Integrand size = 13, antiderivative size = 56 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^5} \, dx=-\frac {1}{3 b x^3}+\frac {a}{2 b^2 x^2}-\frac {a^2}{b^3 x}-\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (b+a x)}{b^4} \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 46} \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^5} \, dx=-\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (a x+b)}{b^4}-\frac {a^2}{b^3 x}+\frac {a}{2 b^2 x^2}-\frac {1}{3 b x^3} \]
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Rule 46
Rule 269
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^4 (b+a x)} \, dx \\ & = \int \left (\frac {1}{b x^4}-\frac {a}{b^2 x^3}+\frac {a^2}{b^3 x^2}-\frac {a^3}{b^4 x}+\frac {a^4}{b^4 (b+a x)}\right ) \, dx \\ & = -\frac {1}{3 b x^3}+\frac {a}{2 b^2 x^2}-\frac {a^2}{b^3 x}-\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (b+a x)}{b^4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^5} \, dx=-\frac {1}{3 b x^3}+\frac {a}{2 b^2 x^2}-\frac {a^2}{b^3 x}-\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (b+a x)}{b^4} \]
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Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {1}{3 b \,x^{3}}+\frac {a}{2 b^{2} x^{2}}-\frac {a^{2}}{b^{3} x}-\frac {a^{3} \ln \left (x \right )}{b^{4}}+\frac {a^{3} \ln \left (a x +b \right )}{b^{4}}\) | \(53\) |
parallelrisch | \(-\frac {6 a^{3} \ln \left (x \right ) x^{3}-6 a^{3} \ln \left (a x +b \right ) x^{3}+6 a^{2} b \,x^{2}-3 a \,b^{2} x +2 b^{3}}{6 b^{4} x^{3}}\) | \(55\) |
norman | \(\frac {-\frac {x}{3 b}+\frac {a \,x^{2}}{2 b^{2}}-\frac {a^{2} x^{3}}{b^{3}}}{x^{4}}+\frac {a^{3} \ln \left (a x +b \right )}{b^{4}}-\frac {a^{3} \ln \left (x \right )}{b^{4}}\) | \(56\) |
risch | \(\frac {-\frac {a^{2} x^{2}}{b^{3}}+\frac {a x}{2 b^{2}}-\frac {1}{3 b}}{x^{3}}+\frac {a^{3} \ln \left (-a x -b \right )}{b^{4}}-\frac {a^{3} \ln \left (x \right )}{b^{4}}\) | \(56\) |
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Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^5} \, dx=\frac {6 \, a^{3} x^{3} \log \left (a x + b\right ) - 6 \, a^{3} x^{3} \log \left (x\right ) - 6 \, a^{2} b x^{2} + 3 \, a b^{2} x - 2 \, b^{3}}{6 \, b^{4} x^{3}} \]
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Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^5} \, dx=\frac {a^{3} \left (- \log {\left (x \right )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{4}} + \frac {- 6 a^{2} x^{2} + 3 a b x - 2 b^{2}}{6 b^{3} x^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^5} \, dx=\frac {a^{3} \log \left (a x + b\right )}{b^{4}} - \frac {a^{3} \log \left (x\right )}{b^{4}} - \frac {6 \, a^{2} x^{2} - 3 \, a b x + 2 \, b^{2}}{6 \, b^{3} x^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^5} \, dx=\frac {a^{3} \log \left ({\left | a x + b \right |}\right )}{b^{4}} - \frac {a^{3} \log \left ({\left | x \right |}\right )}{b^{4}} - \frac {6 \, a^{2} b x^{2} - 3 \, a b^{2} x + 2 \, b^{3}}{6 \, b^{4} x^{3}} \]
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Time = 5.78 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+\frac {b}{x}\right ) x^5} \, dx=\frac {2\,a^3\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^4}-\frac {a^2\,b\,x^2-\frac {a\,b^2\,x}{2}+\frac {b^3}{3}}{b^4\,x^3} \]
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