Optimal. Leaf size=53 \[ -\frac{b c}{2 a^2 \left (a+b x^2\right )}+\frac{b c \log \left (a+b x^2\right )}{a^3}-\frac{2 b c \log (x)}{a^3}-\frac{c}{2 a^2 x^2} \]
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Rubi [A] time = 0.0410684, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {21, 266, 44} \[ -\frac{b c}{2 a^2 \left (a+b x^2\right )}+\frac{b c \log \left (a+b x^2\right )}{a^3}-\frac{2 b c \log (x)}{a^3}-\frac{c}{2 a^2 x^2} \]
Antiderivative was successfully verified.
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Rule 21
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{a c+b c x^2}{x^3 \left (a+b x^2\right )^3} \, dx &=c \int \frac{1}{x^3 \left (a+b x^2\right )^2} \, dx\\ &=\frac{1}{2} c \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} c \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^2}-\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a+b x)^2}+\frac{2 b^2}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{c}{2 a^2 x^2}-\frac{b c}{2 a^2 \left (a+b x^2\right )}-\frac{2 b c \log (x)}{a^3}+\frac{b c \log \left (a+b x^2\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0352622, size = 42, normalized size = 0.79 \[ -\frac{c \left (a \left (\frac{b}{a+b x^2}+\frac{1}{x^2}\right )-2 b \log \left (a+b x^2\right )+4 b \log (x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 50, normalized size = 0.9 \begin{align*} -{\frac{c}{2\,{a}^{2}{x}^{2}}}-{\frac{bc}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{bc\ln \left ( x \right ) }{{a}^{3}}}+{\frac{bc\ln \left ( b{x}^{2}+a \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02034, size = 77, normalized size = 1.45 \begin{align*} -\frac{2 \, b c x^{2} + a c}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac{b c \log \left (b x^{2} + a\right )}{a^{3}} - \frac{b c \log \left (x^{2}\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24731, size = 173, normalized size = 3.26 \begin{align*} -\frac{2 \, a b c x^{2} + a^{2} c - 2 \,{\left (b^{2} c x^{4} + a b c x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left (b^{2} c x^{4} + a b c 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 = 0.529049, size = 51, normalized size = 0.96 \begin{align*} c \left (- \frac{a + 2 b x^{2}}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{b \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15864, size = 76, normalized size = 1.43 \begin{align*} -\frac{b c \log \left (x^{2}\right )}{a^{3}} + \frac{b c \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3}} - \frac{2 \, b c x^{2} + a c}{2 \,{\left (b x^{4} + a x^{2}\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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