Optimal. Leaf size=29 \[ \log (x) (A c+b B)-\frac{A b}{2 x^2}+\frac{1}{2} B c x^2 \]
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Rubi [A] time = 0.0275993, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1584, 446, 76} \[ \log (x) (A c+b B)-\frac{A b}{2 x^2}+\frac{1}{2} B c x^2 \]
Antiderivative was successfully verified.
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Rule 1584
Rule 446
Rule 76
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x^5} \, dx &=\int \frac{\left (A+B x^2\right ) \left (b+c x^2\right )}{x^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) (b+c x)}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (B c+\frac{A b}{x^2}+\frac{b B+A c}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{A b}{2 x^2}+\frac{1}{2} B c x^2+(b B+A c) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0108132, size = 29, normalized size = 1. \[ \log (x) (A c+b B)-\frac{A b}{2 x^2}+\frac{1}{2} B c x^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 26, normalized size = 0.9 \begin{align*}{\frac{Bc{x}^{2}}{2}}+A\ln \left ( x \right ) c+B\ln \left ( x \right ) b-{\frac{Ab}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16114, size = 38, normalized size = 1.31 \begin{align*} \frac{1}{2} \, B c x^{2} + \frac{1}{2} \,{\left (B b + A c\right )} \log \left (x^{2}\right ) - \frac{A b}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.52777, size = 70, normalized size = 2.41 \begin{align*} \frac{B c x^{4} + 2 \,{\left (B b + A c\right )} x^{2} \log \left (x\right ) - A b}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.326274, size = 26, normalized size = 0.9 \begin{align*} - \frac{A b}{2 x^{2}} + \frac{B c x^{2}}{2} + \left (A c + B b\right ) \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17334, size = 57, normalized size = 1.97 \begin{align*} \frac{1}{2} \, B c x^{2} + \frac{1}{2} \,{\left (B b + A c\right )} \log \left (x^{2}\right ) - \frac{B b x^{2} + A c x^{2} + A b}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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