Optimal. Leaf size=49 \[ -\frac{c}{2 b^2 \left (b+c x^2\right )}+\frac{c \log \left (b+c x^2\right )}{b^3}-\frac{2 c \log (x)}{b^3}-\frac{1}{2 b^2 x^2} \]
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Rubi [A] time = 0.0405369, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1584, 266, 44} \[ -\frac{c}{2 b^2 \left (b+c x^2\right )}+\frac{c \log \left (b+c x^2\right )}{b^3}-\frac{2 c \log (x)}{b^3}-\frac{1}{2 b^2 x^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{1}{x^3 \left (b+c x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (b+c x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b^2 x^2}-\frac{2 c}{b^3 x}+\frac{c^2}{b^2 (b+c x)^2}+\frac{2 c^2}{b^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 b^2 x^2}-\frac{c}{2 b^2 \left (b+c x^2\right )}-\frac{2 c \log (x)}{b^3}+\frac{c \log \left (b+c x^2\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0363019, size = 41, normalized size = 0.84 \[ -\frac{b \left (\frac{c}{b+c x^2}+\frac{1}{x^2}\right )-2 c \log \left (b+c x^2\right )+4 c \log (x)}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 46, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{b}^{2}{x}^{2}}}-{\frac{c}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }}-2\,{\frac{c\ln \left ( x \right ) }{{b}^{3}}}+{\frac{c\ln \left ( c{x}^{2}+b \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962003, size = 70, normalized size = 1.43 \begin{align*} -\frac{2 \, c x^{2} + b}{2 \,{\left (b^{2} c x^{4} + b^{3} x^{2}\right )}} + \frac{c \log \left (c x^{2} + b\right )}{b^{3}} - \frac{c \log \left (x^{2}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5285, size = 157, normalized size = 3.2 \begin{align*} -\frac{2 \, b c x^{2} + b^{2} - 2 \,{\left (c^{2} x^{4} + b c x^{2}\right )} \log \left (c x^{2} + b\right ) + 4 \,{\left (c^{2} x^{4} + b c x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{3} c x^{4} + b^{4} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.577194, size = 49, normalized size = 1. \begin{align*} - \frac{b + 2 c x^{2}}{2 b^{3} x^{2} + 2 b^{2} c x^{4}} - \frac{2 c \log{\left (x \right )}}{b^{3}} + \frac{c \log{\left (\frac{b}{c} + x^{2} \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31746, size = 68, normalized size = 1.39 \begin{align*} \frac{c \log \left ({\left | c x^{2} + b \right |}\right )}{b^{3}} - \frac{2 \, c \log \left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, c x^{2} + b}{2 \,{\left (c x^{4} + b x^{2}\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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