Optimal. Leaf size=34 \[ \frac {(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac {A \log (x)}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1584, 446, 72} \[ \frac {(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac {A \log (x)}{b} \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {x \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac {A+B x^2}{x \left (b+c x^2\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x (b+c x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A}{b x}+\frac {b B-A c}{b (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {A \log (x)}{b}+\frac {(b B-A c) \log \left (b+c x^2\right )}{2 b c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \[ \frac {(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac {A \log (x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 32, normalized size = 0.94 \[ \frac {2 \, A c \log \relax (x) + {\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 34, normalized size = 1.00 \[ \frac {A \log \left ({\left | x \right |}\right )}{b} + \frac {{\left (B b - A c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 37, normalized size = 1.09 \[ \frac {A \ln \relax (x )}{b}-\frac {A \ln \left (c \,x^{2}+b \right )}{2 b}+\frac {B \ln \left (c \,x^{2}+b \right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 35, normalized size = 1.03 \[ \frac {A \log \left (x^{2}\right )}{2 \, b} + \frac {{\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 32, normalized size = 0.94 \[ \frac {A\,\ln \relax (x)}{b}-\frac {\ln \left (c\,x^2+b\right )\,\left (A\,c-B\,b\right )}{2\,b\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 26, normalized size = 0.76 \[ \frac {A \log {\relax (x )}}{b} + \frac {\left (- A c + B b\right ) \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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