Integrand size = 15, antiderivative size = 15 \[ \int \frac {x^2}{a x+b x^3} \, dx=\frac {\log \left (a+b x^2\right )}{2 b} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1598, 266} \[ \int \frac {x^2}{a x+b x^3} \, dx=\frac {\log \left (a+b x^2\right )}{2 b} \]
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Rule 266
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{a+b x^2} \, dx \\ & = \frac {\log \left (a+b x^2\right )}{2 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{a x+b x^3} \, dx=\frac {\log \left (a+b x^2\right )}{2 b} \]
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Time = 2.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\ln \left (b \,x^{2}+a \right )}{2 b}\) | \(14\) |
norman | \(\frac {\ln \left (b \,x^{2}+a \right )}{2 b}\) | \(14\) |
risch | \(\frac {\ln \left (b \,x^{2}+a \right )}{2 b}\) | \(14\) |
parallelrisch | \(\frac {\ln \left (b \,x^{2}+a \right )}{2 b}\) | \(14\) |
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none
Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{a x+b x^3} \, dx=\frac {\log \left (b x^{2} + a\right )}{2 \, b} \]
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Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {x^2}{a x+b x^3} \, dx=\frac {\log {\left (a + b x^{2} \right )}}{2 b} \]
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none
Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{a x+b x^3} \, dx=\frac {\log \left (b x^{2} + a\right )}{2 \, b} \]
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none
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{a x+b x^3} \, dx=\frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b} \]
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Time = 10.69 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{a x+b x^3} \, dx=\frac {\ln \left (b\,x^2+a\right )}{2\,b} \]
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