Integrand size = 11, antiderivative size = 117 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=-\frac {1}{4 a^4 x^4}+\frac {4 b}{3 a^5 x^3}-\frac {5 b^2}{a^6 x^2}+\frac {20 b^3}{a^7 x}+\frac {b^4}{3 a^5 (a+b x)^3}+\frac {5 b^4}{2 a^6 (a+b x)^2}+\frac {15 b^4}{a^7 (a+b x)}+\frac {35 b^4 \log (x)}{a^8}-\frac {35 b^4 \log (a+b x)}{a^8} \]
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Time = 0.05 (sec) , antiderivative size = 117, 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} \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {35 b^4 \log (x)}{a^8}-\frac {35 b^4 \log (a+b x)}{a^8}+\frac {15 b^4}{a^7 (a+b x)}+\frac {20 b^3}{a^7 x}+\frac {5 b^4}{2 a^6 (a+b x)^2}-\frac {5 b^2}{a^6 x^2}+\frac {b^4}{3 a^5 (a+b x)^3}+\frac {4 b}{3 a^5 x^3}-\frac {1}{4 a^4 x^4} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^4 x^5}-\frac {4 b}{a^5 x^4}+\frac {10 b^2}{a^6 x^3}-\frac {20 b^3}{a^7 x^2}+\frac {35 b^4}{a^8 x}-\frac {b^5}{a^5 (a+b x)^4}-\frac {5 b^5}{a^6 (a+b x)^3}-\frac {15 b^5}{a^7 (a+b x)^2}-\frac {35 b^5}{a^8 (a+b x)}\right ) \, 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} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {\frac {a \left (-3 a^6+7 a^5 b x-21 a^4 b^2 x^2+105 a^3 b^3 x^3+770 a^2 b^4 x^4+1050 a b^5 x^5+420 b^6 x^6\right )}{x^4 (a+b x)^3}+420 b^4 \log (x)-420 b^4 \log (a+b x)}{12 a^8} \]
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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 | \(\frac {420 b^{7} \ln \left (x \right ) x^{7}-420 \ln \left (b x +a \right ) x^{7} b^{7}+1260 a \,b^{6} \ln \left (x \right ) x^{6}-1260 \ln \left (b x +a \right ) x^{6} a \,b^{6}-770 b^{7} x^{7}+1260 a^{2} b^{5} \ln \left (x \right ) x^{5}-1260 \ln \left (b x +a \right ) x^{5} a^{2} b^{5}-1890 a \,b^{6} x^{6}+420 a^{3} b^{4} \ln \left (x \right ) x^{4}-420 \ln \left (b x +a \right ) x^{4} a^{3} b^{4}-1260 a^{2} b^{5} x^{5}+105 a^{4} b^{3} x^{3}-21 a^{5} b^{2} x^{2}+7 a^{6} b x -3 a^{7}}{12 a^{8} x^{4} \left (b x +a \right )^{3}}\) | \(189\) |
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Time = 0.22 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.68 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {420 \, a b^{6} x^{6} + 1050 \, a^{2} b^{5} x^{5} + 770 \, a^{3} b^{4} x^{4} + 105 \, a^{4} b^{3} x^{3} - 21 \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x - 3 \, a^{7} - 420 \, {\left (b^{7} x^{7} + 3 \, a b^{6} x^{6} + 3 \, a^{2} b^{5} x^{5} + a^{3} b^{4} x^{4}\right )} \log \left (b x + a\right ) + 420 \, {\left (b^{7} x^{7} + 3 \, a b^{6} x^{6} + 3 \, a^{2} b^{5} x^{5} + a^{3} b^{4} x^{4}\right )} \log \left (x\right )}{12 \, {\left (a^{8} b^{3} x^{7} + 3 \, a^{9} b^{2} x^{6} + 3 \, a^{10} b x^{5} + a^{11} x^{4}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {- 3 a^{6} + 7 a^{5} b x - 21 a^{4} b^{2} x^{2} + 105 a^{3} b^{3} x^{3} + 770 a^{2} b^{4} x^{4} + 1050 a b^{5} x^{5} + 420 b^{6} x^{6}}{12 a^{10} x^{4} + 36 a^{9} b x^{5} + 36 a^{8} b^{2} x^{6} + 12 a^{7} b^{3} x^{7}} + \frac {35 b^{4} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{8}} \]
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Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {420 \, b^{6} x^{6} + 1050 \, a b^{5} x^{5} + 770 \, a^{2} b^{4} x^{4} + 105 \, a^{3} b^{3} x^{3} - 21 \, a^{4} b^{2} x^{2} + 7 \, a^{5} b x - 3 \, a^{6}}{12 \, {\left (a^{7} b^{3} x^{7} + 3 \, a^{8} b^{2} x^{6} + 3 \, a^{9} b x^{5} + a^{10} x^{4}\right )}} - \frac {35 \, b^{4} \log \left (b x + a\right )}{a^{8}} + \frac {35 \, b^{4} \log \left (x\right )}{a^{8}} \]
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Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=-\frac {35 \, b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{8}} + \frac {35 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac {420 \, a b^{6} x^{6} + 1050 \, a^{2} b^{5} x^{5} + 770 \, a^{3} b^{4} x^{4} + 105 \, a^{4} b^{3} x^{3} - 21 \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x - 3 \, a^{7}}{12 \, {\left (b x + a\right )}^{3} a^{8} x^{4}} \]
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Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^5 (a+b x)^4} \, dx=\frac {\frac {35\,b^3\,x^3}{4\,a^4}-\frac {7\,b^2\,x^2}{4\,a^3}-\frac {1}{4\,a}+\frac {385\,b^4\,x^4}{6\,a^5}+\frac {175\,b^5\,x^5}{2\,a^6}+\frac {35\,b^6\,x^6}{a^7}+\frac {7\,b\,x}{12\,a^2}}{a^3\,x^4+3\,a^2\,b\,x^5+3\,a\,b^2\,x^6+b^3\,x^7}-\frac {70\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^8} \]
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