Integrand size = 11, antiderivative size = 117 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=-\frac {1}{4 a^4 x^4}+\frac {4 b}{3 a^5 x^3}-\frac {5 b^2}{a^6 x^2}+\frac {20 b^3}{a^7 x}+\frac {b^4}{3 a^5 (a+b x)^3}+\frac {5 b^4}{2 a^6 (a+b x)^2}+\frac {15 b^4}{a^7 (a+b x)}+\frac {35 b^4 \log (x)}{a^8}-\frac {35 b^4 \log (a+b x)}{a^8} \]
-1/4/a^4/x^4+4/3*b/a^5/x^3-5*b^2/a^6/x^2+20*b^3/a^7/x+1/3*b^4/a^5/(b*x+a)^ 3+5/2*b^4/a^6/(b*x+a)^2+15*b^4/a^7/(b*x+a)+35*b^4*ln(x)/a^8-35*b^4*ln(b*x+ a)/a^8
Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {\frac {a \left (-3 a^6+7 a^5 b x-21 a^4 b^2 x^2+105 a^3 b^3 x^3+770 a^2 b^4 x^4+1050 a b^5 x^5+420 b^6 x^6\right )}{x^4 (a+b x)^3}+420 b^4 \log (x)-420 b^4 \log (a+b x)}{12 a^8} \]
((a*(-3*a^6 + 7*a^5*b*x - 21*a^4*b^2*x^2 + 105*a^3*b^3*x^3 + 770*a^2*b^4*x ^4 + 1050*a*b^5*x^5 + 420*b^6*x^6))/(x^4*(a + b*x)^3) + 420*b^4*Log[x] - 4 20*b^4*Log[a + b*x])/(12*a^8)
Time = 0.25 (sec) , antiderivative size = 117, 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.
\(\displaystyle \int \frac {1}{x^5 (a+b x)^4} \, dx\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \int \left (-\frac {35 b^5}{a^8 (a+b x)}+\frac {35 b^4}{a^8 x}-\frac {15 b^5}{a^7 (a+b x)^2}-\frac {20 b^3}{a^7 x^2}-\frac {5 b^5}{a^6 (a+b x)^3}+\frac {10 b^2}{a^6 x^3}-\frac {b^5}{a^5 (a+b x)^4}-\frac {4 b}{a^5 x^4}+\frac {1}{a^4 x^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {35 b^4 \log (x)}{a^8}-\frac {35 b^4 \log (a+b x)}{a^8}+\frac {15 b^4}{a^7 (a+b x)}+\frac {20 b^3}{a^7 x}+\frac {5 b^4}{2 a^6 (a+b x)^2}-\frac {5 b^2}{a^6 x^2}+\frac {b^4}{3 a^5 (a+b x)^3}+\frac {4 b}{3 a^5 x^3}-\frac {1}{4 a^4 x^4}\) |
-1/4*1/(a^4*x^4) + (4*b)/(3*a^5*x^3) - (5*b^2)/(a^6*x^2) + (20*b^3)/(a^7*x ) + b^4/(3*a^5*(a + b*x)^3) + (5*b^4)/(2*a^6*(a + b*x)^2) + (15*b^4)/(a^7* (a + b*x)) + (35*b^4*Log[x])/a^8 - (35*b^4*Log[a + b*x])/a^8
3.3.6.3.1 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 = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.90
method | result | size |
norman | \(\frac {-\frac {1}{4 a}+\frac {7 b x}{12 a^{2}}-\frac {7 b^{2} x^{2}}{4 a^{3}}+\frac {35 b^{3} x^{3}}{4 a^{4}}-\frac {105 b^{5} x^{5}}{a^{6}}-\frac {315 b^{6} x^{6}}{2 a^{7}}-\frac {385 b^{7} x^{7}}{6 a^{8}}}{x^{4} \left (b x +a \right )^{3}}+\frac {35 b^{4} \ln \left (x \right )}{a^{8}}-\frac {35 b^{4} \ln \left (b x +a \right )}{a^{8}}\) | \(105\) |
risch | \(\frac {\frac {35 b^{6} x^{6}}{a^{7}}+\frac {175 b^{5} x^{5}}{2 a^{6}}+\frac {385 b^{4} x^{4}}{6 a^{5}}+\frac {35 b^{3} x^{3}}{4 a^{4}}-\frac {7 b^{2} x^{2}}{4 a^{3}}+\frac {7 b x}{12 a^{2}}-\frac {1}{4 a}}{x^{4} \left (b x +a \right )^{3}}-\frac {35 b^{4} \ln \left (b x +a \right )}{a^{8}}+\frac {35 b^{4} \ln \left (-x \right )}{a^{8}}\) | \(107\) |
default | \(-\frac {1}{4 a^{4} x^{4}}+\frac {4 b}{3 a^{5} x^{3}}-\frac {5 b^{2}}{a^{6} x^{2}}+\frac {20 b^{3}}{a^{7} x}+\frac {b^{4}}{3 a^{5} \left (b x +a \right )^{3}}+\frac {5 b^{4}}{2 a^{6} \left (b x +a \right )^{2}}+\frac {15 b^{4}}{a^{7} \left (b x +a \right )}+\frac {35 b^{4} \ln \left (x \right )}{a^{8}}-\frac {35 b^{4} \ln \left (b x +a \right )}{a^{8}}\) | \(110\) |
parallelrisch | \(\frac {420 b^{7} \ln \left (x \right ) x^{7}-420 \ln \left (b x +a \right ) x^{7} b^{7}+1260 a \,b^{6} \ln \left (x \right ) x^{6}-1260 \ln \left (b x +a \right ) x^{6} a \,b^{6}-770 b^{7} x^{7}+1260 a^{2} b^{5} \ln \left (x \right ) x^{5}-1260 \ln \left (b x +a \right ) x^{5} a^{2} b^{5}-1890 a \,b^{6} x^{6}+420 a^{3} b^{4} \ln \left (x \right ) x^{4}-420 \ln \left (b x +a \right ) x^{4} a^{3} b^{4}-1260 a^{2} b^{5} x^{5}+105 a^{4} b^{3} x^{3}-21 a^{5} b^{2} x^{2}+7 a^{6} b x -3 a^{7}}{12 a^{8} x^{4} \left (b x +a \right )^{3}}\) | \(189\) |
(-1/4/a+7/12*b/a^2*x-7/4*b^2/a^3*x^2+35/4*b^3/a^4*x^3-105*b^5/a^6*x^5-315/ 2*b^6/a^7*x^6-385/6*b^7/a^8*x^7)/x^4/(b*x+a)^3+35*b^4*ln(x)/a^8-35*b^4*ln( b*x+a)/a^8
Time = 0.22 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.68 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {420 \, a b^{6} x^{6} + 1050 \, a^{2} b^{5} x^{5} + 770 \, a^{3} b^{4} x^{4} + 105 \, a^{4} b^{3} x^{3} - 21 \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x - 3 \, a^{7} - 420 \, {\left (b^{7} x^{7} + 3 \, a b^{6} x^{6} + 3 \, a^{2} b^{5} x^{5} + a^{3} b^{4} x^{4}\right )} \log \left (b x + a\right ) + 420 \, {\left (b^{7} x^{7} + 3 \, a b^{6} x^{6} + 3 \, a^{2} b^{5} x^{5} + a^{3} b^{4} x^{4}\right )} \log \left (x\right )}{12 \, {\left (a^{8} b^{3} x^{7} + 3 \, a^{9} b^{2} x^{6} + 3 \, a^{10} b x^{5} + a^{11} x^{4}\right )}} \]
1/12*(420*a*b^6*x^6 + 1050*a^2*b^5*x^5 + 770*a^3*b^4*x^4 + 105*a^4*b^3*x^3 - 21*a^5*b^2*x^2 + 7*a^6*b*x - 3*a^7 - 420*(b^7*x^7 + 3*a*b^6*x^6 + 3*a^2 *b^5*x^5 + a^3*b^4*x^4)*log(b*x + a) + 420*(b^7*x^7 + 3*a*b^6*x^6 + 3*a^2* b^5*x^5 + a^3*b^4*x^4)*log(x))/(a^8*b^3*x^7 + 3*a^9*b^2*x^6 + 3*a^10*b*x^5 + a^11*x^4)
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {- 3 a^{6} + 7 a^{5} b x - 21 a^{4} b^{2} x^{2} + 105 a^{3} b^{3} x^{3} + 770 a^{2} b^{4} x^{4} + 1050 a b^{5} x^{5} + 420 b^{6} x^{6}}{12 a^{10} x^{4} + 36 a^{9} b x^{5} + 36 a^{8} b^{2} x^{6} + 12 a^{7} b^{3} x^{7}} + \frac {35 b^{4} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{8}} \]
(-3*a**6 + 7*a**5*b*x - 21*a**4*b**2*x**2 + 105*a**3*b**3*x**3 + 770*a**2* b**4*x**4 + 1050*a*b**5*x**5 + 420*b**6*x**6)/(12*a**10*x**4 + 36*a**9*b*x **5 + 36*a**8*b**2*x**6 + 12*a**7*b**3*x**7) + 35*b**4*(log(x) - log(a/b + x))/a**8
Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {420 \, b^{6} x^{6} + 1050 \, a b^{5} x^{5} + 770 \, a^{2} b^{4} x^{4} + 105 \, a^{3} b^{3} x^{3} - 21 \, a^{4} b^{2} x^{2} + 7 \, a^{5} b x - 3 \, a^{6}}{12 \, {\left (a^{7} b^{3} x^{7} + 3 \, a^{8} b^{2} x^{6} + 3 \, a^{9} b x^{5} + a^{10} x^{4}\right )}} - \frac {35 \, b^{4} \log \left (b x + a\right )}{a^{8}} + \frac {35 \, b^{4} \log \left (x\right )}{a^{8}} \]
1/12*(420*b^6*x^6 + 1050*a*b^5*x^5 + 770*a^2*b^4*x^4 + 105*a^3*b^3*x^3 - 2 1*a^4*b^2*x^2 + 7*a^5*b*x - 3*a^6)/(a^7*b^3*x^7 + 3*a^8*b^2*x^6 + 3*a^9*b* x^5 + a^10*x^4) - 35*b^4*log(b*x + a)/a^8 + 35*b^4*log(x)/a^8
Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=-\frac {35 \, b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{8}} + \frac {35 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac {420 \, a b^{6} x^{6} + 1050 \, a^{2} b^{5} x^{5} + 770 \, a^{3} b^{4} x^{4} + 105 \, a^{4} b^{3} x^{3} - 21 \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x - 3 \, a^{7}}{12 \, {\left (b x + a\right )}^{3} a^{8} x^{4}} \]
-35*b^4*log(abs(b*x + a))/a^8 + 35*b^4*log(abs(x))/a^8 + 1/12*(420*a*b^6*x ^6 + 1050*a^2*b^5*x^5 + 770*a^3*b^4*x^4 + 105*a^4*b^3*x^3 - 21*a^5*b^2*x^2 + 7*a^6*b*x - 3*a^7)/((b*x + a)^3*a^8*x^4)
Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {\frac {35\,b^3\,x^3}{4\,a^4}-\frac {7\,b^2\,x^2}{4\,a^3}-\frac {1}{4\,a}+\frac {385\,b^4\,x^4}{6\,a^5}+\frac {175\,b^5\,x^5}{2\,a^6}+\frac {35\,b^6\,x^6}{a^7}+\frac {7\,b\,x}{12\,a^2}}{a^3\,x^4+3\,a^2\,b\,x^5+3\,a\,b^2\,x^6+b^3\,x^7}-\frac {70\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^8} \]