Optimal. Leaf size=90 \[ -\frac{a^2 A}{3 x^3}-\frac{A \left (2 a c+b^2\right )+2 a b B}{x}+\log (x) \left (2 a B c+2 A b c+b^2 B\right )-\frac{a (a B+2 A b)}{2 x^2}+c x (A c+2 b B)+\frac{1}{2} B c^2 x^2 \]
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Rubi [A] time = 0.0613583, antiderivative size = 90, 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} \[ -\frac{a^2 A}{3 x^3}-\frac{A \left (2 a c+b^2\right )+2 a b B}{x}+\log (x) \left (2 a B c+2 A b c+b^2 B\right )-\frac{a (a B+2 A b)}{2 x^2}+c x (A c+2 b B)+\frac{1}{2} B c^2 x^2 \]
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^4} \, dx &=\int \left (c (2 b B+A c)+\frac{a^2 A}{x^4}+\frac{a (2 A b+a B)}{x^3}+\frac{2 a b B+A \left (b^2+2 a c\right )}{x^2}+\frac{b^2 B+2 A b c+2 a B c}{x}+B c^2 x\right ) \, dx\\ &=-\frac{a^2 A}{3 x^3}-\frac{a (2 A b+a B)}{2 x^2}-\frac{2 a b B+A \left (b^2+2 a c\right )}{x}+c (2 b B+A c) x+\frac{1}{2} B c^2 x^2+\left (b^2 B+2 A b c+2 a B c\right ) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0532036, size = 90, normalized size = 1. \[ -\frac{a^2 (2 A+3 B x)}{6 x^3}+\log (x) \left (2 a B c+2 A b c+b^2 B\right )-\frac{a (A b+2 A c x+2 b B x)}{x^2}-\frac{A b^2}{x}+A c^2 x+2 b B c x+\frac{1}{2} B c^2 x^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 95, normalized size = 1.1 \begin{align*}{\frac{B{c}^{2}{x}^{2}}{2}}+A{c}^{2}x+2\,Bcbx+2\,A\ln \left ( x \right ) bc+2\,B\ln \left ( x \right ) ac+{b}^{2}B\ln \left ( x \right ) -{\frac{A{a}^{2}}{3\,{x}^{3}}}-{\frac{Aab}{{x}^{2}}}-{\frac{B{a}^{2}}{2\,{x}^{2}}}-2\,{\frac{aAc}{x}}-{\frac{A{b}^{2}}{x}}-2\,{\frac{abB}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16202, size = 120, normalized size = 1.33 \begin{align*} \frac{1}{2} \, B c^{2} x^{2} +{\left (2 \, B b c + A c^{2}\right )} x +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} \log \left (x\right ) - \frac{2 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36503, size = 216, normalized size = 2.4 \begin{align*} \frac{3 \, B c^{2} x^{5} + 6 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 6 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} \log \left (x\right ) - 2 \, A a^{2} - 6 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.66425, size = 95, normalized size = 1.06 \begin{align*} \frac{B c^{2} x^{2}}{2} + x \left (A c^{2} + 2 B b c\right ) + \left (2 A b c + 2 B a c + B b^{2}\right ) \log{\left (x \right )} - \frac{2 A a^{2} + x^{2} \left (12 A a c + 6 A b^{2} + 12 B a b\right ) + x \left (6 A a b + 3 B a^{2}\right )}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27497, size = 120, normalized size = 1.33 \begin{align*} \frac{1}{2} \, B c^{2} x^{2} + 2 \, B b c x + A c^{2} x +{\left (B b^{2} + 2 \, B a c + 2 \, A b c\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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