Integrand size = 20, antiderivative size = 51 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^2} \, dx=\frac {A b-a B}{2 a b \left (a+b x^2\right )}+\frac {A \log (x)}{a^2}-\frac {A \log \left (a+b x^2\right )}{2 a^2} \]
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Time = 0.03 (sec) , antiderivative size = 51, 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} \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^2} \, dx=-\frac {A \log \left (a+b x^2\right )}{2 a^2}+\frac {A \log (x)}{a^2}+\frac {A b-a B}{2 a b \left (a+b x^2\right )} \]
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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)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{a^2 x}+\frac {-A b+a B}{a (a+b x)^2}-\frac {A b}{a^2 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {A b-a B}{2 a b \left (a+b x^2\right )}+\frac {A \log (x)}{a^2}-\frac {A \log \left (a+b x^2\right )}{2 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^2} \, dx=\frac {\frac {a (A b-a B)}{b \left (a+b x^2\right )}+2 A \log (x)-A \log \left (a+b x^2\right )}{2 a^2} \]
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Time = 2.50 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {A \ln \left (x \right )}{a^{2}}-\frac {A \ln \left (b \,x^{2}+a \right )-\frac {a \left (A b -B a \right )}{b \left (b \,x^{2}+a \right )}}{2 a^{2}}\) | \(48\) |
norman | \(-\frac {\left (A b -B a \right ) x^{2}}{2 a^{2} \left (b \,x^{2}+a \right )}+\frac {A \ln \left (x \right )}{a^{2}}-\frac {A \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(48\) |
risch | \(\frac {A}{2 a \left (b \,x^{2}+a \right )}-\frac {B}{2 b \left (b \,x^{2}+a \right )}+\frac {A \ln \left (x \right )}{a^{2}}-\frac {A \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(53\) |
parallelrisch | \(\frac {2 A \ln \left (x \right ) x^{2} b -A \ln \left (b \,x^{2}+a \right ) x^{2} b -A b \,x^{2}+B a \,x^{2}+2 a A \ln \left (x \right )-A \ln \left (b \,x^{2}+a \right ) a}{2 a^{2} \left (b \,x^{2}+a \right )}\) | \(71\) |
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Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^2} \, dx=-\frac {B a^{2} - A a b + {\left (A b^{2} x^{2} + A a b\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (A b^{2} x^{2} + A a b\right )} \log \left (x\right )}{2 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}} \]
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Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^2} \, dx=\frac {A \log {\left (x \right )}}{a^{2}} - \frac {A \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2}} + \frac {A b - B a}{2 a^{2} b + 2 a b^{2} x^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^2} \, dx=-\frac {B a - A b}{2 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} - \frac {A \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {A \log \left (x^{2}\right )}{2 \, a^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^2} \, dx=\frac {A \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {A \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac {A b^{2} x^{2} - B a^{2} + 2 \, A a b}{2 \, {\left (b x^{2} + a\right )} a^{2} b} \]
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Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2}{x \left (a+b x^2\right )^2} \, dx=\frac {A\,\ln \left (x\right )}{a^2}-\frac {A\,\ln \left (b\,x^2+a\right )}{2\,a^2}+\frac {A\,b-B\,a}{2\,a\,b\,\left (b\,x^2+a\right )} \]
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