Optimal. Leaf size=149 \[ -\frac{a^2 A}{6 x^6}-\frac{a^2 B}{5 x^5}-\frac{A \left (2 a c+b^2\right )+2 a b C}{2 x^2}+\log (x) \left (C \left (2 a c+b^2\right )+2 A b c\right )-\frac{a (a C+2 A b)}{4 x^4}-\frac{B \left (2 a c+b^2\right )}{x}-\frac{2 a b B}{3 x^3}+\frac{1}{2} c x^2 (A c+2 b C)+2 b B c x+\frac{1}{3} B c^2 x^3+\frac{1}{4} c^2 C x^4 \]
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Rubi [A] time = 0.143141, 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} \[ -\frac{a^2 A}{6 x^6}-\frac{a^2 B}{5 x^5}-\frac{A \left (2 a c+b^2\right )+2 a b C}{2 x^2}+\log (x) \left (C \left (2 a c+b^2\right )+2 A b c\right )-\frac{a (a C+2 A b)}{4 x^4}-\frac{B \left (2 a c+b^2\right )}{x}-\frac{2 a b B}{3 x^3}+\frac{1}{2} c x^2 (A c+2 b C)+2 b B c x+\frac{1}{3} B c^2 x^3+\frac{1}{4} c^2 C x^4 \]
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^7} \, dx &=\int \left (2 b B c+\frac{a^2 A}{x^7}+\frac{a^2 B}{x^6}+\frac{a (2 A b+a C)}{x^5}+\frac{2 a b B}{x^4}+\frac{A \left (b^2+2 a c\right )+2 a b C}{x^3}+\frac{B \left (b^2+2 a c\right )}{x^2}+\frac{2 A b c+\left (b^2+2 a c\right ) C}{x}+c (A c+2 b C) x+B c^2 x^2+c^2 C x^3\right ) \, dx\\ &=-\frac{a^2 A}{6 x^6}-\frac{a^2 B}{5 x^5}-\frac{a (2 A b+a C)}{4 x^4}-\frac{2 a b B}{3 x^3}-\frac{A \left (b^2+2 a c\right )+2 a b C}{2 x^2}-\frac{B \left (b^2+2 a c\right )}{x}+2 b B c x+\frac{1}{2} c (A c+2 b C) x^2+\frac{1}{3} B c^2 x^3+\frac{1}{4} c^2 C x^4+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) \log (x)\\ \end{align*}
Mathematica [A] time = 0.10065, size = 144, normalized size = 0.97 \[ -\frac{a^2 (10 A+3 x (4 B+5 C x))}{60 x^6}+\log (x) \left (C \left (2 a c+b^2\right )+2 A b c\right )-\frac{a \left (3 A \left (b+2 c x^2\right )+2 x \left (2 b B+3 b C x+6 B c x^2\right )\right )}{6 x^4}+\frac{A \left (c^2 x^4-b^2\right )}{2 x^2}-\frac{b^2 B}{x}+b c x (2 B+C x)+\frac{1}{12} c^2 x^3 (4 B+3 C x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 148, normalized size = 1. \begin{align*}{\frac{{c}^{2}C{x}^{4}}{4}}+{\frac{B{c}^{2}{x}^{3}}{3}}+{\frac{A{x}^{2}{c}^{2}}{2}}+C{x}^{2}bc+2\,bBcx-{\frac{Aab}{2\,{x}^{4}}}-{\frac{C{a}^{2}}{4\,{x}^{4}}}-2\,{\frac{Bac}{x}}-{\frac{B{b}^{2}}{x}}-{\frac{aAc}{{x}^{2}}}-{\frac{A{b}^{2}}{2\,{x}^{2}}}-{\frac{abC}{{x}^{2}}}-{\frac{A{a}^{2}}{6\,{x}^{6}}}-{\frac{B{a}^{2}}{5\,{x}^{5}}}-{\frac{2\,Bab}{3\,{x}^{3}}}+2\,A\ln \left ( x \right ) bc+2\,C\ln \left ( x \right ) ac+C\ln \left ( x \right ){b}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952514, size = 189, normalized size = 1.27 \begin{align*} \frac{1}{4} \, C c^{2} x^{4} + \frac{1}{3} \, B c^{2} x^{3} + 2 \, B b c x + \frac{1}{2} \,{\left (2 \, C b c + A c^{2}\right )} x^{2} +{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} \log \left (x\right ) - \frac{40 \, B a b x^{3} + 60 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} + 30 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + 12 \, B a^{2} x + 10 \, A a^{2} + 15 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21679, size = 346, normalized size = 2.32 \begin{align*} \frac{15 \, C c^{2} x^{10} + 20 \, B c^{2} x^{9} + 120 \, B b c x^{7} + 30 \,{\left (2 \, C b c + A c^{2}\right )} x^{8} + 60 \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{6} \log \left (x\right ) - 40 \, B a b x^{3} - 60 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} - 30 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} - 12 \, B a^{2} x - 10 \, A a^{2} - 15 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 29.7179, size = 153, normalized size = 1.03 \begin{align*} 2 B b c x + \frac{B c^{2} x^{3}}{3} + \frac{C c^{2} x^{4}}{4} + x^{2} \left (\frac{A c^{2}}{2} + C b c\right ) + \left (2 A b c + 2 C a c + C b^{2}\right ) \log{\left (x \right )} - \frac{10 A a^{2} + 12 B a^{2} x + 40 B a b x^{3} + x^{5} \left (120 B a c + 60 B b^{2}\right ) + x^{4} \left (60 A a c + 30 A b^{2} + 60 C a b\right ) + x^{2} \left (30 A a b + 15 C a^{2}\right )}{60 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12944, size = 190, normalized size = 1.28 \begin{align*} \frac{1}{4} \, C c^{2} x^{4} + \frac{1}{3} \, B c^{2} x^{3} + C b c x^{2} + \frac{1}{2} \, A c^{2} x^{2} + 2 \, B b c x +{\left (C b^{2} + 2 \, C a c + 2 \, A b c\right )} \log \left ({\left | x \right |}\right ) - \frac{40 \, B a b x^{3} + 60 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} + 30 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + 12 \, B a^{2} x + 10 \, A a^{2} + 15 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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