Integrand size = 11, antiderivative size = 84 \[ \int \frac {1}{x^5 (a+b x)^2} \, dx=-\frac {1}{4 a^2 x^4}+\frac {2 b}{3 a^3 x^3}-\frac {3 b^2}{2 a^4 x^2}+\frac {4 b^3}{a^5 x}+\frac {b^4}{a^5 (a+b x)}+\frac {5 b^4 \log (x)}{a^6}-\frac {5 b^4 \log (a+b x)}{a^6} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^5 (a+b x)^2} \, dx=\frac {5 b^4 \log (x)}{a^6}-\frac {5 b^4 \log (a+b x)}{a^6}+\frac {b^4}{a^5 (a+b x)}+\frac {4 b^3}{a^5 x}-\frac {3 b^2}{2 a^4 x^2}+\frac {2 b}{3 a^3 x^3}-\frac {1}{4 a^2 x^4} \]
[In]
[Out]
Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^2 x^5}-\frac {2 b}{a^3 x^4}+\frac {3 b^2}{a^4 x^3}-\frac {4 b^3}{a^5 x^2}+\frac {5 b^4}{a^6 x}-\frac {b^5}{a^5 (a+b x)^2}-\frac {5 b^5}{a^6 (a+b x)}\right ) \, dx \\ & = -\frac {1}{4 a^2 x^4}+\frac {2 b}{3 a^3 x^3}-\frac {3 b^2}{2 a^4 x^2}+\frac {4 b^3}{a^5 x}+\frac {b^4}{a^5 (a+b x)}+\frac {5 b^4 \log (x)}{a^6}-\frac {5 b^4 \log (a+b x)}{a^6} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^5 (a+b x)^2} \, dx=\frac {\frac {a \left (-3 a^4+5 a^3 b x-10 a^2 b^2 x^2+30 a b^3 x^3+60 b^4 x^4\right )}{x^4 (a+b x)}+60 b^4 \log (x)-60 b^4 \log (a+b x)}{12 a^6} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {1}{4 a^{2} x^{4}}+\frac {2 b}{3 a^{3} x^{3}}-\frac {3 b^{2}}{2 a^{4} x^{2}}+\frac {4 b^{3}}{a^{5} x}+\frac {b^{4}}{a^{5} \left (b x +a \right )}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}-\frac {5 b^{4} \ln \left (b x +a \right )}{a^{6}}\) | \(79\) |
norman | \(\frac {-\frac {5 b^{5} x^{5}}{a^{6}}-\frac {1}{4 a}+\frac {5 b x}{12 a^{2}}-\frac {5 b^{2} x^{2}}{6 a^{3}}+\frac {5 b^{3} x^{3}}{2 a^{4}}}{x^{4} \left (b x +a \right )}+\frac {5 b^{4} \ln \left (x \right )}{a^{6}}-\frac {5 b^{4} \ln \left (b x +a \right )}{a^{6}}\) | \(83\) |
risch | \(\frac {\frac {5 b^{4} x^{4}}{a^{5}}+\frac {5 b^{3} x^{3}}{2 a^{4}}-\frac {5 b^{2} x^{2}}{6 a^{3}}+\frac {5 b x}{12 a^{2}}-\frac {1}{4 a}}{x^{4} \left (b x +a \right )}+\frac {5 b^{4} \ln \left (-x \right )}{a^{6}}-\frac {5 b^{4} \ln \left (b x +a \right )}{a^{6}}\) | \(85\) |
parallelrisch | \(\frac {60 b^{5} \ln \left (x \right ) x^{5}-60 \ln \left (b x +a \right ) x^{5} b^{5}+60 a \,b^{4} \ln \left (x \right ) x^{4}-60 \ln \left (b x +a \right ) x^{4} a \,b^{4}-60 b^{5} x^{5}+30 a^{2} b^{3} x^{3}-10 a^{3} b^{2} x^{2}+5 a^{4} b x -3 a^{5}}{12 a^{6} x^{4} \left (b x +a \right )}\) | \(109\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^5 (a+b x)^2} \, dx=\frac {60 \, a b^{4} x^{4} + 30 \, a^{2} b^{3} x^{3} - 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x - 3 \, a^{5} - 60 \, {\left (b^{5} x^{5} + a b^{4} x^{4}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} x^{5} + a b^{4} x^{4}\right )} \log \left (x\right )}{12 \, {\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^5 (a+b x)^2} \, dx=\frac {- 3 a^{4} + 5 a^{3} b x - 10 a^{2} b^{2} x^{2} + 30 a b^{3} x^{3} + 60 b^{4} x^{4}}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} + \frac {5 b^{4} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{6}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^5 (a+b x)^2} \, dx=\frac {60 \, b^{4} x^{4} + 30 \, a b^{3} x^{3} - 10 \, a^{2} b^{2} x^{2} + 5 \, a^{3} b x - 3 \, a^{4}}{12 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} - \frac {5 \, b^{4} \log \left (b x + a\right )}{a^{6}} + \frac {5 \, b^{4} \log \left (x\right )}{a^{6}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^5 (a+b x)^2} \, dx=\frac {5 \, b^{4} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{6}} + \frac {b^{4}}{{\left (b x + a\right )} a^{5}} - \frac {\frac {260 \, a b^{4}}{b x + a} - \frac {300 \, a^{2} b^{4}}{{\left (b x + a\right )}^{2}} + \frac {120 \, a^{3} b^{4}}{{\left (b x + a\right )}^{3}} - 77 \, b^{4}}{12 \, a^{6} {\left (\frac {a}{b x + a} - 1\right )}^{4}} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^5 (a+b x)^2} \, dx=\frac {\frac {5\,b^3\,x^3}{2\,a^4}-\frac {5\,b^2\,x^2}{6\,a^3}-\frac {1}{4\,a}+\frac {5\,b^4\,x^4}{a^5}+\frac {5\,b\,x}{12\,a^2}}{b\,x^5+a\,x^4}-\frac {10\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^6} \]
[In]
[Out]