Integrand size = 13, antiderivative size = 38 \[ \int \frac {1}{x \left (a+b x^5\right )^2} \, dx=\frac {1}{5 a \left (a+b x^5\right )}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b x^5\right )}{5 a^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, 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 = {272, 46} \[ \int \frac {1}{x \left (a+b x^5\right )^2} \, dx=-\frac {\log \left (a+b x^5\right )}{5 a^2}+\frac {\log (x)}{a^2}+\frac {1}{5 a \left (a+b x^5\right )} \]
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Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,x^5\right ) \\ & = \frac {1}{5} \text {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,x^5\right ) \\ & = \frac {1}{5 a \left (a+b x^5\right )}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b x^5\right )}{5 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \left (a+b x^5\right )^2} \, dx=\frac {\frac {a}{a+b x^5}+5 \log (x)-\log \left (a+b x^5\right )}{5 a^2} \]
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Time = 4.37 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {1}{5 a \left (b \,x^{5}+a \right )}+\frac {\ln \left (x \right )}{a^{2}}-\frac {\ln \left (b \,x^{5}+a \right )}{5 a^{2}}\) | \(35\) |
norman | \(-\frac {b \,x^{5}}{5 a^{2} \left (b \,x^{5}+a \right )}+\frac {\ln \left (x \right )}{a^{2}}-\frac {\ln \left (b \,x^{5}+a \right )}{5 a^{2}}\) | \(39\) |
default | \(\frac {\ln \left (x \right )}{a^{2}}-\frac {b \left (\frac {\ln \left (b \,x^{5}+a \right )}{b}-\frac {a}{b \left (b \,x^{5}+a \right )}\right )}{5 a^{2}}\) | \(42\) |
parallelrisch | \(\frac {5 b \ln \left (x \right ) x^{5}-b \ln \left (b \,x^{5}+a \right ) x^{5}-b \,x^{5}+5 a \ln \left (x \right )-a \ln \left (b \,x^{5}+a \right )}{5 a^{2} \left (b \,x^{5}+a \right )}\) | \(60\) |
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Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x \left (a+b x^5\right )^2} \, dx=-\frac {{\left (b x^{5} + a\right )} \log \left (b x^{5} + a\right ) - 5 \, {\left (b x^{5} + a\right )} \log \left (x\right ) - a}{5 \, {\left (a^{2} b x^{5} + a^{3}\right )}} \]
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Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (a+b x^5\right )^2} \, dx=\frac {1}{5 a^{2} + 5 a b x^{5}} + \frac {\log {\left (x \right )}}{a^{2}} - \frac {\log {\left (\frac {a}{b} + x^{5} \right )}}{5 a^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \left (a+b x^5\right )^2} \, dx=\frac {1}{5 \, {\left (a b x^{5} + a^{2}\right )}} - \frac {\log \left (b x^{5} + a\right )}{5 \, a^{2}} + \frac {\log \left (x^{5}\right )}{5 \, a^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x \left (a+b x^5\right )^2} \, dx=-\frac {\log \left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a^{2}} + \frac {b x^{5} + 2 \, a}{5 \, {\left (b x^{5} + a\right )} a^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (a+b x^5\right )^2} \, dx=\frac {\ln \left (x\right )}{a^2}+\frac {1}{5\,a\,\left (b\,x^5+a\right )}-\frac {\ln \left (b\,x^5+a\right )}{5\,a^2} \]
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