Integrand size = 20, antiderivative size = 34 \[ \int \frac {c+d x^2}{x \left (a+b x^2\right )} \, dx=\frac {c \log (x)}{a}-\frac {(b c-a d) \log \left (a+b x^2\right )}{2 a b} \]
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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} \[ \int \frac {c+d x^2}{x \left (a+b x^2\right )} \, dx=\frac {c \log (x)}{a}-\frac {(b c-a d) \log \left (a+b x^2\right )}{2 a b} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {c+d x}{x (a+b x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {c}{a x}+\frac {-b c+a d}{a (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {c \log (x)}{a}-\frac {(b c-a d) \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 \[ \int \frac {c+d x^2}{x \left (a+b x^2\right )} \, dx=\frac {c \log (x)}{a}+\frac {(-b c+a d) \log \left (a+b x^2\right )}{2 a b} \]
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Time = 2.57 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {c \ln \left (x \right )}{a}+\frac {\left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a b}\) | \(33\) |
norman | \(\frac {c \ln \left (x \right )}{a}+\frac {\left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a b}\) | \(33\) |
parallelrisch | \(\frac {2 c \ln \left (x \right ) b +\ln \left (b \,x^{2}+a \right ) a d -\ln \left (b \,x^{2}+a \right ) b c}{2 a b}\) | \(39\) |
risch | \(\frac {c \ln \left (x \right )}{a}+\frac {\ln \left (-b \,x^{2}-a \right ) d}{2 b}-\frac {\ln \left (-b \,x^{2}-a \right ) c}{2 a}\) | \(43\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^2}{x \left (a+b x^2\right )} \, dx=\frac {2 \, b c \log \left (x\right ) - {\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \]
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Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {c+d x^2}{x \left (a+b x^2\right )} \, dx=\frac {c \log {\left (x \right )}}{a} + \frac {\left (a d - b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a b} \]
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Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x^2}{x \left (a+b x^2\right )} \, dx=\frac {c \log \left (x^{2}\right )}{2 \, a} - \frac {{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \]
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x^2}{x \left (a+b x^2\right )} \, dx=\frac {c \log \left (x^{2}\right )}{2 \, a} - \frac {{\left (b c - a d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a b} \]
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Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^2}{x \left (a+b x^2\right )} \, dx=\frac {c\,\ln \left (x\right )}{a}+\frac {\ln \left (b\,x^2+a\right )\,\left (a\,d-b\,c\right )}{2\,a\,b} \]
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