Integrand size = 11, antiderivative size = 57 \[ \int \frac {1}{x (a+b x)^4} \, dx=\frac {1}{3 a (a+b x)^3}+\frac {1}{2 a^2 (a+b x)^2}+\frac {1}{a^3 (a+b x)}+\frac {\log (x)}{a^4}-\frac {\log (a+b x)}{a^4} \]
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Time = 0.02 (sec) , antiderivative size = 57, 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} \[ \int \frac {1}{x (a+b x)^4} \, dx=-\frac {\log (a+b x)}{a^4}+\frac {\log (x)}{a^4}+\frac {1}{a^3 (a+b x)}+\frac {1}{2 a^2 (a+b x)^2}+\frac {1}{3 a (a+b x)^3} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^4 x}-\frac {b}{a (a+b x)^4}-\frac {b}{a^2 (a+b x)^3}-\frac {b}{a^3 (a+b x)^2}-\frac {b}{a^4 (a+b x)}\right ) \, dx \\ & = \frac {1}{3 a (a+b x)^3}+\frac {1}{2 a^2 (a+b x)^2}+\frac {1}{a^3 (a+b x)}+\frac {\log (x)}{a^4}-\frac {\log (a+b x)}{a^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x (a+b x)^4} \, dx=\frac {\frac {a \left (11 a^2+15 a b x+6 b^2 x^2\right )}{(a+b x)^3}+6 \log (x)-6 \log (a+b x)}{6 a^4} \]
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Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {\frac {b^{2} x^{2}}{a^{3}}+\frac {5 b x}{2 a^{2}}+\frac {11}{6 a}}{\left (b x +a \right )^{3}}+\frac {\ln \left (-x \right )}{a^{4}}-\frac {\ln \left (b x +a \right )}{a^{4}}\) | \(52\) |
default | \(\frac {1}{3 a \left (b x +a \right )^{3}}+\frac {1}{2 a^{2} \left (b x +a \right )^{2}}+\frac {1}{a^{3} \left (b x +a \right )}+\frac {\ln \left (x \right )}{a^{4}}-\frac {\ln \left (b x +a \right )}{a^{4}}\) | \(54\) |
norman | \(\frac {-\frac {3 b x}{a^{2}}-\frac {9 b^{2} x^{2}}{2 a^{3}}-\frac {11 b^{3} x^{3}}{6 a^{4}}}{\left (b x +a \right )^{3}}+\frac {\ln \left (x \right )}{a^{4}}-\frac {\ln \left (b x +a \right )}{a^{4}}\) | \(57\) |
parallelrisch | \(\frac {6 b^{3} \ln \left (x \right ) x^{3}-6 b^{3} \ln \left (b x +a \right ) x^{3}+18 a \,b^{2} \ln \left (x \right ) x^{2}-18 \ln \left (b x +a \right ) x^{2} a \,b^{2}-11 b^{3} x^{3}+18 a^{2} b \ln \left (x \right ) x -18 \ln \left (b x +a \right ) x \,a^{2} b -27 a \,b^{2} x^{2}+6 a^{3} \ln \left (x \right )-6 a^{3} \ln \left (b x +a \right )-18 a^{2} b x}{6 a^{4} \left (b x +a \right )^{3}}\) | \(128\) |
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (53) = 106\).
Time = 0.23 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.18 \[ \int \frac {1}{x (a+b x)^4} \, dx=\frac {6 \, a b^{2} x^{2} + 15 \, a^{2} b x + 11 \, a^{3} - 6 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (x\right )}{6 \, {\left (a^{4} b^{3} x^{3} + 3 \, a^{5} b^{2} x^{2} + 3 \, a^{6} b x + a^{7}\right )}} \]
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Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x (a+b x)^4} \, dx=\frac {11 a^{2} + 15 a b x + 6 b^{2} x^{2}}{6 a^{6} + 18 a^{5} b x + 18 a^{4} b^{2} x^{2} + 6 a^{3} b^{3} x^{3}} + \frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a^{4}} \]
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none
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x (a+b x)^4} \, dx=\frac {6 \, b^{2} x^{2} + 15 \, a b x + 11 \, a^{2}}{6 \, {\left (a^{3} b^{3} x^{3} + 3 \, a^{4} b^{2} x^{2} + 3 \, a^{5} b x + a^{6}\right )}} - \frac {\log \left (b x + a\right )}{a^{4}} + \frac {\log \left (x\right )}{a^{4}} \]
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none
Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x (a+b x)^4} \, dx=-\frac {\log \left ({\left | b x + a \right |}\right )}{a^{4}} + \frac {\log \left ({\left | x \right |}\right )}{a^{4}} + \frac {6 \, a b^{2} x^{2} + 15 \, a^{2} b x + 11 \, a^{3}}{6 \, {\left (b x + a\right )}^{3} a^{4}} \]
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Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x (a+b x)^4} \, dx=\frac {\frac {\frac {1}{a^2+b\,x\,a}-\frac {\ln \left (\frac {a+b\,x}{x}\right )}{a^2}}{a}+\frac {1}{2\,a\,{\left (a+b\,x\right )}^2}}{a}+\frac {1}{3\,a\,{\left (a+b\,x\right )}^3} \]
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