Optimal. Leaf size=149 \[ -\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{4} x^4 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{2} x^2 \left (A \left (2 a c+b^2\right )+2 a b C\right )+a \log (x) (a C+2 A b)+\frac{1}{3} B x^3 \left (2 a c+b^2\right )+2 a b B x+\frac{1}{6} c x^6 (A c+2 b C)+\frac{2}{5} b B c x^5+\frac{1}{7} B c^2 x^7+\frac{1}{8} c^2 C x^8 \]
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Rubi [A] time = 0.122592, antiderivative size = 149, 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} \[ -\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{4} x^4 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac{1}{2} x^2 \left (A \left (2 a c+b^2\right )+2 a b C\right )+a \log (x) (a C+2 A b)+\frac{1}{3} B x^3 \left (2 a c+b^2\right )+2 a b B x+\frac{1}{6} c x^6 (A c+2 b C)+\frac{2}{5} b B c x^5+\frac{1}{7} B c^2 x^7+\frac{1}{8} c^2 C x^8 \]
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^3} \, dx &=\int \left (2 a b B+\frac{a^2 A}{x^3}+\frac{a^2 B}{x^2}+\frac{a (2 A b+a C)}{x}+\left (A \left (b^2+2 a c\right )+2 a b C\right ) x+B \left (b^2+2 a c\right ) x^2+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^3+2 b B c x^4+c (A c+2 b C) x^5+B c^2 x^6+c^2 C x^7\right ) \, dx\\ &=-\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+2 a b B x+\frac{1}{2} \left (A \left (b^2+2 a c\right )+2 a b C\right ) x^2+\frac{1}{3} B \left (b^2+2 a c\right ) x^3+\frac{1}{4} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^4+\frac{2}{5} b B c x^5+\frac{1}{6} c (A c+2 b C) x^6+\frac{1}{7} B c^2 x^7+\frac{1}{8} c^2 C x^8+a (2 A b+a C) \log (x)\\ \end{align*}
Mathematica [A] time = 0.108635, size = 139, normalized size = 0.93 \[ -\frac{a^2 (A+2 B x)}{2 x^2}+\frac{1}{6} a x \left (c x \left (6 A+4 B x+3 C x^2\right )+6 b (2 B+C x)\right )+a \log (x) (a C+2 A b)+\frac{1}{840} x^2 \left (140 A \left (3 b^2+3 b c x^2+c^2 x^4\right )+70 b^2 x (4 B+3 C x)+56 b c x^3 (6 B+5 C x)+15 c^2 x^5 (8 B+7 C x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 148, normalized size = 1. \begin{align*}{\frac{{c}^{2}C{x}^{8}}{8}}+{\frac{B{c}^{2}{x}^{7}}{7}}+{\frac{A{x}^{6}{c}^{2}}{6}}+{\frac{C{x}^{6}bc}{3}}+{\frac{2\,bBc{x}^{5}}{5}}+{\frac{A{x}^{4}bc}{2}}+{\frac{C{x}^{4}ac}{2}}+{\frac{C{x}^{4}{b}^{2}}{4}}+{\frac{2\,B{x}^{3}ac}{3}}+{\frac{B{x}^{3}{b}^{2}}{3}}+A{x}^{2}ac+{\frac{A{x}^{2}{b}^{2}}{2}}+C{x}^{2}ab+2\,abBx-{\frac{B{a}^{2}}{x}}-{\frac{A{a}^{2}}{2\,{x}^{2}}}+2\,A\ln \left ( x \right ) ab+C\ln \left ( x \right ){a}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943822, size = 188, normalized size = 1.26 \begin{align*} \frac{1}{8} \, C c^{2} x^{8} + \frac{1}{7} \, B c^{2} x^{7} + \frac{2}{5} \, B b c x^{5} + \frac{1}{6} \,{\left (2 \, C b c + A c^{2}\right )} x^{6} + \frac{1}{4} \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{4} + 2 \, B a b x + \frac{1}{3} \,{\left (B b^{2} + 2 \, B a c\right )} x^{3} + \frac{1}{2} \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{2} +{\left (C a^{2} + 2 \, A a b\right )} \log \left (x\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30031, size = 362, normalized size = 2.43 \begin{align*} \frac{105 \, C c^{2} x^{10} + 120 \, B c^{2} x^{9} + 336 \, B b c x^{7} + 140 \,{\left (2 \, C b c + A c^{2}\right )} x^{8} + 210 \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{6} + 1680 \, B a b x^{3} + 280 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} + 420 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} - 840 \, B a^{2} x + 840 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2} \log \left (x\right ) - 420 \, A a^{2}}{840 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.641083, size = 151, normalized size = 1.01 \begin{align*} 2 B a b x + \frac{2 B b c x^{5}}{5} + \frac{B c^{2} x^{7}}{7} + \frac{C c^{2} x^{8}}{8} + a \left (2 A b + C a\right ) \log{\left (x \right )} + x^{6} \left (\frac{A c^{2}}{6} + \frac{C b c}{3}\right ) + x^{4} \left (\frac{A b c}{2} + \frac{C a c}{2} + \frac{C b^{2}}{4}\right ) + x^{3} \left (\frac{2 B a c}{3} + \frac{B b^{2}}{3}\right ) + x^{2} \left (A a c + \frac{A b^{2}}{2} + C a b\right ) - \frac{A a^{2} + 2 B a^{2} x}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13592, size = 200, normalized size = 1.34 \begin{align*} \frac{1}{8} \, C c^{2} x^{8} + \frac{1}{7} \, B c^{2} x^{7} + \frac{1}{3} \, C b c x^{6} + \frac{1}{6} \, A c^{2} x^{6} + \frac{2}{5} \, B b c x^{5} + \frac{1}{4} \, C b^{2} x^{4} + \frac{1}{2} \, C a c x^{4} + \frac{1}{2} \, A b c x^{4} + \frac{1}{3} \, B b^{2} x^{3} + \frac{2}{3} \, B a c x^{3} + C a b x^{2} + \frac{1}{2} \, A b^{2} x^{2} + A a c x^{2} + 2 \, B a b x +{\left (C a^{2} + 2 \, A a b\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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