Integrand size = 19, antiderivative size = 12 \[ \int \frac {a+b x^2}{\left (-a+b x^2\right )^2} \, dx=\frac {x}{a-b x^2} \]
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Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {391} \[ \int \frac {a+b x^2}{\left (-a+b x^2\right )^2} \, dx=\frac {x}{a-b x^2} \]
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Rule 391
Rubi steps \begin{align*} \text {integral}& = \frac {x}{a-b x^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2}{\left (-a+b x^2\right )^2} \, dx=-\frac {x}{-a+b x^2} \]
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Time = 2.51 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
method | result | size |
gosper | \(\frac {x}{-b \,x^{2}+a}\) | \(13\) |
default | \(\frac {x}{-b \,x^{2}+a}\) | \(13\) |
norman | \(\frac {x}{-b \,x^{2}+a}\) | \(13\) |
risch | \(\frac {x}{-b \,x^{2}+a}\) | \(13\) |
parallelrisch | \(-\frac {x}{b \,x^{2}-a}\) | \(15\) |
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none
Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2}{\left (-a+b x^2\right )^2} \, dx=-\frac {x}{b x^{2} - a} \]
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Time = 0.09 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {a+b x^2}{\left (-a+b x^2\right )^2} \, dx=- \frac {x}{- a + b x^{2}} \]
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none
Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2}{\left (-a+b x^2\right )^2} \, dx=-\frac {x}{b x^{2} - a} \]
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none
Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2}{\left (-a+b x^2\right )^2} \, dx=-\frac {x}{b x^{2} - a} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2}{\left (-a+b x^2\right )^2} \, dx=\frac {x}{a-b\,x^2} \]
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