Optimal. Leaf size=15 \[ \frac{1}{2} \log \left (b+c x^2\right )+\log (x) \]
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Rubi [A] time = 0.0151964, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {446, 72} \[ \frac{1}{2} \log \left (b+c x^2\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{b+2 c x^2}{x \left (b+c x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{b+2 c x}{x (b+c x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{x}+\frac{c}{b+c x}\right ) \, dx,x,x^2\right )\\ &=\log (x)+\frac{1}{2} \log \left (b+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0068275, size = 15, normalized size = 1. \[ \frac{1}{2} \log \left (b+c x^2\right )+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 14, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) +{\frac{\ln \left ( c{x}^{2}+b \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.934556, size = 23, normalized size = 1.53 \begin{align*} \frac{1}{2} \, \log \left (c x^{2} + b\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.909115, size = 39, normalized size = 2.6 \begin{align*} \frac{1}{2} \, \log \left (c x^{2} + b\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.313993, size = 12, normalized size = 0.8 \begin{align*} \log{\left (x \right )} + \frac{\log{\left (\frac{b}{c} + x^{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06698, size = 24, normalized size = 1.6 \begin{align*} \frac{1}{2} \, \log \left (x^{2}\right ) + \frac{1}{2} \, \log \left ({\left | c x^{2} + b \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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