Optimal. Leaf size=35 \[ \frac{c \log \left (b+c x^2\right )}{2 b^2}-\frac{c \log (x)}{b^2}-\frac{1}{2 b x^2} \]
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Rubi [A] time = 0.0275042, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1584, 266, 44} \[ \frac{c \log \left (b+c x^2\right )}{2 b^2}-\frac{c \log (x)}{b^2}-\frac{1}{2 b x^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (b x^2+c x^4\right )} \, dx &=\int \frac{1}{x^3 \left (b+c x^2\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (b+c x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{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*}
Mathematica [A] time = 0.0068712, size = 35, normalized size = 1. \[ \frac{c \log \left (b+c x^2\right )}{2 b^2}-\frac{c \log (x)}{b^2}-\frac{1}{2 b x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 32, normalized size = 0.9 \begin{align*} -{\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955562, size = 45, normalized size = 1.29 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50013, size = 80, normalized size = 2.29 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.468961, size = 31, normalized size = 0.89 \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22322, size = 58, normalized size = 1.66 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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