Integrand size = 11, antiderivative size = 163 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=-\frac {1}{9 a^3 x^9}+\frac {3 b}{8 a^4 x^8}-\frac {6 b^2}{7 a^5 x^7}+\frac {5 b^3}{3 a^6 x^6}-\frac {3 b^4}{a^7 x^5}+\frac {21 b^5}{4 a^8 x^4}-\frac {28 b^6}{3 a^9 x^3}+\frac {18 b^7}{a^{10} x^2}-\frac {45 b^8}{a^{11} x}-\frac {b^9}{2 a^{10} (a+b x)^2}-\frac {10 b^9}{a^{11} (a+b x)}-\frac {55 b^9 \log (x)}{a^{12}}+\frac {55 b^9 \log (a+b x)}{a^{12}} \] Output:
-1/9/a^3/x^9+3/8*b/a^4/x^8-6/7*b^2/a^5/x^7+5/3*b^3/a^6/x^6-3*b^4/a^7/x^5+2 1/4*b^5/a^8/x^4-28/3*b^6/a^9/x^3+18*b^7/a^10/x^2-45*b^8/a^11/x-1/2*b^9/a^1 0/(b*x+a)^2-10*b^9/a^11/(b*x+a)-55*b^9*ln(x)/a^12+55*b^9*ln(b*x+a)/a^12
Time = 0.06 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=-\frac {\frac {a \left (56 a^{10}-77 a^9 b x+110 a^8 b^2 x^2-165 a^7 b^3 x^3+264 a^6 b^4 x^4-462 a^5 b^5 x^5+924 a^4 b^6 x^6-2310 a^3 b^7 x^7+9240 a^2 b^8 x^8+41580 a b^9 x^9+27720 b^{10} x^{10}\right )}{x^9 (a+b x)^2}+27720 b^9 \log (x)-27720 b^9 \log (a+b x)}{504 a^{12}} \] Input:
Integrate[1/(x^10*(a + b*x)^3),x]
Output:
-1/504*((a*(56*a^10 - 77*a^9*b*x + 110*a^8*b^2*x^2 - 165*a^7*b^3*x^3 + 264 *a^6*b^4*x^4 - 462*a^5*b^5*x^5 + 924*a^4*b^6*x^6 - 2310*a^3*b^7*x^7 + 9240 *a^2*b^8*x^8 + 41580*a*b^9*x^9 + 27720*b^10*x^10))/(x^9*(a + b*x)^2) + 277 20*b^9*Log[x] - 27720*b^9*Log[a + b*x])/a^12
Time = 0.28 (sec) , antiderivative size = 163, 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.
| \(\displaystyle \int \frac {1}{x^{10} (a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \int \left (\frac {55 b^{10}}{a^{12} (a+b x)}-\frac {55 b^9}{a^{12} x}+\frac {10 b^{10}}{a^{11} (a+b x)^2}+\frac {45 b^8}{a^{11} x^2}+\frac {b^{10}}{a^{10} (a+b x)^3}-\frac {36 b^7}{a^{10} x^3}+\frac {28 b^6}{a^9 x^4}-\frac {21 b^5}{a^8 x^5}+\frac {15 b^4}{a^7 x^6}-\frac {10 b^3}{a^6 x^7}+\frac {6 b^2}{a^5 x^8}-\frac {3 b}{a^4 x^9}+\frac {1}{a^3 x^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {55 b^9 \log (x)}{a^{12}}+\frac {55 b^9 \log (a+b x)}{a^{12}}-\frac {10 b^9}{a^{11} (a+b x)}-\frac {45 b^8}{a^{11} x}-\frac {b^9}{2 a^{10} (a+b x)^2}+\frac {18 b^7}{a^{10} x^2}-\frac {28 b^6}{3 a^9 x^3}+\frac {21 b^5}{4 a^8 x^4}-\frac {3 b^4}{a^7 x^5}+\frac {5 b^3}{3 a^6 x^6}-\frac {6 b^2}{7 a^5 x^7}+\frac {3 b}{8 a^4 x^8}-\frac {1}{9 a^3 x^9}\) |
Input:
Int[1/(x^10*(a + b*x)^3),x]
Output:
-1/9*1/(a^3*x^9) + (3*b)/(8*a^4*x^8) - (6*b^2)/(7*a^5*x^7) + (5*b^3)/(3*a^ 6*x^6) - (3*b^4)/(a^7*x^5) + (21*b^5)/(4*a^8*x^4) - (28*b^6)/(3*a^9*x^3) + (18*b^7)/(a^10*x^2) - (45*b^8)/(a^11*x) - b^9/(2*a^10*(a + b*x)^2) - (10* b^9)/(a^11*(a + b*x)) - (55*b^9*Log[x])/a^12 + (55*b^9*Log[a + b*x])/a^12
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 = 0.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.91
| method | result | size |
| norman | \(\frac {-\frac {1}{9 a}+\frac {11 b x}{72 a^{2}}-\frac {55 b^{2} x^{2}}{252 a^{3}}+\frac {55 b^{3} x^{3}}{168 a^{4}}-\frac {11 b^{4} x^{4}}{21 a^{5}}+\frac {11 b^{5} x^{5}}{12 a^{6}}-\frac {11 b^{6} x^{6}}{6 a^{7}}+\frac {55 b^{7} x^{7}}{12 a^{8}}-\frac {55 b^{8} x^{8}}{3 a^{9}}+\frac {110 b^{10} x^{10}}{a^{11}}+\frac {165 b^{11} x^{11}}{2 a^{12}}}{x^{9} \left (b x +a \right )^{2}}-\frac {55 b^{9} \ln \left (x \right )}{a^{12}}+\frac {55 b^{9} \ln \left (b x +a \right )}{a^{12}}\) | \(149\) |
| default | \(-\frac {1}{9 a^{3} x^{9}}+\frac {3 b}{8 a^{4} x^{8}}-\frac {6 b^{2}}{7 a^{5} x^{7}}+\frac {5 b^{3}}{3 a^{6} x^{6}}-\frac {3 b^{4}}{a^{7} x^{5}}+\frac {21 b^{5}}{4 a^{8} x^{4}}-\frac {28 b^{6}}{3 a^{9} x^{3}}+\frac {18 b^{7}}{a^{10} x^{2}}-\frac {45 b^{8}}{a^{11} x}-\frac {b^{9}}{2 a^{10} \left (b x +a \right )^{2}}-\frac {10 b^{9}}{a^{11} \left (b x +a \right )}-\frac {55 b^{9} \ln \left (x \right )}{a^{12}}+\frac {55 b^{9} \ln \left (b x +a \right )}{a^{12}}\) | \(150\) |
| risch | \(\frac {-\frac {55 b^{10} x^{10}}{a^{11}}-\frac {165 b^{9} x^{9}}{2 a^{10}}-\frac {55 b^{8} x^{8}}{3 a^{9}}+\frac {55 b^{7} x^{7}}{12 a^{8}}-\frac {11 b^{6} x^{6}}{6 a^{7}}+\frac {11 b^{5} x^{5}}{12 a^{6}}-\frac {11 b^{4} x^{4}}{21 a^{5}}+\frac {55 b^{3} x^{3}}{168 a^{4}}-\frac {55 b^{2} x^{2}}{252 a^{3}}+\frac {11 b x}{72 a^{2}}-\frac {1}{9 a}}{x^{9} \left (b x +a \right )^{2}}-\frac {55 b^{9} \ln \left (x \right )}{a^{12}}+\frac {55 b^{9} \ln \left (-b x -a \right )}{a^{12}}\) | \(152\) |
| parallelrisch | \(-\frac {27720 \ln \left (x \right ) x^{11} b^{11}-27720 \ln \left (b x +a \right ) x^{11} b^{11}+55440 \ln \left (x \right ) x^{10} a \,b^{10}-55440 \ln \left (b x +a \right ) x^{10} a \,b^{10}-41580 b^{11} x^{11}+27720 \ln \left (x \right ) x^{9} a^{2} b^{9}-27720 \ln \left (b x +a \right ) x^{9} a^{2} b^{9}-55440 a \,x^{10} b^{10}+9240 a^{3} x^{8} b^{8}-2310 a^{4} b^{7} x^{7}+924 a^{5} b^{6} x^{6}-462 a^{6} b^{5} x^{5}+264 a^{7} b^{4} x^{4}-165 b^{3} a^{8} x^{3}+110 b^{2} a^{9} x^{2}-77 a^{10} b x +56 a^{11}}{504 a^{12} x^{9} \left (b x +a \right )^{2}}\) | \(203\) |
Input:
int(1/x^10/(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
(-1/9/a+11/72*b/a^2*x-55/252*b^2/a^3*x^2+55/168*b^3/a^4*x^3-11/21*b^4/a^5* x^4+11/12*b^5/a^6*x^5-11/6*b^6/a^7*x^6+55/12*b^7/a^8*x^7-55/3*b^8/a^9*x^8+ 110*b^10/a^11*x^10+165/2*b^11/a^12*x^11)/x^9/(b*x+a)^2-55*b^9*ln(x)/a^12+5 5*b^9*ln(b*x+a)/a^12
Time = 0.09 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=-\frac {27720 \, a b^{10} x^{10} + 41580 \, a^{2} b^{9} x^{9} + 9240 \, a^{3} b^{8} x^{8} - 2310 \, a^{4} b^{7} x^{7} + 924 \, a^{5} b^{6} x^{6} - 462 \, a^{6} b^{5} x^{5} + 264 \, a^{7} b^{4} x^{4} - 165 \, a^{8} b^{3} x^{3} + 110 \, a^{9} b^{2} x^{2} - 77 \, a^{10} b x + 56 \, a^{11} - 27720 \, {\left (b^{11} x^{11} + 2 \, a b^{10} x^{10} + a^{2} b^{9} x^{9}\right )} \log \left (b x + a\right ) + 27720 \, {\left (b^{11} x^{11} + 2 \, a b^{10} x^{10} + a^{2} b^{9} x^{9}\right )} \log \left (x\right )}{504 \, {\left (a^{12} b^{2} x^{11} + 2 \, a^{13} b x^{10} + a^{14} x^{9}\right )}} \] Input:
integrate(1/x^10/(b*x+a)^3,x, algorithm="fricas")
Output:
-1/504*(27720*a*b^10*x^10 + 41580*a^2*b^9*x^9 + 9240*a^3*b^8*x^8 - 2310*a^ 4*b^7*x^7 + 924*a^5*b^6*x^6 - 462*a^6*b^5*x^5 + 264*a^7*b^4*x^4 - 165*a^8* b^3*x^3 + 110*a^9*b^2*x^2 - 77*a^10*b*x + 56*a^11 - 27720*(b^11*x^11 + 2*a *b^10*x^10 + a^2*b^9*x^9)*log(b*x + a) + 27720*(b^11*x^11 + 2*a*b^10*x^10 + a^2*b^9*x^9)*log(x))/(a^12*b^2*x^11 + 2*a^13*b*x^10 + a^14*x^9)
Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=\frac {- 56 a^{10} + 77 a^{9} b x - 110 a^{8} b^{2} x^{2} + 165 a^{7} b^{3} x^{3} - 264 a^{6} b^{4} x^{4} + 462 a^{5} b^{5} x^{5} - 924 a^{4} b^{6} x^{6} + 2310 a^{3} b^{7} x^{7} - 9240 a^{2} b^{8} x^{8} - 41580 a b^{9} x^{9} - 27720 b^{10} x^{10}}{504 a^{13} x^{9} + 1008 a^{12} b x^{10} + 504 a^{11} b^{2} x^{11}} + \frac {55 b^{9} \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{12}} \] Input:
integrate(1/x**10/(b*x+a)**3,x)
Output:
(-56*a**10 + 77*a**9*b*x - 110*a**8*b**2*x**2 + 165*a**7*b**3*x**3 - 264*a **6*b**4*x**4 + 462*a**5*b**5*x**5 - 924*a**4*b**6*x**6 + 2310*a**3*b**7*x **7 - 9240*a**2*b**8*x**8 - 41580*a*b**9*x**9 - 27720*b**10*x**10)/(504*a* *13*x**9 + 1008*a**12*b*x**10 + 504*a**11*b**2*x**11) + 55*b**9*(-log(x) + log(a/b + x))/a**12
Time = 0.03 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=-\frac {27720 \, b^{10} x^{10} + 41580 \, a b^{9} x^{9} + 9240 \, a^{2} b^{8} x^{8} - 2310 \, a^{3} b^{7} x^{7} + 924 \, a^{4} b^{6} x^{6} - 462 \, a^{5} b^{5} x^{5} + 264 \, a^{6} b^{4} x^{4} - 165 \, a^{7} b^{3} x^{3} + 110 \, a^{8} b^{2} x^{2} - 77 \, a^{9} b x + 56 \, a^{10}}{504 \, {\left (a^{11} b^{2} x^{11} + 2 \, a^{12} b x^{10} + a^{13} x^{9}\right )}} + \frac {55 \, b^{9} \log \left (b x + a\right )}{a^{12}} - \frac {55 \, b^{9} \log \left (x\right )}{a^{12}} \] Input:
integrate(1/x^10/(b*x+a)^3,x, algorithm="maxima")
Output:
-1/504*(27720*b^10*x^10 + 41580*a*b^9*x^9 + 9240*a^2*b^8*x^8 - 2310*a^3*b^ 7*x^7 + 924*a^4*b^6*x^6 - 462*a^5*b^5*x^5 + 264*a^6*b^4*x^4 - 165*a^7*b^3* x^3 + 110*a^8*b^2*x^2 - 77*a^9*b*x + 56*a^10)/(a^11*b^2*x^11 + 2*a^12*b*x^ 10 + a^13*x^9) + 55*b^9*log(b*x + a)/a^12 - 55*b^9*log(x)/a^12
Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=\frac {55 \, b^{9} \log \left ({\left | b x + a \right |}\right )}{a^{12}} - \frac {55 \, b^{9} \log \left ({\left | x \right |}\right )}{a^{12}} - \frac {27720 \, a b^{10} x^{10} + 41580 \, a^{2} b^{9} x^{9} + 9240 \, a^{3} b^{8} x^{8} - 2310 \, a^{4} b^{7} x^{7} + 924 \, a^{5} b^{6} x^{6} - 462 \, a^{6} b^{5} x^{5} + 264 \, a^{7} b^{4} x^{4} - 165 \, a^{8} b^{3} x^{3} + 110 \, a^{9} b^{2} x^{2} - 77 \, a^{10} b x + 56 \, a^{11}}{504 \, {\left (b x + a\right )}^{2} a^{12} x^{9}} \] Input:
integrate(1/x^10/(b*x+a)^3,x, algorithm="giac")
Output:
55*b^9*log(abs(b*x + a))/a^12 - 55*b^9*log(abs(x))/a^12 - 1/504*(27720*a*b ^10*x^10 + 41580*a^2*b^9*x^9 + 9240*a^3*b^8*x^8 - 2310*a^4*b^7*x^7 + 924*a ^5*b^6*x^6 - 462*a^6*b^5*x^5 + 264*a^7*b^4*x^4 - 165*a^8*b^3*x^3 + 110*a^9 *b^2*x^2 - 77*a^10*b*x + 56*a^11)/((b*x + a)^2*a^12*x^9)
Time = 0.15 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=\frac {110\,b^9\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{12}}-\frac {\frac {1}{9\,a}+\frac {55\,b^2\,x^2}{252\,a^3}-\frac {55\,b^3\,x^3}{168\,a^4}+\frac {11\,b^4\,x^4}{21\,a^5}-\frac {11\,b^5\,x^5}{12\,a^6}+\frac {11\,b^6\,x^6}{6\,a^7}-\frac {55\,b^7\,x^7}{12\,a^8}+\frac {55\,b^8\,x^8}{3\,a^9}+\frac {165\,b^9\,x^9}{2\,a^{10}}+\frac {55\,b^{10}\,x^{10}}{a^{11}}-\frac {11\,b\,x}{72\,a^2}}{a^2\,x^9+2\,a\,b\,x^{10}+b^2\,x^{11}} \] Input:
int(1/(x^10*(a + b*x)^3),x)
Output:
(110*b^9*atanh((2*b*x)/a + 1))/a^12 - (1/(9*a) + (55*b^2*x^2)/(252*a^3) - (55*b^3*x^3)/(168*a^4) + (11*b^4*x^4)/(21*a^5) - (11*b^5*x^5)/(12*a^6) + ( 11*b^6*x^6)/(6*a^7) - (55*b^7*x^7)/(12*a^8) + (55*b^8*x^8)/(3*a^9) + (165* b^9*x^9)/(2*a^10) + (55*b^10*x^10)/a^11 - (11*b*x)/(72*a^2))/(a^2*x^9 + b^ 2*x^11 + 2*a*b*x^10)
Time = 0.16 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=\frac {27720 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{9} x^{9}+55440 \,\mathrm {log}\left (b x +a \right ) a \,b^{10} x^{10}+27720 \,\mathrm {log}\left (b x +a \right ) b^{11} x^{11}-27720 \,\mathrm {log}\left (x \right ) a^{2} b^{9} x^{9}-55440 \,\mathrm {log}\left (x \right ) a \,b^{10} x^{10}-27720 \,\mathrm {log}\left (x \right ) b^{11} x^{11}-56 a^{11}+77 a^{10} b x -110 a^{9} b^{2} x^{2}+165 a^{8} b^{3} x^{3}-264 a^{7} b^{4} x^{4}+462 a^{6} b^{5} x^{5}-924 a^{5} b^{6} x^{6}+2310 a^{4} b^{7} x^{7}-9240 a^{3} b^{8} x^{8}-27720 a^{2} b^{9} x^{9}+13860 b^{11} x^{11}}{504 a^{12} x^{9} \left (b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:
int(1/x^10/(b*x+a)^3,x)
Output:
(27720*log(a + b*x)*a**2*b**9*x**9 + 55440*log(a + b*x)*a*b**10*x**10 + 27 720*log(a + b*x)*b**11*x**11 - 27720*log(x)*a**2*b**9*x**9 - 55440*log(x)* a*b**10*x**10 - 27720*log(x)*b**11*x**11 - 56*a**11 + 77*a**10*b*x - 110*a **9*b**2*x**2 + 165*a**8*b**3*x**3 - 264*a**7*b**4*x**4 + 462*a**6*b**5*x* *5 - 924*a**5*b**6*x**6 + 2310*a**4*b**7*x**7 - 9240*a**3*b**8*x**8 - 2772 0*a**2*b**9*x**9 + 13860*b**11*x**11)/(504*a**12*x**9*(a**2 + 2*a*b*x + b* *2*x**2))