} + 84 \, a^{14} b^{6} x^{7} + 126 \, a^{15} b^{5} x^{6} + 126 \, a^{16} b^{4} x^{5} + 84 \, a^{17} b^{3} x^{4} + 36 \, a^{18} b^{2} x^{3} + 9 \, a^{19} b x^{2} + a^{20} x\right )}} \]
-1/252*(2520*a*b^9*x^9 + 21420*a^2*b^8*x^8 + 80220*a^3*b^7*x^7 + 173250*a^ 4*b^6*x^6 + 236754*a^5*b^5*x^5 + 210756*a^6*b^4*x^4 + 120564*a^7*b^3*x^3 + 41481*a^8*b^2*x^2 + 7129*a^9*b*x + 252*a^10 - 2520*(b^10*x^10 + 9*a*b^9*x ^9 + 36*a^2*b^8*x^8 + 84*a^3*b^7*x^7 + 126*a^4*b^6*x^6 + 126*a^5*b^5*x^5 + 84*a^6*b^4*x^4 + 36*a^7*b^3*x^3 + 9*a^8*b^2*x^2 + a^9*b*x)*log(b*x + a) + 2520*(b^10*x^10 + 9*a*b^9*x^9 + 36*a^2*b^8*x^8 + 84*a^3*b^7*x^7 + 126*a^4 *b^6*x^6 + 126*a^5*b^5*x^5 + 84*a^6*b^4*x^4 + 36*a^7*b^3*x^3 + 9*a^8*b^2*x ^2 + a^9*b*x)*log(x))/(a^11*b^9*x^10 + 9*a^12*b^8*x^9 + 36*a^13*b^7*x^8 + 84*a^14*b^6*x^7 + 126*a^15*b^5*x^6 + 126*a^16*b^4*x^5 + 84*a^17*b^3*x^4 + 36*a^18*b^2*x^3 + 9*a^19*b*x^2 + a^20*x)
Time = 0.66 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=\frac {- 252 a^{9} - 7129 a^{8} b x - 41481 a^{7} b^{2} x^{2} - 120564 a^{6} b^{3} x^{3} - 210756 a^{5} b^{4} x^{4} - 236754 a^{4} b^{5} x^{5} - 173250 a^{3} b^{6} x^{6} - 80220 a^{2} b^{7} x^{7} - 21420 a b^{8} x^{8} - 2520 b^{9} x^{9}}{252 a^{19} x + 2268 a^{18} b x^{2} + 9072 a^{17} b^{2} x^{3} + 21168 a^{16} b^{3} x^{4} + 31752 a^{15} b^{4} x^{5} + 31752 a^{14} b^{5} x^{6} + 21168 a^{13} b^{6} x^{7} + 9072 a^{12} b^{7} x^{8} + 2268 a^{11} b^{8} x^{9} + 252 a^{10} b^{9} x^{10}} + \frac {10 b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{11}} \]
(-252*a**9 - 7129*a**8*b*x - 41481*a**7*b**2*x**2 - 120564*a**6*b**3*x**3 - 210756*a**5*b**4*x**4 - 236754*a**4*b**5*x**5 - 173250*a**3*b**6*x**6 - 80220*a**2*b**7*x**7 - 21420*a*b**8*x**8 - 2520*b**9*x**9)/(252*a**19*x + 2268*a**18*b*x**2 + 9072*a**17*b**2*x**3 + 21168*a**16*b**3*x**4 + 31752*a **15*b**4*x**5 + 31752*a**14*b**5*x**6 + 21168*a**13*b**6*x**7 + 9072*a**1 2*b**7*x**8 + 2268*a**11*b**8*x**9 + 252*a**10*b**9*x**10) + 10*b*(-log(x) + log(a/b + x))/a**11
Time = 0.22 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=-\frac {2520 \, b^{9} x^{9} + 21420 \, a b^{8} x^{8} + 80220 \, a^{2} b^{7} x^{7} + 173250 \, a^{3} b^{6} x^{6} + 236754 \, a^{4} b^{5} x^{5} + 210756 \, a^{5} b^{4} x^{4} + 120564 \, a^{6} b^{3} x^{3} + 41481 \, a^{7} b^{2} x^{2} + 7129 \, a^{8} b x + 252 \, a^{9}}{252 \, {\left (a^{10} b^{9} x^{10} + 9 \, a^{11} b^{8} x^{9} + 36 \, a^{12} b^{7} x^{8} + 84 \, a^{13} b^{6} x^{7} + 126 \, a^{14} b^{5} x^{6} + 126 \, a^{15} b^{4} x^{5} + 84 \, a^{16} b^{3} x^{4} + 36 \, a^{17} b^{2} x^{3} + 9 \, a^{18} b x^{2} + a^{19} x\right )}} + \frac {10 \, b \log \left (b x + a\right )}{a^{11}} - \frac {10 \, b \log \left (x\right )}{a^{11}} \]
-1/252*(2520*b^9*x^9 + 21420*a*b^8*x^8 + 80220*a^2*b^7*x^7 + 173250*a^3*b^ 6*x^6 + 236754*a^4*b^5*x^5 + 210756*a^5*b^4*x^4 + 120564*a^6*b^3*x^3 + 414 81*a^7*b^2*x^2 + 7129*a^8*b*x + 252*a^9)/(a^10*b^9*x^10 + 9*a^11*b^8*x^9 + 36*a^12*b^7*x^8 + 84*a^13*b^6*x^7 + 126*a^14*b^5*x^6 + 126*a^15*b^4*x^5 + 84*a^16*b^3*x^4 + 36*a^17*b^2*x^3 + 9*a^18*b*x^2 + a^19*x) + 10*b*log(b*x + a)/a^11 - 10*b*log(x)/a^11
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=\frac {10 \, b \log \left ({\left | b x + a \right |}\right )}{a^{11}} - \frac {10 \, b \log \left ({\left | x \right |}\right )}{a^{11}} - \frac {2520 \, a b^{9} x^{9} + 21420 \, a^{2} b^{8} x^{8} + 80220 \, a^{3} b^{7} x^{7} + 173250 \, a^{4} b^{6} x^{6} + 236754 \, a^{5} b^{5} x^{5} + 210756 \, a^{6} b^{4} x^{4} + 120564 \, a^{7} b^{3} x^{3} + 41481 \, a^{8} b^{2} x^{2} + 7129 \, a^{9} b x + 252 \, a^{10}}{252 \, {\left (b x + a\right )}^{9} a^{11} x} \]
10*b*log(abs(b*x + a))/a^11 - 10*b*log(abs(x))/a^11 - 1/252*(2520*a*b^9*x^ 9 + 21420*a^2*b^8*x^8 + 80220*a^3*b^7*x^7 + 173250*a^4*b^6*x^6 + 236754*a^ 5*b^5*x^5 + 210756*a^6*b^4*x^4 + 120564*a^7*b^3*x^3 + 41481*a^8*b^2*x^2 + 7129*a^9*b*x + 252*a^10)/((b*x + a)^9*a^11*x)
Time = 0.40 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=\frac {20\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{11}}-\frac {\frac {1}{a}+\frac {4609\,b^2\,x^2}{28\,a^3}+\frac {3349\,b^3\,x^3}{7\,a^4}+\frac {2509\,b^4\,x^4}{3\,a^5}+\frac {1879\,b^5\,x^5}{2\,a^6}+\frac {1375\,b^6\,x^6}{2\,a^7}+\frac {955\,b^7\,x^7}{3\,a^8}+\frac {85\,b^8\,x^8}{a^9}+\frac {10\,b^9\,x^9}{a^{10}}+\frac {7129\,b\,x}{252\,a^2}}{a^9\,x+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^3+84\,a^6\,b^3\,x^4+126\,a^5\,b^4\,x^5+126\,a^4\,b^5\,x^6+84\,a^3\,b^6\,x^7+36\,a^2\,b^7\,x^8+9\,a\,b^8\,x^9+b^9\,x^{10}} \]
(20*b*atanh((2*b*x)/a + 1))/a^11 - (1/a + (4609*b^2*x^2)/(28*a^3) + (3349* b^3*x^3)/(7*a^4) + (2509*b^4*x^4)/(3*a^5) + (1879*b^5*x^5)/(2*a^6) + (1375 *b^6*x^6)/(2*a^7) + (955*b^7*x^7)/(3*a^8) + (85*b^8*x^8)/a^9 + (10*b^9*x^9 )/a^10 + (7129*b*x)/(252*a^2))/(a^9*x + b^9*x^10 + 9*a^8*b*x^2 + 9*a*b^8*x ^9 + 36*a^7*b^2*x^3 + 84*a^6*b^3*x^4 + 126*a^5*b^4*x^5 + 126*a^4*b^5*x^6 + 84*a^3*b^6*x^7 + 36*a^2*b^7*x^8)