Optimal. Leaf size=97 \[ \frac{b B-2 A c}{2 b^3 \left (b+c x^2\right )}+\frac{b B-A c}{4 b^2 \left (b+c x^2\right )^2}-\frac{(b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}+\frac{\log (x) (b B-3 A c)}{b^4}-\frac{A}{2 b^3 x^2} \]
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Rubi [A] time = 0.117311, antiderivative size = 97, 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} \[ \frac{b B-2 A c}{2 b^3 \left (b+c x^2\right )}+\frac{b B-A c}{4 b^2 \left (b+c x^2\right )^2}-\frac{(b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}+\frac{\log (x) (b B-3 A c)}{b^4}-\frac{A}{2 b^3 x^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^3 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{A+B x^2}{x^3 \left (b+c x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 (b+c x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A}{b^3 x^2}+\frac{b B-3 A c}{b^4 x}-\frac{c (b B-A c)}{b^2 (b+c x)^3}-\frac{c (b B-2 A c)}{b^3 (b+c x)^2}-\frac{c (b B-3 A c)}{b^4 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{A}{2 b^3 x^2}+\frac{b B-A c}{4 b^2 \left (b+c x^2\right )^2}+\frac{b B-2 A c}{2 b^3 \left (b+c x^2\right )}+\frac{(b B-3 A c) \log (x)}{b^4}-\frac{(b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}\\ \end{align*}
Mathematica [A] time = 0.0595325, size = 86, normalized size = 0.89 \[ \frac{\frac{b^2 (b B-A c)}{\left (b+c x^2\right )^2}+\frac{2 b (b B-2 A c)}{b+c x^2}-2 (b B-3 A c) \log \left (b+c x^2\right )+4 \log (x) (b B-3 A c)-\frac{2 A b}{x^2}}{4 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 118, normalized size = 1.2 \begin{align*} -{\frac{A}{2\,{b}^{3}{x}^{2}}}-3\,{\frac{A\ln \left ( x \right ) c}{{b}^{4}}}+{\frac{\ln \left ( x \right ) B}{{b}^{3}}}+{\frac{3\,c\ln \left ( c{x}^{2}+b \right ) A}{2\,{b}^{4}}}-{\frac{\ln \left ( c{x}^{2}+b \right ) B}{2\,{b}^{3}}}-{\frac{Ac}{{b}^{3} \left ( c{x}^{2}+b \right ) }}+{\frac{B}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }}-{\frac{Ac}{4\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{B}{4\,b \left ( c{x}^{2}+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0992, size = 147, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (B b c - 3 \, A c^{2}\right )} x^{4} - 2 \, A b^{2} + 3 \,{\left (B b^{2} - 3 \, A b c\right )} x^{2}}{4 \,{\left (b^{3} c^{2} x^{6} + 2 \, b^{4} c x^{4} + b^{5} x^{2}\right )}} - \frac{{\left (B b - 3 \, A c\right )} \log \left (c x^{2} + b\right )}{2 \, b^{4}} + \frac{{\left (B b - 3 \, A c\right )} \log \left (x^{2}\right )}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.05911, size = 412, normalized size = 4.25 \begin{align*} \frac{2 \,{\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} - 2 \, A b^{3} + 3 \,{\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2} - 2 \,{\left ({\left (B b c^{2} - 3 \, A c^{3}\right )} x^{6} + 2 \,{\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} +{\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2}\right )} \log \left (c x^{2} + b\right ) + 4 \,{\left ({\left (B b c^{2} - 3 \, A c^{3}\right )} x^{6} + 2 \,{\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} +{\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2}\right )} \log \left (x\right )}{4 \,{\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.36029, size = 107, normalized size = 1.1 \begin{align*} \frac{- 2 A b^{2} + x^{4} \left (- 6 A c^{2} + 2 B b c\right ) + x^{2} \left (- 9 A b c + 3 B b^{2}\right )}{4 b^{5} x^{2} + 8 b^{4} c x^{4} + 4 b^{3} c^{2} x^{6}} + \frac{\left (- 3 A c + B b\right ) \log{\left (x \right )}}{b^{4}} - \frac{\left (- 3 A c + B b\right ) \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19155, size = 142, normalized size = 1.46 \begin{align*} \frac{{\left (B b - 3 \, A c\right )} \log \left ({\left | x \right |}\right )}{b^{4}} - \frac{{\left (B b c - 3 \, A c^{2}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4} c} + \frac{2 \,{\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{4} - 2 \, A b^{3} + 3 \,{\left (B b^{3} - 3 \, A b^{2} c\right )} x^{2}}{4 \,{\left (c x^{2} + b\right )}^{2} b^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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