Integrand size = 11, antiderivative size = 22 \[ \int \frac {1}{b x+d x^3} \, dx=\frac {\log (x)}{b}-\frac {\log \left (b+d x^2\right )}{2 b} \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1607, 272, 36, 29, 31} \[ \int \frac {1}{b x+d x^3} \, dx=\frac {\log (x)}{b}-\frac {\log \left (b+d x^2\right )}{2 b} \]
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (b+d x^2\right )} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x (b+d x)} \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 b}-\frac {d \text {Subst}\left (\int \frac {1}{b+d x} \, dx,x,x^2\right )}{2 b} \\ & = \frac {\log (x)}{b}-\frac {\log \left (b+d x^2\right )}{2 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{b x+d x^3} \, dx=\frac {\log (x)}{b}-\frac {\log \left (b+d x^2\right )}{2 b} \]
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Time = 0.68 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (d \,x^{2}+b \right )}{2 b}\) | \(21\) |
norman | \(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (d \,x^{2}+b \right )}{2 b}\) | \(21\) |
risch | \(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (d \,x^{2}+b \right )}{2 b}\) | \(21\) |
parallelrisch | \(\frac {2 \ln \left (x \right )-\ln \left (d \,x^{2}+b \right )}{2 b}\) | \(21\) |
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none
Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{b x+d x^3} \, dx=-\frac {\log \left (d x^{2} + b\right ) - 2 \, \log \left (x\right )}{2 \, b} \]
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Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {1}{b x+d x^3} \, dx=\frac {\log {\left (x \right )}}{b} - \frac {\log {\left (\frac {b}{d} + x^{2} \right )}}{2 b} \]
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Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{b x+d x^3} \, dx=-\frac {\log \left (d x^{2} + b\right )}{2 \, b} + \frac {\log \left (x\right )}{b} \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{b x+d x^3} \, dx=\frac {\log \left (x^{2}\right )}{2 \, b} - \frac {\log \left ({\left | d x^{2} + b \right |}\right )}{2 \, b} \]
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Time = 9.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{b x+d x^3} \, dx=-\frac {\ln \left (d\,x^2+b\right )-2\,\ln \left (x\right )}{2\,b} \]
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