Optimal. Leaf size=63 \[ x (a C+A b)-\frac{a A}{x}+a B \log (x)+\frac{1}{3} x^3 (A c+b C)+\frac{1}{2} b B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5 \]
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Rubi [A] time = 0.0505412, 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} \[ x (a C+A b)-\frac{a A}{x}+a B \log (x)+\frac{1}{3} x^3 (A c+b C)+\frac{1}{2} b B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5 \]
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^2} \, dx &=\int \left (A b \left (1+\frac{a C}{A b}\right )+\frac{a A}{x^2}+\frac{a B}{x}+b B x+(A c+b C) x^2+B c x^3+c C x^4\right ) \, dx\\ &=-\frac{a A}{x}+(A b+a C) x+\frac{1}{2} b B x^2+\frac{1}{3} (A c+b C) x^3+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5+a B \log (x)\\ \end{align*}
Mathematica [A] time = 0.0230996, size = 63, normalized size = 1. \[ x (a C+A b)-\frac{a A}{x}+a B \log (x)+\frac{1}{3} x^3 (A c+b C)+\frac{1}{2} b B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 57, normalized size = 0.9 \begin{align*}{\frac{cC{x}^{5}}{5}}+{\frac{Bc{x}^{4}}{4}}+{\frac{A{x}^{3}c}{3}}+{\frac{C{x}^{3}b}{3}}+{\frac{bB{x}^{2}}{2}}+Abx+aCx-{\frac{Aa}{x}}+aB\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965118, size = 74, normalized size = 1.17 \begin{align*} \frac{1}{5} \, C c x^{5} + \frac{1}{4} \, B c x^{4} + \frac{1}{2} \, B b x^{2} + \frac{1}{3} \,{\left (C b + A c\right )} x^{3} + B a \log \left (x\right ) +{\left (C a + A b\right )} x - \frac{A a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28934, size = 157, normalized size = 2.49 \begin{align*} \frac{12 \, C c x^{6} + 15 \, B c x^{5} + 30 \, B b x^{3} + 20 \,{\left (C b + A c\right )} x^{4} + 60 \, B a x \log \left (x\right ) + 60 \,{\left (C a + A b\right )} x^{2} - 60 \, A a}{60 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.49202, size = 58, normalized size = 0.92 \begin{align*} - \frac{A a}{x} + B a \log{\left (x \right )} + \frac{B b x^{2}}{2} + \frac{B c x^{4}}{4} + \frac{C c x^{5}}{5} + x^{3} \left (\frac{A c}{3} + \frac{C b}{3}\right ) + x \left (A b + C a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09517, size = 77, normalized size = 1.22 \begin{align*} \frac{1}{5} \, C c x^{5} + \frac{1}{4} \, B c x^{4} + \frac{1}{3} \, C b x^{3} + \frac{1}{3} \, A c x^{3} + \frac{1}{2} \, B b x^{2} + C a x + A b x + B a \log \left ({\left | x \right |}\right ) - \frac{A a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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