Integrand size = 11, antiderivative size = 118 \[ \int \frac {x^7}{(a+b x)^7} \, dx=\frac {x}{b^7}+\frac {a^7}{6 b^8 (a+b x)^6}-\frac {7 a^6}{5 b^8 (a+b x)^5}+\frac {21 a^5}{4 b^8 (a+b x)^4}-\frac {35 a^4}{3 b^8 (a+b x)^3}+\frac {35 a^3}{2 b^8 (a+b x)^2}-\frac {21 a^2}{b^8 (a+b x)}-\frac {7 a \log (a+b x)}{b^8} \]
x/b^7+1/6*a^7/b^8/(b*x+a)^6-7/5*a^6/b^8/(b*x+a)^5+21/4*a^5/b^8/(b*x+a)^4-3 5/3*a^4/b^8/(b*x+a)^3+35/2*a^3/b^8/(b*x+a)^2-21*a^2/b^8/(b*x+a)-7*a*ln(b*x +a)/b^8
Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.88 \[ \int \frac {x^7}{(a+b x)^7} \, dx=-\frac {669 a^7+3594 a^6 b x+7725 a^5 b^2 x^2+8200 a^4 b^3 x^3+4050 a^3 b^4 x^4+360 a^2 b^5 x^5-360 a b^6 x^6-60 b^7 x^7+420 a (a+b x)^6 \log (a+b x)}{60 b^8 (a+b x)^6} \]
-1/60*(669*a^7 + 3594*a^6*b*x + 7725*a^5*b^2*x^2 + 8200*a^4*b^3*x^3 + 4050 *a^3*b^4*x^4 + 360*a^2*b^5*x^5 - 360*a*b^6*x^6 - 60*b^7*x^7 + 420*a*(a + b *x)^6*Log[a + b*x])/(b^8*(a + b*x)^6)
Time = 0.26 (sec) , antiderivative size = 118, 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.
\(\displaystyle \int \frac {x^7}{(a+b x)^7} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (-\frac {a^7}{b^7 (a+b x)^7}+\frac {7 a^6}{b^7 (a+b x)^6}-\frac {21 a^5}{b^7 (a+b x)^5}+\frac {35 a^4}{b^7 (a+b x)^4}-\frac {35 a^3}{b^7 (a+b x)^3}+\frac {21 a^2}{b^7 (a+b x)^2}-\frac {7 a}{b^7 (a+b x)}+\frac {1}{b^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^7}{6 b^8 (a+b x)^6}-\frac {7 a^6}{5 b^8 (a+b x)^5}+\frac {21 a^5}{4 b^8 (a+b x)^4}-\frac {35 a^4}{3 b^8 (a+b x)^3}+\frac {35 a^3}{2 b^8 (a+b x)^2}-\frac {21 a^2}{b^8 (a+b x)}-\frac {7 a \log (a+b x)}{b^8}+\frac {x}{b^7}\) |
x/b^7 + a^7/(6*b^8*(a + b*x)^6) - (7*a^6)/(5*b^8*(a + b*x)^5) + (21*a^5)/( 4*b^8*(a + b*x)^4) - (35*a^4)/(3*b^8*(a + b*x)^3) + (35*a^3)/(2*b^8*(a + b *x)^2) - (21*a^2)/(b^8*(a + b*x)) - (7*a*Log[a + b*x])/b^8
3.3.10.3.1 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 = 87, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {x}{b^{7}}+\frac {-21 a^{2} b^{4} x^{5}-\frac {175 a^{3} b^{3} x^{4}}{2}-\frac {455 a^{4} b^{2} x^{3}}{3}-\frac {539 a^{5} b \,x^{2}}{4}-\frac {609 a^{6} x}{10}-\frac {223 a^{7}}{20 b}}{b^{7} \left (b x +a \right )^{6}}-\frac {7 a \ln \left (b x +a \right )}{b^{8}}\) | \(87\) |
norman | \(\frac {\frac {x^{7}}{b}-\frac {343 a^{7}}{20 b^{8}}-\frac {42 a^{2} x^{5}}{b^{3}}-\frac {315 a^{3} x^{4}}{2 b^{4}}-\frac {770 a^{4} x^{3}}{3 b^{5}}-\frac {875 a^{5} x^{2}}{4 b^{6}}-\frac {959 a^{6} x}{10 b^{7}}}{\left (b x +a \right )^{6}}-\frac {7 a \ln \left (b x +a \right )}{b^{8}}\) | \(91\) |
default | \(\frac {x}{b^{7}}+\frac {a^{7}}{6 b^{8} \left (b x +a \right )^{6}}-\frac {7 a^{6}}{5 b^{8} \left (b x +a \right )^{5}}+\frac {21 a^{5}}{4 b^{8} \left (b x +a \right )^{4}}-\frac {35 a^{4}}{3 b^{8} \left (b x +a \right )^{3}}+\frac {35 a^{3}}{2 b^{8} \left (b x +a \right )^{2}}-\frac {21 a^{2}}{b^{8} \left (b x +a \right )}-\frac {7 a \ln \left (b x +a \right )}{b^{8}}\) | \(109\) |
parallelrisch | \(-\frac {420 \ln \left (b x +a \right ) x^{6} a \,b^{6}-60 b^{7} x^{7}+2520 \ln \left (b x +a \right ) x^{5} a^{2} b^{5}+6300 \ln \left (b x +a \right ) x^{4} a^{3} b^{4}+2520 a^{2} b^{5} x^{5}+8400 \ln \left (b x +a \right ) x^{3} a^{4} b^{3}+9450 a^{3} b^{4} x^{4}+6300 \ln \left (b x +a \right ) x^{2} a^{5} b^{2}+15400 a^{4} b^{3} x^{3}+2520 \ln \left (b x +a \right ) x \,a^{6} b +13125 a^{5} b^{2} x^{2}+420 \ln \left (b x +a \right ) a^{7}+5754 a^{6} b x +1029 a^{7}}{60 b^{8} \left (b x +a \right )^{6}}\) | \(185\) |
x/b^7+(-21*a^2*b^4*x^5-175/2*a^3*b^3*x^4-455/3*a^4*b^2*x^3-539/4*a^5*b*x^2 -609/10*a^6*x-223/20*a^7/b)/b^7/(b*x+a)^6-7*a*ln(b*x+a)/b^8
Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.82 \[ \int \frac {x^7}{(a+b x)^7} \, dx=\frac {60 \, b^{7} x^{7} + 360 \, a b^{6} x^{6} - 360 \, a^{2} b^{5} x^{5} - 4050 \, a^{3} b^{4} x^{4} - 8200 \, a^{4} b^{3} x^{3} - 7725 \, a^{5} b^{2} x^{2} - 3594 \, a^{6} b x - 669 \, a^{7} - 420 \, {\left (a b^{6} x^{6} + 6 \, a^{2} b^{5} x^{5} + 15 \, a^{3} b^{4} x^{4} + 20 \, a^{4} b^{3} x^{3} + 15 \, a^{5} b^{2} x^{2} + 6 \, a^{6} b x + a^{7}\right )} \log \left (b x + a\right )}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} \]
1/60*(60*b^7*x^7 + 360*a*b^6*x^6 - 360*a^2*b^5*x^5 - 4050*a^3*b^4*x^4 - 82 00*a^4*b^3*x^3 - 7725*a^5*b^2*x^2 - 3594*a^6*b*x - 669*a^7 - 420*(a*b^6*x^ 6 + 6*a^2*b^5*x^5 + 15*a^3*b^4*x^4 + 20*a^4*b^3*x^3 + 15*a^5*b^2*x^2 + 6*a ^6*b*x + a^7)*log(b*x + a))/(b^14*x^6 + 6*a*b^13*x^5 + 15*a^2*b^12*x^4 + 2 0*a^3*b^11*x^3 + 15*a^4*b^10*x^2 + 6*a^5*b^9*x + a^6*b^8)
Time = 0.36 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.30 \[ \int \frac {x^7}{(a+b x)^7} \, dx=- \frac {7 a \log {\left (a + b x \right )}}{b^{8}} + \frac {- 669 a^{7} - 3654 a^{6} b x - 8085 a^{5} b^{2} x^{2} - 9100 a^{4} b^{3} x^{3} - 5250 a^{3} b^{4} x^{4} - 1260 a^{2} b^{5} x^{5}}{60 a^{6} b^{8} + 360 a^{5} b^{9} x + 900 a^{4} b^{10} x^{2} + 1200 a^{3} b^{11} x^{3} + 900 a^{2} b^{12} x^{4} + 360 a b^{13} x^{5} + 60 b^{14} x^{6}} + \frac {x}{b^{7}} \]
-7*a*log(a + b*x)/b**8 + (-669*a**7 - 3654*a**6*b*x - 8085*a**5*b**2*x**2 - 9100*a**4*b**3*x**3 - 5250*a**3*b**4*x**4 - 1260*a**2*b**5*x**5)/(60*a** 6*b**8 + 360*a**5*b**9*x + 900*a**4*b**10*x**2 + 1200*a**3*b**11*x**3 + 90 0*a**2*b**12*x**4 + 360*a*b**13*x**5 + 60*b**14*x**6) + x/b**7
Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.23 \[ \int \frac {x^7}{(a+b x)^7} \, dx=-\frac {1260 \, a^{2} b^{5} x^{5} + 5250 \, a^{3} b^{4} x^{4} + 9100 \, a^{4} b^{3} x^{3} + 8085 \, a^{5} b^{2} x^{2} + 3654 \, a^{6} b x + 669 \, a^{7}}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} + \frac {x}{b^{7}} - \frac {7 \, a \log \left (b x + a\right )}{b^{8}} \]
-1/60*(1260*a^2*b^5*x^5 + 5250*a^3*b^4*x^4 + 9100*a^4*b^3*x^3 + 8085*a^5*b ^2*x^2 + 3654*a^6*b*x + 669*a^7)/(b^14*x^6 + 6*a*b^13*x^5 + 15*a^2*b^12*x^ 4 + 20*a^3*b^11*x^3 + 15*a^4*b^10*x^2 + 6*a^5*b^9*x + a^6*b^8) + x/b^7 - 7 *a*log(b*x + a)/b^8
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75 \[ \int \frac {x^7}{(a+b x)^7} \, dx=\frac {x}{b^{7}} - \frac {7 \, a \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {1260 \, a^{2} b^{5} x^{5} + 5250 \, a^{3} b^{4} x^{4} + 9100 \, a^{4} b^{3} x^{3} + 8085 \, a^{5} b^{2} x^{2} + 3654 \, a^{6} b x + 669 \, a^{7}}{60 \, {\left (b x + a\right )}^{6} b^{8}} \]
x/b^7 - 7*a*log(abs(b*x + a))/b^8 - 1/60*(1260*a^2*b^5*x^5 + 5250*a^3*b^4* x^4 + 9100*a^4*b^3*x^3 + 8085*a^5*b^2*x^2 + 3654*a^6*b*x + 669*a^7)/((b*x + a)^6*b^8)
Time = 0.45 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int \frac {x^7}{(a+b x)^7} \, dx=-\frac {7\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {21\,a^2}{a+b\,x}-\frac {35\,a^3}{2\,{\left (a+b\,x\right )}^2}+\frac {35\,a^4}{3\,{\left (a+b\,x\right )}^3}-\frac {21\,a^5}{4\,{\left (a+b\,x\right )}^4}+\frac {7\,a^6}{5\,{\left (a+b\,x\right )}^5}-\frac {a^7}{6\,{\left (a+b\,x\right )}^6}}{b^8} \]
-(7*a*log(a + b*x) - b*x + (21*a^2)/(a + b*x) - (35*a^3)/(2*(a + b*x)^2) + (35*a^4)/(3*(a + b*x)^3) - (21*a^5)/(4*(a + b*x)^4) + (7*a^6)/(5*(a + b*x )^5) - a^7/(6*(a + b*x)^6))/b^8
Time = 0.00 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.01 \[ \int \frac {x^7}{(a+b x)^7} \, dx=\frac {-420 \,\mathrm {log}\left (b x +a \right ) a^{7}-2520 \,\mathrm {log}\left (b x +a \right ) a^{6} b x -6300 \,\mathrm {log}\left (b x +a \right ) a^{5} b^{2} x^{2}-8400 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{3} x^{3}-6300 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{4} x^{4}-2520 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{5} x^{5}-420 \,\mathrm {log}\left (b x +a \right ) a \,b^{6} x^{6}-609 a^{7}-3234 a^{6} b x -6825 a^{5} b^{2} x^{2}-7000 a^{4} b^{3} x^{3}-3150 a^{3} b^{4} x^{4}+420 a \,b^{6} x^{6}+60 b^{7} x^{7}}{60 b^{8} \left (b^{6} x^{6}+6 a \,b^{5} x^{5}+15 a^{2} b^{4} x^{4}+20 a^{3} b^{3} x^{3}+15 a^{4} b^{2} x^{2}+6 a^{5} b x +a^{6}\right )} \]
int(x**7/(a**7 + 7*a**6*b*x + 21*a**5*b**2*x**2 + 35*a**4*b**3*x**3 + 35*a **3*b**4*x**4 + 21*a**2*b**5*x**5 + 7*a*b**6*x**6 + b**7*x**7),x)
( - 420*log(a + b*x)*a**7 - 2520*log(a + b*x)*a**6*b*x - 6300*log(a + b*x) *a**5*b**2*x**2 - 8400*log(a + b*x)*a**4*b**3*x**3 - 6300*log(a + b*x)*a** 3*b**4*x**4 - 2520*log(a + b*x)*a**2*b**5*x**5 - 420*log(a + b*x)*a*b**6*x **6 - 609*a**7 - 3234*a**6*b*x - 6825*a**5*b**2*x**2 - 7000*a**4*b**3*x**3 - 3150*a**3*b**4*x**4 + 420*a*b**6*x**6 + 60*b**7*x**7)/(60*b**8*(a**6 + 6*a**5*b*x + 15*a**4*b**2*x**2 + 20*a**3*b**3*x**3 + 15*a**2*b**4*x**4 + 6 *a*b**5*x**5 + b**6*x**6))