Integrand size = 23, antiderivative size = 24 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^2} \, dx=\frac {c \log (x)}{a}-\frac {c \log \left (a+b x^2\right )}{2 a} \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {21, 272, 36, 29, 31} \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^2} \, dx=\frac {c \log (x)}{a}-\frac {c \log \left (a+b x^2\right )}{2 a} \]
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Rule 21
Rule 29
Rule 31
Rule 36
Rule 272
Rubi steps \begin{align*} \text {integral}& = c \int \frac {1}{x \left (a+b x^2\right )} \, dx \\ & = \frac {1}{2} c \text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,x^2\right ) \\ & = \frac {c \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 a}-\frac {(b c) \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^2\right )}{2 a} \\ & = \frac {c \log (x)}{a}-\frac {c \log \left (a+b x^2\right )}{2 a} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^2} \, dx=c \left (\frac {\log (x)}{a}-\frac {\log \left (a+b x^2\right )}{2 a}\right ) \]
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Time = 2.54 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
default | \(c \left (\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a}\right )\) | \(23\) |
norman | \(\frac {c \ln \left (x \right )}{a}-\frac {c \ln \left (b \,x^{2}+a \right )}{2 a}\) | \(23\) |
risch | \(\frac {c \ln \left (x \right )}{a}-\frac {c \ln \left (b \,x^{2}+a \right )}{2 a}\) | \(23\) |
parallelrisch | \(\frac {2 c \ln \left (x \right )-c \ln \left (b \,x^{2}+a \right )}{2 a}\) | \(23\) |
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none
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^2} \, dx=-\frac {c \log \left (b x^{2} + a\right ) - 2 \, c \log \left (x\right )}{2 \, a} \]
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Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^2} \, dx=c \left (\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a}\right ) \]
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none
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^2} \, dx=-\frac {c \log \left (b x^{2} + a\right )}{2 \, a} + \frac {c \log \left (x^{2}\right )}{2 \, a} \]
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none
Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^2} \, dx=\frac {c \log \left (x^{2}\right )}{2 \, a} - \frac {c \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a} \]
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Time = 5.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^2} \, dx=-\frac {c\,\left (\ln \left (b\,x^2+a\right )-2\,\ln \left (x\right )\right )}{2\,a} \]
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