Optimal. Leaf size=63 \[ -\frac{a C+A b}{x}-\frac{a A}{3 x^3}-\frac{a B}{2 x^2}+x (A c+b C)+b B \log (x)+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3 \]
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Rubi [A] time = 0.0509742, 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} \[ -\frac{a C+A b}{x}-\frac{a A}{3 x^3}-\frac{a B}{2 x^2}+x (A c+b C)+b B \log (x)+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3 \]
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^4} \, dx &=\int \left (A c \left (1+\frac{b C}{A c}\right )+\frac{a A}{x^4}+\frac{a B}{x^3}+\frac{A b+a C}{x^2}+\frac{b B}{x}+B c x+c C x^2\right ) \, dx\\ &=-\frac{a A}{3 x^3}-\frac{a B}{2 x^2}-\frac{A b+a C}{x}+(A c+b C) x+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3+b B \log (x)\\ \end{align*}
Mathematica [A] time = 0.0466659, size = 60, normalized size = 0.95 \[ -\frac{a (2 A+3 x (B+2 C x))}{6 x^3}-\frac{A b}{x}+A c x+b B \log (x)+b C x+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 57, normalized size = 0.9 \begin{align*}{\frac{cC{x}^{3}}{3}}+{\frac{Bc{x}^{2}}{2}}+Acx+bCx-{\frac{Ab}{x}}-{\frac{aC}{x}}-{\frac{Ba}{2\,{x}^{2}}}-{\frac{Aa}{3\,{x}^{3}}}+bB\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959429, size = 76, normalized size = 1.21 \begin{align*} \frac{1}{3} \, C c x^{3} + \frac{1}{2} \, B c x^{2} + B b \log \left (x\right ) +{\left (C b + A c\right )} x - \frac{3 \, B a x + 6 \,{\left (C a + A b\right )} x^{2} + 2 \, A a}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26255, size = 149, normalized size = 2.37 \begin{align*} \frac{2 \, C c x^{6} + 3 \, B c x^{5} + 6 \, B b x^{3} \log \left (x\right ) + 6 \,{\left (C b + A c\right )} x^{4} - 3 \, B a x - 6 \,{\left (C a + A b\right )} x^{2} - 2 \, A a}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.01312, size = 61, normalized size = 0.97 \begin{align*} B b \log{\left (x \right )} + \frac{B c x^{2}}{2} + \frac{C c x^{3}}{3} + x \left (A c + C b\right ) - \frac{2 A a + 3 B a x + x^{2} \left (6 A b + 6 C a\right )}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10642, size = 76, normalized size = 1.21 \begin{align*} \frac{1}{3} \, C c x^{3} + \frac{1}{2} \, B c x^{2} + C b x + A c x + B b \log \left ({\left | x \right |}\right ) - \frac{3 \, B a x + 6 \,{\left (C a + A b\right )} x^{2} + 2 \, A a}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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