Integrand size = 13, antiderivative size = 15 \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\frac {\sqrt {b x^2} \log (x)}{x} \]
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Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 29} \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\frac {\sqrt {b x^2} \log (x)}{x} \]
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Rule 15
Rule 29
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b x^2} \int \frac {1}{x} \, dx}{x} \\ & = \frac {\sqrt {b x^2} \log (x)}{x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\frac {b x \log (x)}{\sqrt {b x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\ln \left (x \right ) \sqrt {b \,x^{2}}}{x}\) | \(14\) |
risch | \(\frac {\ln \left (x \right ) \sqrt {b \,x^{2}}}{x}\) | \(14\) |
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none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\frac {\sqrt {b x^{2}} \log \left (x\right )}{x} \]
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Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\frac {\sqrt {b x^{2}} \log {\left (x \right )}}{x} \]
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Exception generated. \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\sqrt {b} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) \]
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Timed out. \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\int \frac {\sqrt {b\,x^2}}{x^2} \,d x \]
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