Integrand size = 17, antiderivative size = 35 \[ \int \frac {1}{x \left (b x^2+c x^4\right )} \, dx=-\frac {1}{2 b x^2}-\frac {c \log (x)}{b^2}+\frac {c \log \left (b+c x^2\right )}{2 b^2} \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 272, 46} \[ \int \frac {1}{x \left (b x^2+c x^4\right )} \, dx=\frac {c \log \left (b+c x^2\right )}{2 b^2}-\frac {c \log (x)}{b^2}-\frac {1}{2 b x^2} \]
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Rule 46
Rule 272
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^3 \left (b+c x^2\right )} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (b+c x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b x^2}-\frac {c}{b^2 x}+\frac {c^2}{b^2 (b+c x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {1}{2 b x^2}-\frac {c \log (x)}{b^2}+\frac {c \log \left (b+c x^2\right )}{2 b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (b x^2+c x^4\right )} \, dx=-\frac {1}{2 b x^2}-\frac {c \log (x)}{b^2}+\frac {c \log \left (b+c x^2\right )}{2 b^2} \]
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Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {1}{2 b \,x^{2}}-\frac {c \ln \left (x \right )}{b^{2}}+\frac {c \ln \left (c \,x^{2}+b \right )}{2 b^{2}}\) | \(32\) |
norman | \(-\frac {1}{2 b \,x^{2}}-\frac {c \ln \left (x \right )}{b^{2}}+\frac {c \ln \left (c \,x^{2}+b \right )}{2 b^{2}}\) | \(32\) |
parallelrisch | \(-\frac {2 c \ln \left (x \right ) x^{2}-c \ln \left (c \,x^{2}+b \right ) x^{2}+b}{2 b^{2} x^{2}}\) | \(33\) |
risch | \(-\frac {1}{2 b \,x^{2}}-\frac {c \ln \left (x \right )}{b^{2}}+\frac {c \ln \left (-c \,x^{2}-b \right )}{2 b^{2}}\) | \(35\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x \left (b x^2+c x^4\right )} \, dx=\frac {c x^{2} \log \left (c x^{2} + b\right ) - 2 \, c x^{2} \log \left (x\right ) - b}{2 \, b^{2} x^{2}} \]
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Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (b x^2+c x^4\right )} \, dx=- \frac {1}{2 b x^{2}} - \frac {c \log {\left (x \right )}}{b^{2}} + \frac {c \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x \left (b x^2+c x^4\right )} \, dx=\frac {c \log \left (c x^{2} + b\right )}{2 \, b^{2}} - \frac {c \log \left (x^{2}\right )}{2 \, b^{2}} - \frac {1}{2 \, b x^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x \left (b x^2+c x^4\right )} \, dx=-\frac {c \log \left (x^{2}\right )}{2 \, b^{2}} + \frac {c \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac {c x^{2} - b}{2 \, b^{2} x^{2}} \]
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Time = 12.95 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (b x^2+c x^4\right )} \, dx=\frac {c\,\ln \left (c\,x^2+b\right )}{2\,b^2}-\frac {1}{2\,b\,x^2}-\frac {c\,\ln \left (x\right )}{b^2} \]
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