Integrand size = 20, antiderivative size = 34 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )} \, dx=\frac {A \log (x)}{a}-\frac {(A b-a B) \log \left (a+b x^2\right )}{2 a b} \]
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Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 78} \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )} \, dx=\frac {A \log (x)}{a}-\frac {(A b-a B) \log \left (a+b x^2\right )}{2 a b} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x (a+b x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{a x}+\frac {-A b+a B}{a (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {A \log (x)}{a}-\frac {(A b-a B) \log \left (a+b x^2\right )}{2 a b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )} \, dx=\frac {A \log (x)}{a}+\frac {(-A b+a B) \log \left (a+b x^2\right )}{2 a b} \]
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Time = 2.49 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {A \ln \left (x \right )}{a}-\frac {\left (A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a b}\) | \(33\) |
norman | \(\frac {A \ln \left (x \right )}{a}-\frac {\left (A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a b}\) | \(33\) |
risch | \(\frac {A \ln \left (x \right )}{a}-\frac {\ln \left (b \,x^{2}+a \right ) A}{2 a}+\frac {\ln \left (b \,x^{2}+a \right ) B}{2 b}\) | \(37\) |
parallelrisch | \(\frac {2 A \ln \left (x \right ) b -A \ln \left (b \,x^{2}+a \right ) b +B \ln \left (b \,x^{2}+a \right ) a}{2 a b}\) | \(39\) |
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )} \, dx=\frac {2 \, A b \log \left (x\right ) + {\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \]
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Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )} \, dx=\frac {A \log {\left (x \right )}}{a} + \frac {\left (- A b + B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a b} \]
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Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )} \, dx=\frac {A \log \left (x^{2}\right )}{2 \, a} + \frac {{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \]
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )} \, dx=\frac {A \log \left (x^{2}\right )}{2 \, a} + \frac {{\left (B a - A b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a b} \]
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Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )} \, dx=\frac {A\,\ln \left (x\right )}{a}-\frac {\ln \left (b\,x^2+a\right )\,\left (A\,b-B\,a\right )}{2\,a\,b} \]
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