Integrand size = 23, antiderivative size = 8 \[ \int \frac {a c+b c x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c}{2 x^2} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {21, 30} \[ \int \frac {a c+b c x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c}{2 x^2} \]
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Rule 21
Rule 30
Rubi steps \begin{align*} \text {integral}& = c \int \frac {1}{x^3} \, dx \\ & = -\frac {c}{2 x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {a c+b c x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c}{2 x^2} \]
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Time = 2.50 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(-\frac {c}{2 x^{2}}\) | \(7\) |
default | \(-\frac {c}{2 x^{2}}\) | \(7\) |
norman | \(-\frac {c}{2 x^{2}}\) | \(7\) |
risch | \(-\frac {c}{2 x^{2}}\) | \(7\) |
parallelrisch | \(-\frac {c}{2 x^{2}}\) | \(7\) |
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none
Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {a c+b c x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c}{2 \, x^{2}} \]
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Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \frac {a c+b c x^2}{x^3 \left (a+b x^2\right )} \, dx=- \frac {c}{2 x^{2}} \]
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none
Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {a c+b c x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c}{2 \, x^{2}} \]
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none
Time = 0.27 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {a c+b c x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c}{2 \, x^{2}} \]
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Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {a c+b c x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c}{2\,x^2} \]
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