Optimal. Leaf size=63 \[ -\frac{a C+A b}{2 x^2}-\frac{a A}{4 x^4}-\frac{a B}{3 x^3}+\log (x) (A c+b C)-\frac{b B}{x}+B c x+\frac{1}{2} c C x^2 \]
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Rubi [A] time = 0.0510479, 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} \[ -\frac{a C+A b}{2 x^2}-\frac{a A}{4 x^4}-\frac{a B}{3 x^3}+\log (x) (A c+b C)-\frac{b B}{x}+B c x+\frac{1}{2} c C x^2 \]
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^5} \, dx &=\int \left (B c+\frac{a A}{x^5}+\frac{a B}{x^4}+\frac{A b+a C}{x^3}+\frac{b B}{x^2}+\frac{A c+b C}{x}+c C x\right ) \, dx\\ &=-\frac{a A}{4 x^4}-\frac{a B}{3 x^3}-\frac{A b+a C}{2 x^2}-\frac{b B}{x}+B c x+\frac{1}{2} c C x^2+(A c+b C) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0296124, size = 62, normalized size = 0.98 \[ -\frac{a \left (3 A+4 B x+6 C x^2\right )}{12 x^4}+\frac{-A b-2 b B x+c x^3 (2 B+C x)}{2 x^2}+\log (x) (A c+b C) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 58, normalized size = 0.9 \begin{align*}{\frac{cC{x}^{2}}{2}}+Bcx-{\frac{Bb}{x}}-{\frac{Aa}{4\,{x}^{4}}}-{\frac{Ab}{2\,{x}^{2}}}-{\frac{aC}{2\,{x}^{2}}}-{\frac{Ba}{3\,{x}^{3}}}+A\ln \left ( x \right ) c+C\ln \left ( x \right ) b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969201, size = 76, normalized size = 1.21 \begin{align*} \frac{1}{2} \, C c x^{2} + B c x +{\left (C b + A c\right )} \log \left (x\right ) - \frac{12 \, B b x^{3} + 4 \, B a x + 6 \,{\left (C a + A b\right )} x^{2} + 3 \, A a}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29137, size = 154, normalized size = 2.44 \begin{align*} \frac{6 \, C c x^{6} + 12 \, B c x^{5} + 12 \,{\left (C b + A c\right )} x^{4} \log \left (x\right ) - 12 \, B b x^{3} - 4 \, B a x - 6 \,{\left (C a + A b\right )} x^{2} - 3 \, A a}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.98161, size = 61, normalized size = 0.97 \begin{align*} B c x + \frac{C c x^{2}}{2} + \left (A c + C b\right ) \log{\left (x \right )} - \frac{3 A a + 4 B a x + 12 B b x^{3} + x^{2} \left (6 A b + 6 C a\right )}{12 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09652, size = 77, normalized size = 1.22 \begin{align*} \frac{1}{2} \, C c x^{2} + B c x +{\left (C b + A c\right )} \log \left ({\left | x \right |}\right ) - \frac{12 \, B b x^{3} + 4 \, B a x + 6 \,{\left (C a + A b\right )} x^{2} + 3 \, A a}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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