Integrand size = 13, antiderivative size = 16 \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\frac {b \sqrt {b x^2} \log (x)}{x} \]
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Time = 0.00 (sec) , antiderivative size = 16, 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} \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\frac {b \sqrt {b x^2} \log (x)}{x} \]
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Rule 15
Rule 29
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b \sqrt {b x^2}\right ) \int \frac {1}{x} \, dx}{x} \\ & = \frac {b \sqrt {b x^2} \log (x)}{x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\frac {\left (b x^2\right )^{3/2} \log (x)}{x^3} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\left (b \,x^{2}\right )^{\frac {3}{2}} \ln \left (x \right )}{x^{3}}\) | \(14\) |
risch | \(\frac {b \ln \left (x \right ) \sqrt {b \,x^{2}}}{x}\) | \(15\) |
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none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\frac {\sqrt {b x^{2}} b \log \left (x\right )}{x} \]
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Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\frac {\left (b x^{2}\right )^{\frac {3}{2}} \log {\left (x \right )}}{x^{3}} \]
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Exception generated. \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=b^{\frac {3}{2}} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) \]
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Timed out. \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (b\,x^2\right )}^{3/2}}{x^4} \,d x \]
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