Integrand size = 11, antiderivative size = 144 \[ \int \frac {1}{x^3 (a+b x)^7} \, dx=-\frac {1}{2 a^7 x^2}+\frac {7 b}{a^8 x}+\frac {b^2}{6 a^3 (a+b x)^6}+\frac {3 b^2}{5 a^4 (a+b x)^5}+\frac {3 b^2}{2 a^5 (a+b x)^4}+\frac {10 b^2}{3 a^6 (a+b x)^3}+\frac {15 b^2}{2 a^7 (a+b x)^2}+\frac {21 b^2}{a^8 (a+b x)}+\frac {28 b^2 \log (x)}{a^9}-\frac {28 b^2 \log (a+b x)}{a^9} \]
-1/2/a^7/x^2+7*b/a^8/x+1/6*b^2/a^3/(b*x+a)^6+3/5*b^2/a^4/(b*x+a)^5+3/2*b^2 /a^5/(b*x+a)^4+10/3*b^2/a^6/(b*x+a)^3+15/2*b^2/a^7/(b*x+a)^2+21*b^2/a^8/(b *x+a)+28*b^2*ln(x)/a^9-28*b^2*ln(b*x+a)/a^9
Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 (a+b x)^7} \, dx=\frac {\frac {a \left (-15 a^7+120 a^6 b x+2058 a^5 b^2 x^2+7308 a^4 b^3 x^3+11970 a^3 b^4 x^4+10360 a^2 b^5 x^5+4620 a b^6 x^6+840 b^7 x^7\right )}{x^2 (a+b x)^6}+840 b^2 \log (x)-840 b^2 \log (a+b x)}{30 a^9} \]
((a*(-15*a^7 + 120*a^6*b*x + 2058*a^5*b^2*x^2 + 7308*a^4*b^3*x^3 + 11970*a ^3*b^4*x^4 + 10360*a^2*b^5*x^5 + 4620*a*b^6*x^6 + 840*b^7*x^7))/(x^2*(a + b*x)^6) + 840*b^2*Log[x] - 840*b^2*Log[a + b*x])/(30*a^9)
Time = 0.29 (sec) , antiderivative size = 144, 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.
\(\displaystyle \int \frac {1}{x^3 (a+b x)^7} \, dx\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \int \left (-\frac {28 b^3}{a^9 (a+b x)}+\frac {28 b^2}{a^9 x}-\frac {21 b^3}{a^8 (a+b x)^2}-\frac {7 b}{a^8 x^2}-\frac {15 b^3}{a^7 (a+b x)^3}+\frac {1}{a^7 x^3}-\frac {10 b^3}{a^6 (a+b x)^4}-\frac {6 b^3}{a^5 (a+b x)^5}-\frac {3 b^3}{a^4 (a+b x)^6}-\frac {b^3}{a^3 (a+b x)^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {28 b^2 \log (x)}{a^9}-\frac {28 b^2 \log (a+b x)}{a^9}+\frac {21 b^2}{a^8 (a+b x)}+\frac {7 b}{a^8 x}+\frac {15 b^2}{2 a^7 (a+b x)^2}-\frac {1}{2 a^7 x^2}+\frac {10 b^2}{3 a^6 (a+b x)^3}+\frac {3 b^2}{2 a^5 (a+b x)^4}+\frac {3 b^2}{5 a^4 (a+b x)^5}+\frac {b^2}{6 a^3 (a+b x)^6}\) |
-1/2*1/(a^7*x^2) + (7*b)/(a^8*x) + b^2/(6*a^3*(a + b*x)^6) + (3*b^2)/(5*a^ 4*(a + b*x)^5) + (3*b^2)/(2*a^5*(a + b*x)^4) + (10*b^2)/(3*a^6*(a + b*x)^3 ) + (15*b^2)/(2*a^7*(a + b*x)^2) + (21*b^2)/(a^8*(a + b*x)) + (28*b^2*Log[ x])/a^9 - (28*b^2*Log[a + b*x])/a^9
3.3.20.3.1 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 = 0.05 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.81
method | result | size |
norman | \(\frac {-\frac {1}{2 a}+\frac {4 b x}{a^{2}}-\frac {168 b^{3} x^{3}}{a^{4}}-\frac {630 b^{4} x^{4}}{a^{5}}-\frac {3080 b^{5} x^{5}}{3 a^{6}}-\frac {875 b^{6} x^{6}}{a^{7}}-\frac {1918 b^{7} x^{7}}{5 a^{8}}-\frac {343 b^{8} x^{8}}{5 a^{9}}}{x^{2} \left (b x +a \right )^{6}}+\frac {28 b^{2} \ln \left (x \right )}{a^{9}}-\frac {28 b^{2} \ln \left (b x +a \right )}{a^{9}}\) | \(116\) |
risch | \(\frac {\frac {28 b^{7} x^{7}}{a^{8}}+\frac {154 b^{6} x^{6}}{a^{7}}+\frac {1036 b^{5} x^{5}}{3 a^{6}}+\frac {399 b^{4} x^{4}}{a^{5}}+\frac {1218 b^{3} x^{3}}{5 a^{4}}+\frac {343 b^{2} x^{2}}{5 a^{3}}+\frac {4 b x}{a^{2}}-\frac {1}{2 a}}{x^{2} \left (b x +a \right )^{6}}+\frac {28 b^{2} \ln \left (-x \right )}{a^{9}}-\frac {28 b^{2} \ln \left (b x +a \right )}{a^{9}}\) | \(118\) |
default | \(-\frac {1}{2 a^{7} x^{2}}+\frac {7 b}{a^{8} x}+\frac {b^{2}}{6 a^{3} \left (b x +a \right )^{6}}+\frac {3 b^{2}}{5 a^{4} \left (b x +a \right )^{5}}+\frac {3 b^{2}}{2 a^{5} \left (b x +a \right )^{4}}+\frac {10 b^{2}}{3 a^{6} \left (b x +a \right )^{3}}+\frac {15 b^{2}}{2 a^{7} \left (b x +a \right )^{2}}+\frac {21 b^{2}}{a^{8} \left (b x +a \right )}+\frac {28 b^{2} \ln \left (x \right )}{a^{9}}-\frac {28 b^{2} \ln \left (b x +a \right )}{a^{9}}\) | \(133\) |
parallelrisch | \(\frac {-15 a^{8}+5040 \ln \left (x \right ) x^{7} a \,b^{7}-5040 \ln \left (b x +a \right ) x^{7} a \,b^{7}+12600 \ln \left (x \right ) x^{6} a^{2} b^{6}+16800 \ln \left (x \right ) x^{5} a^{3} b^{5}+12600 \ln \left (x \right ) x^{4} a^{4} b^{4}+5040 \ln \left (x \right ) x^{3} a^{5} b^{3}+840 \ln \left (x \right ) x^{2} a^{6} b^{2}-12600 \ln \left (b x +a \right ) x^{4} a^{4} b^{4}-12600 \ln \left (b x +a \right ) x^{6} a^{2} b^{6}-16800 \ln \left (b x +a \right ) x^{5} a^{3} b^{5}-5040 \ln \left (b x +a \right ) x^{3} a^{5} b^{3}-840 \ln \left (b x +a \right ) x^{2} a^{6} b^{2}-2058 b^{8} x^{8}+120 a^{7} x b -5040 a^{5} b^{3} x^{3}-26250 a^{2} x^{6} b^{6}-30800 a^{3} x^{5} b^{5}-18900 a^{4} x^{4} b^{4}-11508 a \,x^{7} b^{7}+840 \ln \left (x \right ) x^{8} b^{8}-840 \ln \left (b x +a \right ) x^{8} b^{8}}{30 a^{9} x^{2} \left (b x +a \right )^{6}}\) | \(290\) |
(-1/2/a+4*b/a^2*x-168*b^3/a^4*x^3-630*b^4/a^5*x^4-3080/3*b^5/a^6*x^5-875*b ^6/a^7*x^6-1918/5*b^7/a^8*x^7-343/5*b^8/a^9*x^8)/x^2/(b*x+a)^6+28*b^2*ln(x )/a^9-28*b^2*ln(b*x+a)/a^9
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (132) = 264\).
Time = 0.23 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.12 \[ \int \frac {1}{x^3 (a+b x)^7} \, dx=\frac {840 \, a b^{7} x^{7} + 4620 \, a^{2} b^{6} x^{6} + 10360 \, a^{3} b^{5} x^{5} + 11970 \, a^{4} b^{4} x^{4} + 7308 \, a^{5} b^{3} x^{3} + 2058 \, a^{6} b^{2} x^{2} + 120 \, a^{7} b x - 15 \, a^{8} - 840 \, {\left (b^{8} x^{8} + 6 \, a b^{7} x^{7} + 15 \, a^{2} b^{6} x^{6} + 20 \, a^{3} b^{5} x^{5} + 15 \, a^{4} b^{4} x^{4} + 6 \, a^{5} b^{3} x^{3} + a^{6} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 840 \, {\left (b^{8} x^{8} + 6 \, a b^{7} x^{7} + 15 \, a^{2} b^{6} x^{6} + 20 \, a^{3} b^{5} x^{5} + 15 \, a^{4} b^{4} x^{4} + 6 \, a^{5} b^{3} x^{3} + a^{6} b^{2} x^{2}\right )} \log \left (x\right )}{30 \, {\left (a^{9} b^{6} x^{8} + 6 \, a^{10} b^{5} x^{7} + 15 \, a^{11} b^{4} x^{6} + 20 \, a^{12} b^{3} x^{5} + 15 \, a^{13} b^{2} x^{4} + 6 \, a^{14} b x^{3} + a^{15} x^{2}\right )}} \]
1/30*(840*a*b^7*x^7 + 4620*a^2*b^6*x^6 + 10360*a^3*b^5*x^5 + 11970*a^4*b^4 *x^4 + 7308*a^5*b^3*x^3 + 2058*a^6*b^2*x^2 + 120*a^7*b*x - 15*a^8 - 840*(b ^8*x^8 + 6*a*b^7*x^7 + 15*a^2*b^6*x^6 + 20*a^3*b^5*x^5 + 15*a^4*b^4*x^4 + 6*a^5*b^3*x^3 + a^6*b^2*x^2)*log(b*x + a) + 840*(b^8*x^8 + 6*a*b^7*x^7 + 1 5*a^2*b^6*x^6 + 20*a^3*b^5*x^5 + 15*a^4*b^4*x^4 + 6*a^5*b^3*x^3 + a^6*b^2* x^2)*log(x))/(a^9*b^6*x^8 + 6*a^10*b^5*x^7 + 15*a^11*b^4*x^6 + 20*a^12*b^3 *x^5 + 15*a^13*b^2*x^4 + 6*a^14*b*x^3 + a^15*x^2)
Time = 0.39 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^3 (a+b x)^7} \, dx=\frac {- 15 a^{7} + 120 a^{6} b x + 2058 a^{5} b^{2} x^{2} + 7308 a^{4} b^{3} x^{3} + 11970 a^{3} b^{4} x^{4} + 10360 a^{2} b^{5} x^{5} + 4620 a b^{6} x^{6} + 840 b^{7} x^{7}}{30 a^{14} x^{2} + 180 a^{13} b x^{3} + 450 a^{12} b^{2} x^{4} + 600 a^{11} b^{3} x^{5} + 450 a^{10} b^{4} x^{6} + 180 a^{9} b^{5} x^{7} + 30 a^{8} b^{6} x^{8}} + \frac {28 b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{9}} \]
(-15*a**7 + 120*a**6*b*x + 2058*a**5*b**2*x**2 + 7308*a**4*b**3*x**3 + 119 70*a**3*b**4*x**4 + 10360*a**2*b**5*x**5 + 4620*a*b**6*x**6 + 840*b**7*x** 7)/(30*a**14*x**2 + 180*a**13*b*x**3 + 450*a**12*b**2*x**4 + 600*a**11*b** 3*x**5 + 450*a**10*b**4*x**6 + 180*a**9*b**5*x**7 + 30*a**8*b**6*x**8) + 2 8*b**2*(log(x) - log(a/b + x))/a**9
Time = 0.26 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^3 (a+b x)^7} \, dx=\frac {840 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 10360 \, a^{2} b^{5} x^{5} + 11970 \, a^{3} b^{4} x^{4} + 7308 \, a^{4} b^{3} x^{3} + 2058 \, a^{5} b^{2} x^{2} + 120 \, a^{6} b x - 15 \, a^{7}}{30 \, {\left (a^{8} b^{6} x^{8} + 6 \, a^{9} b^{5} x^{7} + 15 \, a^{10} b^{4} x^{6} + 20 \, a^{11} b^{3} x^{5} + 15 \, a^{12} b^{2} x^{4} + 6 \, a^{13} b x^{3} + a^{14} x^{2}\right )}} - \frac {28 \, b^{2} \log \left (b x + a\right )}{a^{9}} + \frac {28 \, b^{2} \log \left (x\right )}{a^{9}} \]
1/30*(840*b^7*x^7 + 4620*a*b^6*x^6 + 10360*a^2*b^5*x^5 + 11970*a^3*b^4*x^4 + 7308*a^4*b^3*x^3 + 2058*a^5*b^2*x^2 + 120*a^6*b*x - 15*a^7)/(a^8*b^6*x^ 8 + 6*a^9*b^5*x^7 + 15*a^10*b^4*x^6 + 20*a^11*b^3*x^5 + 15*a^12*b^2*x^4 + 6*a^13*b*x^3 + a^14*x^2) - 28*b^2*log(b*x + a)/a^9 + 28*b^2*log(x)/a^9
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^3 (a+b x)^7} \, dx=-\frac {28 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{9}} + \frac {28 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{9}} + \frac {840 \, a b^{7} x^{7} + 4620 \, a^{2} b^{6} x^{6} + 10360 \, a^{3} b^{5} x^{5} + 11970 \, a^{4} b^{4} x^{4} + 7308 \, a^{5} b^{3} x^{3} + 2058 \, a^{6} b^{2} x^{2} + 120 \, a^{7} b x - 15 \, a^{8}}{30 \, {\left (b x + a\right )}^{6} a^{9} x^{2}} \]
-28*b^2*log(abs(b*x + a))/a^9 + 28*b^2*log(abs(x))/a^9 + 1/30*(840*a*b^7*x ^7 + 4620*a^2*b^6*x^6 + 10360*a^3*b^5*x^5 + 11970*a^4*b^4*x^4 + 7308*a^5*b ^3*x^3 + 2058*a^6*b^2*x^2 + 120*a^7*b*x - 15*a^8)/((b*x + a)^6*a^9*x^2)
Time = 0.22 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 (a+b x)^7} \, dx=\frac {\frac {343\,b^2\,x^2}{5\,a^3}-\frac {1}{2\,a}+\frac {1218\,b^3\,x^3}{5\,a^4}+\frac {399\,b^4\,x^4}{a^5}+\frac {1036\,b^5\,x^5}{3\,a^6}+\frac {154\,b^6\,x^6}{a^7}+\frac {28\,b^7\,x^7}{a^8}+\frac {4\,b\,x}{a^2}}{a^6\,x^2+6\,a^5\,b\,x^3+15\,a^4\,b^2\,x^4+20\,a^3\,b^3\,x^5+15\,a^2\,b^4\,x^6+6\,a\,b^5\,x^7+b^6\,x^8}-\frac {56\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^9} \]
((343*b^2*x^2)/(5*a^3) - 1/(2*a) + (1218*b^3*x^3)/(5*a^4) + (399*b^4*x^4)/ a^5 + (1036*b^5*x^5)/(3*a^6) + (154*b^6*x^6)/a^7 + (28*b^7*x^7)/a^8 + (4*b *x)/a^2)/(a^6*x^2 + b^6*x^8 + 6*a^5*b*x^3 + 6*a*b^5*x^7 + 15*a^4*b^2*x^4 + 20*a^3*b^3*x^5 + 15*a^2*b^4*x^6) - (56*b^2*atanh((2*b*x)/a + 1))/a^9