Optimal. Leaf size=29 \[ -\frac{A c+b B}{2 x^2}-\frac{A b}{4 x^4}+B c \log (x) \]
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Rubi [A] time = 0.0269632, 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} \[ -\frac{A c+b B}{2 x^2}-\frac{A b}{4 x^4}+B c \log (x) \]
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^7} \, dx &=\int \frac{\left (A+B x^2\right ) \left (b+c x^2\right )}{x^5} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) (b+c x)}{x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A b}{x^3}+\frac{b B+A c}{x^2}+\frac{B c}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{A b}{4 x^4}-\frac{b B+A c}{2 x^2}+B c \log (x)\\ \end{align*}
Mathematica [A] time = 0.0168233, size = 31, normalized size = 1.07 \[ \frac{-A c-b B}{2 x^2}-\frac{A b}{4 x^4}+B c \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 28, normalized size = 1. \begin{align*} Bc\ln \left ( x \right ) -{\frac{Ab}{4\,{x}^{4}}}-{\frac{Ac}{2\,{x}^{2}}}-{\frac{Bb}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09194, size = 41, normalized size = 1.41 \begin{align*} \frac{1}{2} \, B c \log \left (x^{2}\right ) - \frac{2 \,{\left (B b + A c\right )} x^{2} + A b}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.495456, size = 73, normalized size = 2.52 \begin{align*} \frac{4 \, B c x^{4} \log \left (x\right ) - 2 \,{\left (B b + A c\right )} x^{2} - A b}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.496881, size = 27, normalized size = 0.93 \begin{align*} B c \log{\left (x \right )} - \frac{A b + x^{2} \left (2 A c + 2 B b\right )}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21355, size = 53, normalized size = 1.83 \begin{align*} \frac{1}{2} \, B c \log \left (x^{2}\right ) - \frac{3 \, B c x^{4} + 2 \, B b x^{2} + 2 \, A c x^{2} + A b}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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