Integrand size = 11, antiderivative size = 93 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=-\frac {1}{2 a^4 x^2}+\frac {4 b}{a^5 x}+\frac {b^2}{3 a^3 (a+b x)^3}+\frac {3 b^2}{2 a^4 (a+b x)^2}+\frac {6 b^2}{a^5 (a+b x)}+\frac {10 b^2 \log (x)}{a^6}-\frac {10 b^2 \log (a+b x)}{a^6} \]
-1/2/a^4/x^2+4*b/a^5/x+1/3*b^2/a^3/(b*x+a)^3+3/2*b^2/a^4/(b*x+a)^2+6*b^2/a ^5/(b*x+a)+10*b^2*ln(x)/a^6-10*b^2*ln(b*x+a)/a^6
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {\frac {a \left (-3 a^4+15 a^3 b x+110 a^2 b^2 x^2+150 a b^3 x^3+60 b^4 x^4\right )}{x^2 (a+b x)^3}+60 b^2 \log (x)-60 b^2 \log (a+b x)}{6 a^6} \]
((a*(-3*a^4 + 15*a^3*b*x + 110*a^2*b^2*x^2 + 150*a*b^3*x^3 + 60*b^4*x^4))/ (x^2*(a + b*x)^3) + 60*b^2*Log[x] - 60*b^2*Log[a + b*x])/(6*a^6)
Time = 0.22 (sec) , antiderivative size = 93, 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.
\(\displaystyle \int \frac {1}{x^3 (a+b x)^4} \, dx\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \int \left (-\frac {10 b^3}{a^6 (a+b x)}+\frac {10 b^2}{a^6 x}-\frac {6 b^3}{a^5 (a+b x)^2}-\frac {4 b}{a^5 x^2}-\frac {3 b^3}{a^4 (a+b x)^3}+\frac {1}{a^4 x^3}-\frac {b^3}{a^3 (a+b x)^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {10 b^2 \log (x)}{a^6}-\frac {10 b^2 \log (a+b x)}{a^6}+\frac {6 b^2}{a^5 (a+b x)}+\frac {4 b}{a^5 x}+\frac {3 b^2}{2 a^4 (a+b x)^2}-\frac {1}{2 a^4 x^2}+\frac {b^2}{3 a^3 (a+b x)^3}\) |
-1/2*1/(a^4*x^2) + (4*b)/(a^5*x) + b^2/(3*a^3*(a + b*x)^3) + (3*b^2)/(2*a^ 4*(a + b*x)^2) + (6*b^2)/(a^5*(a + b*x)) + (10*b^2*Log[x])/a^6 - (10*b^2*L og[a + b*x])/a^6
3.3.4.3.1 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 = 83, normalized size of antiderivative = 0.89
method | result | size |
norman | \(\frac {-\frac {1}{2 a}+\frac {5 b x}{2 a^{2}}-\frac {30 b^{3} x^{3}}{a^{4}}-\frac {45 b^{4} x^{4}}{a^{5}}-\frac {55 b^{5} x^{5}}{3 a^{6}}}{x^{2} \left (b x +a \right )^{3}}+\frac {10 b^{2} \ln \left (x \right )}{a^{6}}-\frac {10 b^{2} \ln \left (b x +a \right )}{a^{6}}\) | \(83\) |
risch | \(\frac {\frac {10 b^{4} x^{4}}{a^{5}}+\frac {25 b^{3} x^{3}}{a^{4}}+\frac {55 b^{2} x^{2}}{3 a^{3}}+\frac {5 b x}{2 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b x +a \right )^{3}}-\frac {10 b^{2} \ln \left (b x +a \right )}{a^{6}}+\frac {10 b^{2} \ln \left (-x \right )}{a^{6}}\) | \(85\) |
default | \(-\frac {1}{2 a^{4} x^{2}}+\frac {4 b}{a^{5} x}+\frac {b^{2}}{3 a^{3} \left (b x +a \right )^{3}}+\frac {3 b^{2}}{2 a^{4} \left (b x +a \right )^{2}}+\frac {6 b^{2}}{a^{5} \left (b x +a \right )}+\frac {10 b^{2} \ln \left (x \right )}{a^{6}}-\frac {10 b^{2} \ln \left (b x +a \right )}{a^{6}}\) | \(88\) |
parallelrisch | \(\frac {60 b^{5} \ln \left (x \right ) x^{5}-60 \ln \left (b x +a \right ) x^{5} b^{5}+180 a \,b^{4} \ln \left (x \right ) x^{4}-180 \ln \left (b x +a \right ) x^{4} a \,b^{4}-110 b^{5} x^{5}+180 a^{2} b^{3} \ln \left (x \right ) x^{3}-180 \ln \left (b x +a \right ) x^{3} a^{2} b^{3}-270 a \,b^{4} x^{4}+60 a^{3} b^{2} \ln \left (x \right ) x^{2}-60 \ln \left (b x +a \right ) x^{2} a^{3} b^{2}-180 a^{2} b^{3} x^{3}+15 a^{4} b x -3 a^{5}}{6 a^{6} x^{2} \left (b x +a \right )^{3}}\) | \(167\) |
(-1/2/a+5/2*b/a^2*x-30*b^3/a^4*x^3-45*b^4/a^5*x^4-55/3*b^5/a^6*x^5)/x^2/(b *x+a)^3+10*b^2*ln(x)/a^6-10*b^2*ln(b*x+a)/a^6
Time = 0.22 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.87 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {60 \, a b^{4} x^{4} + 150 \, a^{2} b^{3} x^{3} + 110 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x - 3 \, a^{5} - 60 \, {\left (b^{5} x^{5} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + a^{3} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} x^{5} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + a^{3} b^{2} x^{2}\right )} \log \left (x\right )}{6 \, {\left (a^{6} b^{3} x^{5} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{3} + a^{9} x^{2}\right )}} \]
1/6*(60*a*b^4*x^4 + 150*a^2*b^3*x^3 + 110*a^3*b^2*x^2 + 15*a^4*b*x - 3*a^5 - 60*(b^5*x^5 + 3*a*b^4*x^4 + 3*a^2*b^3*x^3 + a^3*b^2*x^2)*log(b*x + a) + 60*(b^5*x^5 + 3*a*b^4*x^4 + 3*a^2*b^3*x^3 + a^3*b^2*x^2)*log(x))/(a^6*b^3 *x^5 + 3*a^7*b^2*x^4 + 3*a^8*b*x^3 + a^9*x^2)
Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {- 3 a^{4} + 15 a^{3} b x + 110 a^{2} b^{2} x^{2} + 150 a b^{3} x^{3} + 60 b^{4} x^{4}}{6 a^{8} x^{2} + 18 a^{7} b x^{3} + 18 a^{6} b^{2} x^{4} + 6 a^{5} b^{3} x^{5}} + \frac {10 b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{6}} \]
(-3*a**4 + 15*a**3*b*x + 110*a**2*b**2*x**2 + 150*a*b**3*x**3 + 60*b**4*x* *4)/(6*a**8*x**2 + 18*a**7*b*x**3 + 18*a**6*b**2*x**4 + 6*a**5*b**3*x**5) + 10*b**2*(log(x) - log(a/b + x))/a**6
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {60 \, b^{4} x^{4} + 150 \, a b^{3} x^{3} + 110 \, a^{2} b^{2} x^{2} + 15 \, a^{3} b x - 3 \, a^{4}}{6 \, {\left (a^{5} b^{3} x^{5} + 3 \, a^{6} b^{2} x^{4} + 3 \, a^{7} b x^{3} + a^{8} x^{2}\right )}} - \frac {10 \, b^{2} \log \left (b x + a\right )}{a^{6}} + \frac {10 \, b^{2} \log \left (x\right )}{a^{6}} \]
1/6*(60*b^4*x^4 + 150*a*b^3*x^3 + 110*a^2*b^2*x^2 + 15*a^3*b*x - 3*a^4)/(a ^5*b^3*x^5 + 3*a^6*b^2*x^4 + 3*a^7*b*x^3 + a^8*x^2) - 10*b^2*log(b*x + a)/ a^6 + 10*b^2*log(x)/a^6
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=-\frac {10 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{6}} + \frac {10 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {60 \, a b^{4} x^{4} + 150 \, a^{2} b^{3} x^{3} + 110 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x - 3 \, a^{5}}{6 \, {\left (b x + a\right )}^{3} a^{6} x^{2}} \]
-10*b^2*log(abs(b*x + a))/a^6 + 10*b^2*log(abs(x))/a^6 + 1/6*(60*a*b^4*x^4 + 150*a^2*b^3*x^3 + 110*a^3*b^2*x^2 + 15*a^4*b*x - 3*a^5)/((b*x + a)^3*a^ 6*x^2)
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {\frac {55\,b^2\,x^2}{3\,a^3}-\frac {1}{2\,a}+\frac {25\,b^3\,x^3}{a^4}+\frac {10\,b^4\,x^4}{a^5}+\frac {5\,b\,x}{2\,a^2}}{a^3\,x^2+3\,a^2\,b\,x^3+3\,a\,b^2\,x^4+b^3\,x^5}-\frac {20\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^6} \]
((55*b^2*x^2)/(3*a^3) - 1/(2*a) + (25*b^3*x^3)/a^4 + (10*b^4*x^4)/a^5 + (5 *b*x)/(2*a^2))/(a^3*x^2 + b^3*x^5 + 3*a^2*b*x^3 + 3*a*b^2*x^4) - (20*b^2*a tanh((2*b*x)/a + 1))/a^6
Time = 0.00 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.04 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {-60 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} x^{2}-180 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} x^{3}-180 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} x^{4}-60 \,\mathrm {log}\left (b x +a \right ) b^{5} x^{5}+60 \,\mathrm {log}\left (x \right ) a^{3} b^{2} x^{2}+180 \,\mathrm {log}\left (x \right ) a^{2} b^{3} x^{3}+180 \,\mathrm {log}\left (x \right ) a \,b^{4} x^{4}+60 \,\mathrm {log}\left (x \right ) b^{5} x^{5}-3 a^{5}+15 a^{4} b x +90 a^{3} b^{2} x^{2}+90 a^{2} b^{3} x^{3}-20 b^{5} x^{5}}{6 a^{6} x^{2} \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )} \]
( - 60*log(a + b*x)*a**3*b**2*x**2 - 180*log(a + b*x)*a**2*b**3*x**3 - 180 *log(a + b*x)*a*b**4*x**4 - 60*log(a + b*x)*b**5*x**5 + 60*log(x)*a**3*b** 2*x**2 + 180*log(x)*a**2*b**3*x**3 + 180*log(x)*a*b**4*x**4 + 60*log(x)*b* *5*x**5 - 3*a**5 + 15*a**4*b*x + 90*a**3*b**2*x**2 + 90*a**2*b**3*x**3 - 2 0*b**5*x**5)/(6*a**6*x**2*(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))