Optimal. Leaf size=76 \[ -\frac{b c-a d}{2 a^2 \left (a+b x^2\right )}+\frac{(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3}-\frac{\log (x) (2 b c-a d)}{a^3}-\frac{c}{2 a^2 x^2} \]
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Rubi [A] time = 0.0740186, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ -\frac{b c-a d}{2 a^2 \left (a+b x^2\right )}+\frac{(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3}-\frac{\log (x) (2 b c-a d)}{a^3}-\frac{c}{2 a^2 x^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{c+d x^2}{x^3 \left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{c+d x}{x^2 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{c}{a^2 x^2}+\frac{-2 b c+a d}{a^3 x}-\frac{b (-b c+a d)}{a^2 (a+b x)^2}-\frac{b (-2 b c+a d)}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{c}{2 a^2 x^2}-\frac{b c-a d}{2 a^2 \left (a+b x^2\right )}-\frac{(2 b c-a d) \log (x)}{a^3}+\frac{(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0458944, size = 64, normalized size = 0.84 \[ \frac{\frac{a (a d-b c)}{a+b x^2}+(2 b c-a d) \log \left (a+b x^2\right )+2 \log (x) (a d-2 b c)-\frac{a c}{x^2}}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 86, normalized size = 1.1 \begin{align*} -{\frac{c}{2\,{a}^{2}{x}^{2}}}+{\frac{\ln \left ( x \right ) d}{{a}^{2}}}-2\,{\frac{bc\ln \left ( x \right ) }{{a}^{3}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) d}{2\,{a}^{2}}}+{\frac{bc\ln \left ( b{x}^{2}+a \right ) }{{a}^{3}}}+{\frac{d}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bc}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09233, size = 105, normalized size = 1.38 \begin{align*} -\frac{{\left (2 \, b c - a d\right )} x^{2} + a c}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac{{\left (2 \, b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3}} - \frac{{\left (2 \, b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55478, size = 248, normalized size = 3.26 \begin{align*} -\frac{a^{2} c +{\left (2 \, a b c - a^{2} d\right )} x^{2} -{\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} +{\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} +{\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00116, size = 70, normalized size = 0.92 \begin{align*} \frac{- a c + x^{2} \left (a d - 2 b c\right )}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} + \frac{\left (a d - 2 b c\right ) \log{\left (x \right )}}{a^{3}} - \frac{\left (a d - 2 b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16426, size = 113, normalized size = 1.49 \begin{align*} -\frac{{\left (2 \, b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac{2 \, b c x^{2} - a d x^{2} + a c}{2 \,{\left (b x^{4} + a x^{2}\right )} a^{2}} + \frac{{\left (2 \, b^{2} c - a b d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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