Optimal. Leaf size=29 \[ \frac{1}{2} x^2 (A c+b B)+A b \log (x)+\frac{1}{4} B c x^4 \]
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Rubi [A] time = 0.027211, 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} \[ \frac{1}{2} x^2 (A c+b B)+A b \log (x)+\frac{1}{4} B c x^4 \]
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^3} \, dx &=\int \frac{\left (A+B x^2\right ) \left (b+c x^2\right )}{x} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) (b+c x)}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (b B+A c+\frac{A b}{x}+B c x\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} (b B+A c) x^2+\frac{1}{4} B c x^4+A b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0086492, size = 29, normalized size = 1. \[ \frac{1}{2} x^2 (A c+b B)+A b \log (x)+\frac{1}{4} B c x^4 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 28, normalized size = 1. \begin{align*}{\frac{Bc{x}^{4}}{4}}+{\frac{A{x}^{2}c}{2}}+{\frac{B{x}^{2}b}{2}}+Ab\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.32364, size = 38, normalized size = 1.31 \begin{align*} \frac{1}{4} \, B c x^{4} + \frac{1}{2} \,{\left (B b + A c\right )} x^{2} + \frac{1}{2} \, A b \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.495015, size = 65, normalized size = 2.24 \begin{align*} \frac{1}{4} \, B c x^{4} + \frac{1}{2} \,{\left (B b + A c\right )} x^{2} + A b \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.255003, size = 27, normalized size = 0.93 \begin{align*} A b \log{\left (x \right )} + \frac{B c x^{4}}{4} + x^{2} \left (\frac{A c}{2} + \frac{B b}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26228, size = 41, normalized size = 1.41 \begin{align*} \frac{1}{4} \, B c x^{4} + \frac{1}{2} \, B b x^{2} + \frac{1}{2} \, A c x^{2} + \frac{1}{2} \, A b \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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