Optimal. Leaf size=63 \[ \log (x) (a C+A b)-\frac{a A}{2 x^2}-\frac{a B}{x}+\frac{1}{2} x^2 (A c+b C)+b B x+\frac{1}{3} B c x^3+\frac{1}{4} c C x^4 \]
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Rubi [A] time = 0.048346, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1628} \[ \log (x) (a C+A b)-\frac{a A}{2 x^2}-\frac{a B}{x}+\frac{1}{2} x^2 (A c+b C)+b B x+\frac{1}{3} B c x^3+\frac{1}{4} c C x^4 \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin{align*} \int \frac{\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^3} \, dx &=\int \left (b B+\frac{a A}{x^3}+\frac{a B}{x^2}+\frac{A b+a C}{x}+(A c+b C) x+B c x^2+c C x^3\right ) \, dx\\ &=-\frac{a A}{2 x^2}-\frac{a B}{x}+b B x+\frac{1}{2} (A c+b C) x^2+\frac{1}{3} B c x^3+\frac{1}{4} c C x^4+(A b+a C) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0398518, size = 58, normalized size = 0.92 \[ \log (x) (a C+A b)-\frac{a (A+2 B x)}{2 x^2}+\frac{1}{12} x \left (c x \left (6 A+4 B x+3 C x^2\right )+6 b (2 B+C x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 58, normalized size = 0.9 \begin{align*}{\frac{cC{x}^{4}}{4}}+{\frac{Bc{x}^{3}}{3}}+{\frac{A{x}^{2}c}{2}}+{\frac{C{x}^{2}b}{2}}+bBx-{\frac{Ba}{x}}-{\frac{Aa}{2\,{x}^{2}}}+A\ln \left ( x \right ) b+C\ln \left ( x \right ) a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945481, size = 74, normalized size = 1.17 \begin{align*} \frac{1}{4} \, C c x^{4} + \frac{1}{3} \, B c x^{3} + B b x + \frac{1}{2} \,{\left (C b + A c\right )} x^{2} +{\left (C a + A b\right )} \log \left (x\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21064, size = 154, normalized size = 2.44 \begin{align*} \frac{3 \, C c x^{6} + 4 \, B c x^{5} + 12 \, B b x^{3} + 6 \,{\left (C b + A c\right )} x^{4} + 12 \,{\left (C a + A b\right )} x^{2} \log \left (x\right ) - 12 \, B a x - 6 \, A a}{12 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.670233, size = 60, normalized size = 0.95 \begin{align*} B b x + \frac{B c x^{3}}{3} + \frac{C c x^{4}}{4} + x^{2} \left (\frac{A c}{2} + \frac{C b}{2}\right ) + \left (A b + C a\right ) \log{\left (x \right )} - \frac{A a + 2 B a x}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08334, size = 78, normalized size = 1.24 \begin{align*} \frac{1}{4} \, C c x^{4} + \frac{1}{3} \, B c x^{3} + \frac{1}{2} \, C b x^{2} + \frac{1}{2} \, A c x^{2} + B b x +{\left (C a + A b\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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