Integrand size = 26, antiderivative size = 19 \[ \int \frac {x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx=\frac {1}{2} \log \left (a-b x^2-c x^4\right ) \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1261, 642} \[ \int \frac {x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx=\frac {1}{2} \log \left (a-b x^2-c x^4\right ) \]
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Rule 642
Rule 1261
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {b+2 c x}{-a+b x+c x^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \log \left (a-b x^2-c x^4\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx=\frac {1}{2} \log \left (-a+b x^2+c x^4\right ) \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\ln \left (-c \,x^{4}-b \,x^{2}+a \right )}{2}\) | \(18\) |
norman | \(\frac {\ln \left (-c \,x^{4}-b \,x^{2}+a \right )}{2}\) | \(18\) |
risch | \(\frac {\ln \left (-c \,x^{4}-b \,x^{2}+a \right )}{2}\) | \(18\) |
parallelrisch | \(\frac {\ln \left (c \,x^{4}+b \,x^{2}-a \right )}{2}\) | \(18\) |
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none
Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx=\frac {1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \]
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Time = 0.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx=\frac {\log {\left (- a + b x^{2} + c x^{4} \right )}}{2} \]
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none
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx=\frac {1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \]
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none
Time = 0.58 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx=\frac {1}{2} \, \log \left ({\left | c x^{4} + b x^{2} - a \right |}\right ) \]
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Time = 8.57 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx=\frac {\ln \left (c\,x^4+b\,x^2-a\right )}{2} \]
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