Optimal. Leaf size=15 \[ \frac {1}{2} \log \left (b+c x^2\right )+\log (x) \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1593, 446, 72} \[ \frac {1}{2} \log \left (b+c x^2\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 1593
Rubi steps
\begin {align*} \int \frac {b+2 c x^2}{b x+c x^3} \, dx &=\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*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {1}{2} \log \left (b+c x^2\right )+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 13, normalized size = 0.87 \[ \frac {1}{2} \, \log \left (c x^{2} + b\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 18, normalized size = 1.20 \[ \frac {1}{2} \, \log \left (x^{2}\right ) + \frac {1}{2} \, \log \left ({\left | c x^{2} + b \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.93 \[ \ln \relax (x )+\frac {\ln \left (c \,x^{2}+b \right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 13, normalized size = 0.87 \[ \frac {1}{2} \, \log \left (c x^{2} + b\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.08, size = 13, normalized size = 0.87 \[ \frac {\ln \left (c\,x^2+b\right )}{2}+\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 12, normalized size = 0.80 \[ \log {\relax (x )} + \frac {\log {\left (\frac {b}{c} + x^{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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