Optimal. Leaf size=145 \[ -\frac{a^2 A}{x}+a^2 B \log (x)+\frac{1}{5} x^5 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{3} x^3 \left (A \left (2 a c+b^2\right )+2 a b C\right )+a x (a C+2 A b)+\frac{1}{4} B x^4 \left (2 a c+b^2\right )+a b B x^2+\frac{1}{7} c x^7 (A c+2 b C)+\frac{1}{3} b B c x^6+\frac{1}{8} B c^2 x^8+\frac{1}{9} c^2 C x^9 \]
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Rubi [A] time = 0.121105, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1628} \[ -\frac{a^2 A}{x}+a^2 B \log (x)+\frac{1}{5} x^5 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{3} x^3 \left (A \left (2 a c+b^2\right )+2 a b C\right )+a x (a C+2 A b)+\frac{1}{4} B x^4 \left (2 a c+b^2\right )+a b B x^2+\frac{1}{7} c x^7 (A c+2 b C)+\frac{1}{3} b B c x^6+\frac{1}{8} B c^2 x^8+\frac{1}{9} c^2 C x^9 \]
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 )^2}{x^2} \, dx &=\int \left (a (2 A b+a C)+\frac{a^2 A}{x^2}+\frac{a^2 B}{x}+2 a b B x+\left (A \left (b^2+2 a c\right )+2 a b C\right ) x^2+B \left (b^2+2 a c\right ) x^3+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^4+2 b B c x^5+c (A c+2 b C) x^6+B c^2 x^7+c^2 C x^8\right ) \, dx\\ &=-\frac{a^2 A}{x}+a (2 A b+a C) x+a b B x^2+\frac{1}{3} \left (A \left (b^2+2 a c\right )+2 a b C\right ) x^3+\frac{1}{4} B \left (b^2+2 a c\right ) x^4+\frac{1}{5} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^5+\frac{1}{3} b B c x^6+\frac{1}{7} c (A c+2 b C) x^7+\frac{1}{8} B c^2 x^8+\frac{1}{9} c^2 C x^9+a^2 B \log (x)\\ \end{align*}
Mathematica [A] time = 0.0969736, size = 145, normalized size = 1. \[ -\frac{a^2 A}{x}+a^2 B \log (x)+\frac{1}{5} x^5 \left (2 a c C+2 A b c+b^2 C\right )+\frac{1}{3} x^3 \left (2 a A c+2 a b C+A b^2\right )+a x (a C+2 A b)+\frac{1}{4} B x^4 \left (2 a c+b^2\right )+a b B x^2+\frac{1}{7} c x^7 (A c+2 b C)+\frac{1}{3} b B c x^6+\frac{1}{8} B c^2 x^8+\frac{1}{9} c^2 C x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 147, normalized size = 1. \begin{align*}{\frac{{c}^{2}C{x}^{9}}{9}}+{\frac{B{c}^{2}{x}^{8}}{8}}+{\frac{A{x}^{7}{c}^{2}}{7}}+{\frac{2\,C{x}^{7}bc}{7}}+{\frac{bBc{x}^{6}}{3}}+{\frac{2\,A{x}^{5}bc}{5}}+{\frac{2\,C{x}^{5}ac}{5}}+{\frac{C{x}^{5}{b}^{2}}{5}}+{\frac{B{x}^{4}ac}{2}}+{\frac{B{x}^{4}{b}^{2}}{4}}+{\frac{2\,A{x}^{3}ac}{3}}+{\frac{A{x}^{3}{b}^{2}}{3}}+{\frac{2\,C{x}^{3}ab}{3}}+abB{x}^{2}+2\,Aabx+C{a}^{2}x-{\frac{A{a}^{2}}{x}}+{a}^{2}B\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.930936, size = 185, normalized size = 1.28 \begin{align*} \frac{1}{9} \, C c^{2} x^{9} + \frac{1}{8} \, B c^{2} x^{8} + \frac{1}{3} \, B b c x^{6} + \frac{1}{7} \,{\left (2 \, C b c + A c^{2}\right )} x^{7} + \frac{1}{5} \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{5} + B a b x^{2} + \frac{1}{4} \,{\left (B b^{2} + 2 \, B a c\right )} x^{4} + \frac{1}{3} \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{3} + B a^{2} \log \left (x\right ) - \frac{A a^{2}}{x} +{\left (C a^{2} + 2 \, A a b\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25102, size = 365, normalized size = 2.52 \begin{align*} \frac{280 \, C c^{2} x^{10} + 315 \, B c^{2} x^{9} + 840 \, B b c x^{7} + 360 \,{\left (2 \, C b c + A c^{2}\right )} x^{8} + 504 \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{6} + 2520 \, B a b x^{3} + 630 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} + 840 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + 2520 \, B a^{2} x \log \left (x\right ) - 2520 \, A a^{2} + 2520 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{2520 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.503331, size = 156, normalized size = 1.08 \begin{align*} - \frac{A a^{2}}{x} + B a^{2} \log{\left (x \right )} + B a b x^{2} + \frac{B b c x^{6}}{3} + \frac{B c^{2} x^{8}}{8} + \frac{C c^{2} x^{9}}{9} + x^{7} \left (\frac{A c^{2}}{7} + \frac{2 C b c}{7}\right ) + x^{5} \left (\frac{2 A b c}{5} + \frac{2 C a c}{5} + \frac{C b^{2}}{5}\right ) + x^{4} \left (\frac{B a c}{2} + \frac{B b^{2}}{4}\right ) + x^{3} \left (\frac{2 A a c}{3} + \frac{A b^{2}}{3} + \frac{2 C a b}{3}\right ) + x \left (2 A a b + C a^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08822, size = 198, normalized size = 1.37 \begin{align*} \frac{1}{9} \, C c^{2} x^{9} + \frac{1}{8} \, B c^{2} x^{8} + \frac{2}{7} \, C b c x^{7} + \frac{1}{7} \, A c^{2} x^{7} + \frac{1}{3} \, B b c x^{6} + \frac{1}{5} \, C b^{2} x^{5} + \frac{2}{5} \, C a c x^{5} + \frac{2}{5} \, A b c x^{5} + \frac{1}{4} \, B b^{2} x^{4} + \frac{1}{2} \, B a c x^{4} + \frac{2}{3} \, C a b x^{3} + \frac{1}{3} \, A b^{2} x^{3} + \frac{2}{3} \, A a c x^{3} + B a b x^{2} + C a^{2} x + 2 \, A a b x + B a^{2} \log \left ({\left | x \right |}\right ) - \frac{A a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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