Optimal. Leaf size=95 \[ -\frac{a^2 A}{5 x^5}-\frac{2 a B c+2 A b c+b^2 B}{2 x^2}-\frac{A \left (2 a c+b^2\right )+2 a b B}{3 x^3}-\frac{a (a B+2 A b)}{4 x^4}-\frac{c (A c+2 b B)}{x}+B c^2 \log (x) \]
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Rubi [A] time = 0.0544118, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ -\frac{a^2 A}{5 x^5}-\frac{2 a B c+2 A b c+b^2 B}{2 x^2}-\frac{A \left (2 a c+b^2\right )+2 a b B}{3 x^3}-\frac{a (a B+2 A b)}{4 x^4}-\frac{c (A c+2 b B)}{x}+B c^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )^2}{x^6} \, dx &=\int \left (\frac{a^2 A}{x^6}+\frac{a (2 A b+a B)}{x^5}+\frac{2 a b B+A \left (b^2+2 a c\right )}{x^4}+\frac{b^2 B+2 A b c+2 a B c}{x^3}+\frac{c (2 b B+A c)}{x^2}+\frac{B c^2}{x}\right ) \, dx\\ &=-\frac{a^2 A}{5 x^5}-\frac{a (2 A b+a B)}{4 x^4}-\frac{2 a b B+A \left (b^2+2 a c\right )}{3 x^3}-\frac{b^2 B+2 A b c+2 a B c}{2 x^2}-\frac{c (2 b B+A c)}{x}+B c^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0622706, size = 92, normalized size = 0.97 \[ B c^2 \log (x)-\frac{3 a^2 (4 A+5 B x)+10 a x \left (3 A b+4 A c x+4 b B x+6 B c x^2\right )+10 x^2 \left (2 A \left (b^2+3 b c x+3 c^2 x^2\right )+3 b B x (b+4 c x)\right )}{60 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 102, normalized size = 1.1 \begin{align*} B{c}^{2}\ln \left ( x \right ) -{\frac{2\,aAc}{3\,{x}^{3}}}-{\frac{A{b}^{2}}{3\,{x}^{3}}}-{\frac{2\,abB}{3\,{x}^{3}}}-{\frac{Abc}{{x}^{2}}}-{\frac{aBc}{{x}^{2}}}-{\frac{{b}^{2}B}{2\,{x}^{2}}}-{\frac{A{c}^{2}}{x}}-2\,{\frac{Bcb}{x}}-{\frac{A{a}^{2}}{5\,{x}^{5}}}-{\frac{Aab}{2\,{x}^{4}}}-{\frac{B{a}^{2}}{4\,{x}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09443, size = 124, normalized size = 1.31 \begin{align*} B c^{2} \log \left (x\right ) - \frac{60 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 30 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + 12 \, A a^{2} + 20 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20106, size = 225, normalized size = 2.37 \begin{align*} \frac{60 \, B c^{2} x^{5} \log \left (x\right ) - 60 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} - 30 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} - 12 \, A a^{2} - 20 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 15 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.8477, size = 99, normalized size = 1.04 \begin{align*} B c^{2} \log{\left (x \right )} - \frac{12 A a^{2} + x^{4} \left (60 A c^{2} + 120 B b c\right ) + x^{3} \left (60 A b c + 60 B a c + 30 B b^{2}\right ) + x^{2} \left (40 A a c + 20 A b^{2} + 40 B a b\right ) + x \left (30 A a b + 15 B a^{2}\right )}{60 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25355, size = 126, normalized size = 1.33 \begin{align*} B c^{2} \log \left ({\left | x \right |}\right ) - \frac{60 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 30 \,{\left (B b^{2} + 2 \, B a c + 2 \, A b c\right )} x^{3} + 12 \, A a^{2} + 20 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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