Integrand size = 11, antiderivative size = 97 \[ \int \frac {1}{x^5 (a+b x)^3} \, dx=-\frac {1}{4 a^3 x^4}+\frac {b}{a^4 x^3}-\frac {3 b^2}{a^5 x^2}+\frac {10 b^3}{a^6 x}+\frac {b^4}{2 a^5 (a+b x)^2}+\frac {5 b^4}{a^6 (a+b x)}+\frac {15 b^4 \log (x)}{a^7}-\frac {15 b^4 \log (a+b x)}{a^7} \]
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Time = 0.04 (sec) , antiderivative size = 97, 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} \[ \int \frac {1}{x^5 (a+b x)^3} \, dx=\frac {15 b^4 \log (x)}{a^7}-\frac {15 b^4 \log (a+b x)}{a^7}+\frac {5 b^4}{a^6 (a+b x)}+\frac {10 b^3}{a^6 x}+\frac {b^4}{2 a^5 (a+b x)^2}-\frac {3 b^2}{a^5 x^2}+\frac {b}{a^4 x^3}-\frac {1}{4 a^3 x^4} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^3 x^5}-\frac {3 b}{a^4 x^4}+\frac {6 b^2}{a^5 x^3}-\frac {10 b^3}{a^6 x^2}+\frac {15 b^4}{a^7 x}-\frac {b^5}{a^5 (a+b x)^3}-\frac {5 b^5}{a^6 (a+b x)^2}-\frac {15 b^5}{a^7 (a+b x)}\right ) \, dx \\ & = -\frac {1}{4 a^3 x^4}+\frac {b}{a^4 x^3}-\frac {3 b^2}{a^5 x^2}+\frac {10 b^3}{a^6 x}+\frac {b^4}{2 a^5 (a+b x)^2}+\frac {5 b^4}{a^6 (a+b x)}+\frac {15 b^4 \log (x)}{a^7}-\frac {15 b^4 \log (a+b x)}{a^7} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^5 (a+b x)^3} \, dx=\frac {\frac {a \left (-a^5+2 a^4 b x-5 a^3 b^2 x^2+20 a^2 b^3 x^3+90 a b^4 x^4+60 b^5 x^5\right )}{x^4 (a+b x)^2}+60 b^4 \log (x)-60 b^4 \log (a+b x)}{4 a^7} \]
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Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {1}{4 a^{3} x^{4}}+\frac {b}{a^{4} x^{3}}-\frac {3 b^{2}}{a^{5} x^{2}}+\frac {10 b^{3}}{a^{6} x}+\frac {b^{4}}{2 a^{5} \left (b x +a \right )^{2}}+\frac {5 b^{4}}{a^{6} \left (b x +a \right )}+\frac {15 b^{4} \ln \left (x \right )}{a^{7}}-\frac {15 b^{4} \ln \left (b x +a \right )}{a^{7}}\) | \(94\) |
norman | \(\frac {-\frac {1}{4 a}+\frac {b x}{2 a^{2}}-\frac {5 b^{2} x^{2}}{4 a^{3}}+\frac {5 b^{3} x^{3}}{a^{4}}-\frac {30 b^{5} x^{5}}{a^{6}}-\frac {45 b^{6} x^{6}}{2 a^{7}}}{x^{4} \left (b x +a \right )^{2}}+\frac {15 b^{4} \ln \left (x \right )}{a^{7}}-\frac {15 b^{4} \ln \left (b x +a \right )}{a^{7}}\) | \(94\) |
risch | \(\frac {\frac {15 b^{5} x^{5}}{a^{6}}+\frac {45 b^{4} x^{4}}{2 a^{5}}+\frac {5 b^{3} x^{3}}{a^{4}}-\frac {5 b^{2} x^{2}}{4 a^{3}}+\frac {b x}{2 a^{2}}-\frac {1}{4 a}}{x^{4} \left (b x +a \right )^{2}}-\frac {15 b^{4} \ln \left (b x +a \right )}{a^{7}}+\frac {15 b^{4} \ln \left (-x \right )}{a^{7}}\) | \(96\) |
parallelrisch | \(\frac {60 \ln \left (x \right ) x^{6} b^{6}-60 \ln \left (b x +a \right ) x^{6} b^{6}+120 \ln \left (x \right ) x^{5} a \,b^{5}-120 \ln \left (b x +a \right ) x^{5} a \,b^{5}-90 b^{6} x^{6}+60 \ln \left (x \right ) x^{4} a^{2} b^{4}-60 \ln \left (b x +a \right ) x^{4} a^{2} b^{4}-120 a \,x^{5} b^{5}+20 a^{3} x^{3} b^{3}-5 a^{4} x^{2} b^{2}+2 a^{5} x b -a^{6}}{4 a^{7} x^{4} \left (b x +a \right )^{2}}\) | \(148\) |
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Time = 0.22 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.57 \[ \int \frac {1}{x^5 (a+b x)^3} \, dx=\frac {60 \, a b^{5} x^{5} + 90 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{4} b^{2} x^{2} + 2 \, a^{5} b x - a^{6} - 60 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} + a^{2} b^{4} x^{4}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} + a^{2} b^{4} x^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{7} b^{2} x^{6} + 2 \, a^{8} b x^{5} + a^{9} x^{4}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^5 (a+b x)^3} \, dx=\frac {- a^{5} + 2 a^{4} b x - 5 a^{3} b^{2} x^{2} + 20 a^{2} b^{3} x^{3} + 90 a b^{4} x^{4} + 60 b^{5} x^{5}}{4 a^{8} x^{4} + 8 a^{7} b x^{5} + 4 a^{6} b^{2} x^{6}} + \frac {15 b^{4} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{7}} \]
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Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^5 (a+b x)^3} \, dx=\frac {60 \, b^{5} x^{5} + 90 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} - 5 \, a^{3} b^{2} x^{2} + 2 \, a^{4} b x - a^{5}}{4 \, {\left (a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{5} + a^{8} x^{4}\right )}} - \frac {15 \, b^{4} \log \left (b x + a\right )}{a^{7}} + \frac {15 \, b^{4} \log \left (x\right )}{a^{7}} \]
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Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 (a+b x)^3} \, dx=-\frac {15 \, b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{7}} + \frac {15 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac {60 \, a b^{5} x^{5} + 90 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{4} b^{2} x^{2} + 2 \, a^{5} b x - a^{6}}{4 \, {\left (b x + a\right )}^{2} a^{7} x^{4}} \]
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Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^5 (a+b x)^3} \, dx=\frac {\frac {5\,b^3\,x^3}{a^4}-\frac {5\,b^2\,x^2}{4\,a^3}-\frac {1}{4\,a}+\frac {45\,b^4\,x^4}{2\,a^5}+\frac {15\,b^5\,x^5}{a^6}+\frac {b\,x}{2\,a^2}}{a^2\,x^4+2\,a\,b\,x^5+b^2\,x^6}-\frac {30\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^7} \]
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