Optimal. Leaf size=22 \[ \frac {\log (x)}{b}-\frac {\log \left (b+c x^2\right )}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1584, 266, 36, 29, 31} \begin {gather*} \frac {\log (x)}{b}-\frac {\log \left (b+c x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {x}{b x^2+c x^4} \, dx &=\int \frac {1}{x \left (b+c x^2\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (b+c x)} \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 b}-\frac {c \operatorname {Subst}\left (\int \frac {1}{b+c x} \, dx,x,x^2\right )}{2 b}\\ &=\frac {\log (x)}{b}-\frac {\log \left (b+c x^2\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{b}-\frac {\log \left (b+c x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{b x^2+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.62, size = 18, normalized size = 0.82 \begin {gather*} -\frac {\log \left (c x^{2} + b\right ) - 2 \, \log \relax (x)}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 22, normalized size = 1.00 \begin {gather*} -\frac {\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b} + \frac {\log \left ({\left | x \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.95 \begin {gather*} \frac {\ln \relax (x )}{b}-\frac {\ln \left (c \,x^{2}+b \right )}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 23, normalized size = 1.05 \begin {gather*} -\frac {\log \left (c x^{2} + b\right )}{2 \, b} + \frac {\log \left (x^{2}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 18, normalized size = 0.82 \begin {gather*} -\frac {\ln \left (c\,x^2+b\right )-2\,\ln \relax (x)}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 15, normalized size = 0.68 \begin {gather*} \frac {\log {\relax (x )}}{b} - \frac {\log {\left (\frac {b}{c} + x^{2} \right )}}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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