Optimal. Leaf size=49 \[ -\frac{(b B-A c) \log \left (b+c x^2\right )}{2 b^2}+\frac{\log (x) (b B-A c)}{b^2}-\frac{A}{2 b x^2} \]
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Rubi [A] time = 0.0557434, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1584, 446, 77} \[ -\frac{(b B-A c) \log \left (b+c x^2\right )}{2 b^2}+\frac{\log (x) (b B-A c)}{b^2}-\frac{A}{2 b x^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x \left (b x^2+c x^4\right )} \, dx &=\int \frac{A+B x^2}{x^3 \left (b+c x^2\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 (b+c x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A}{b x^2}+\frac{b B-A c}{b^2 x}-\frac{c (b B-A c)}{b^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{A}{2 b x^2}+\frac{(b B-A c) \log (x)}{b^2}-\frac{(b B-A c) \log \left (b+c x^2\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.020931, size = 49, normalized size = 1. \[ \frac{(A c-b B) \log \left (b+c x^2\right )}{2 b^2}+\frac{\log (x) (b B-A c)}{b^2}-\frac{A}{2 b x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 1.1 \begin{align*} -{\frac{A}{2\,b{x}^{2}}}-{\frac{A\ln \left ( x \right ) c}{{b}^{2}}}+{\frac{\ln \left ( x \right ) B}{b}}+{\frac{\ln \left ( c{x}^{2}+b \right ) Ac}{2\,{b}^{2}}}-{\frac{\ln \left ( c{x}^{2}+b \right ) B}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16714, size = 65, normalized size = 1.33 \begin{align*} -\frac{{\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, b^{2}} + \frac{{\left (B b - A c\right )} \log \left (x^{2}\right )}{2 \, b^{2}} - \frac{A}{2 \, b x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.619287, size = 111, normalized size = 2.27 \begin{align*} -\frac{{\left (B b - A c\right )} x^{2} \log \left (c x^{2} + b\right ) - 2 \,{\left (B b - A c\right )} x^{2} \log \left (x\right ) + A 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 = 0.815655, size = 41, normalized size = 0.84 \begin{align*} - \frac{A}{2 b x^{2}} + \frac{\left (- A c + B b\right ) \log{\left (x \right )}}{b^{2}} - \frac{\left (- A c + B b\right ) \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 = 1.20614, size = 96, normalized size = 1.96 \begin{align*} \frac{{\left (B b - A c\right )} \log \left (x^{2}\right )}{2 \, b^{2}} - \frac{{\left (B b c - A c^{2}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{2} c} - \frac{B b x^{2} - A c x^{2} + A b}{2 \, b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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