Integrand size = 21, antiderivative size = 8 \[ \int \frac {x \left (a c+b c x^2\right )}{a+b x^2} \, dx=\frac {c x^2}{2} \]
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Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {21, 30} \[ \int \frac {x \left (a c+b c x^2\right )}{a+b x^2} \, dx=\frac {c x^2}{2} \]
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Rule 21
Rule 30
Rubi steps \begin{align*} \text {integral}& = c \int x \, dx \\ & = \frac {c x^2}{2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (a c+b c x^2\right )}{a+b x^2} \, dx=\frac {c x^2}{2} \]
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Time = 2.52 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(\frac {c \,x^{2}}{2}\) | \(7\) |
default | \(\frac {c \,x^{2}}{2}\) | \(7\) |
norman | \(\frac {c \,x^{2}}{2}\) | \(7\) |
risch | \(\frac {c \,x^{2}}{2}\) | \(7\) |
parallelrisch | \(\frac {c \,x^{2}}{2}\) | \(7\) |
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none
Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {x \left (a c+b c x^2\right )}{a+b x^2} \, dx=\frac {1}{2} \, c x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.62 \[ \int \frac {x \left (a c+b c x^2\right )}{a+b x^2} \, dx=\frac {c x^{2}}{2} \]
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none
Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {x \left (a c+b c x^2\right )}{a+b x^2} \, dx=\frac {1}{2} \, c x^{2} \]
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none
Time = 0.27 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {x \left (a c+b c x^2\right )}{a+b x^2} \, dx=\frac {1}{2} \, c x^{2} \]
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Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {x \left (a c+b c x^2\right )}{a+b x^2} \, dx=\frac {c\,x^2}{2} \]
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