Integrand size = 13, antiderivative size = 38 \[ \int \frac {x}{\left (a x+b x^3\right )^2} \, dx=\frac {1}{2 a \left (a+b x^2\right )}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b x^2\right )}{2 a^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1598, 272, 46} \[ \int \frac {x}{\left (a x+b x^3\right )^2} \, dx=-\frac {\log \left (a+b x^2\right )}{2 a^2}+\frac {\log (x)}{a^2}+\frac {1}{2 a \left (a+b x^2\right )} \]
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Rule 46
Rule 272
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (a+b x^2\right )^2} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \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^2\right ) \\ & = \frac {1}{2 a \left (a+b x^2\right )}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b x^2\right )}{2 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {x}{\left (a x+b x^3\right )^2} \, dx=\frac {\frac {a}{a+b x^2}+2 \log (x)-\log \left (a+b x^2\right )}{2 a^2} \]
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Time = 2.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {1}{2 a \left (b \,x^{2}+a \right )}+\frac {\ln \left (x \right )}{a^{2}}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(35\) |
norman | \(-\frac {b \,x^{2}}{2 a^{2} \left (b \,x^{2}+a \right )}+\frac {\ln \left (x \right )}{a^{2}}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(39\) |
default | \(\frac {\ln \left (x \right )}{a^{2}}-\frac {b \left (\frac {\ln \left (b \,x^{2}+a \right )}{b}-\frac {a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{2}}\) | \(42\) |
parallelrisch | \(\frac {2 b \ln \left (x \right ) x^{2}-b \ln \left (b \,x^{2}+a \right ) x^{2}-b \,x^{2}+2 a \ln \left (x \right )-a \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (b \,x^{2}+a \right )}\) | \(60\) |
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Time = 0.37 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24 \[ \int \frac {x}{\left (a x+b x^3\right )^2} \, dx=-\frac {{\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b x^{2} + a\right )} \log \left (x\right ) - a}{2 \, {\left (a^{2} b x^{2} + a^{3}\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\left (a x+b x^3\right )^2} \, dx=\frac {1}{2 a^{2} + 2 a b x^{2}} + \frac {\log {\left (x \right )}}{a^{2}} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\left (a x+b x^3\right )^2} \, dx=\frac {1}{2 \, {\left (a b x^{2} + a^{2}\right )}} - \frac {\log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {\log \left (x\right )}{a^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24 \[ \int \frac {x}{\left (a x+b x^3\right )^2} \, dx=\frac {\log \left (x^{2}\right )}{2 \, a^{2}} - \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac {b x^{2} + 2 \, a}{2 \, {\left (b x^{2} + a\right )} a^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\left (a x+b x^3\right )^2} \, dx=\frac {\ln \left (x\right )}{a^2}+\frac {1}{2\,a\,\left (b\,x^2+a\right )}-\frac {\ln \left (b\,x^2+a\right )}{2\,a^2} \]
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