Optimal. Leaf size=92 \[ a^2 A \log (x)+\frac{1}{3} x^3 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{2} x^2 \left (A \left (2 a c+b^2\right )+2 a b B\right )+a x (a B+2 A b)+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{5} B c^2 x^5 \]
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Rubi [A] time = 0.052385, antiderivative size = 92, 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} \[ a^2 A \log (x)+\frac{1}{3} x^3 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{2} x^2 \left (A \left (2 a c+b^2\right )+2 a b B\right )+a x (a B+2 A b)+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{5} B c^2 x^5 \]
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} \, dx &=\int \left (a (2 A b+a B)+\frac{a^2 A}{x}+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x+\left (b^2 B+2 A b c+2 a B c\right ) x^2+c (2 b B+A c) x^3+B c^2 x^4\right ) \, dx\\ &=a (2 A b+a B) x+\frac{1}{2} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^2+\frac{1}{3} \left (b^2 B+2 A b c+2 a B c\right ) x^3+\frac{1}{4} c (2 b B+A c) x^4+\frac{1}{5} B c^2 x^5+a^2 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.0329251, size = 92, normalized size = 1. \[ a^2 A \log (x)+\frac{1}{3} x^3 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{2} x^2 \left (2 a A c+2 a b B+A b^2\right )+a x (a B+2 A b)+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{5} B c^2 x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 95, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{5}}{5}}+{\frac{A{c}^{2}{x}^{4}}{4}}+{\frac{B{x}^{4}bc}{2}}+{\frac{2\,A{x}^{3}bc}{3}}+{\frac{2\,aBc{x}^{3}}{3}}+{\frac{{b}^{2}B{x}^{3}}{3}}+aAc{x}^{2}+{\frac{A{b}^{2}{x}^{2}}{2}}+B{x}^{2}ab+2\,aAbx+{a}^{2}Bx+{a}^{2}A\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02091, size = 119, normalized size = 1.29 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} +{\left (B 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 = 1.21463, size = 208, normalized size = 2.26 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} +{\left (B a^{2} + 2 \, A a b\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.648749, size = 95, normalized size = 1.03 \begin{align*} A a^{2} \log{\left (x \right )} + \frac{B c^{2} x^{5}}{5} + x^{4} \left (\frac{A c^{2}}{4} + \frac{B b c}{2}\right ) + x^{3} \left (\frac{2 A b c}{3} + \frac{2 B a c}{3} + \frac{B b^{2}}{3}\right ) + x^{2} \left (A a c + \frac{A b^{2}}{2} + B a b\right ) + x \left (2 A a b + B a^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2817, size = 128, normalized size = 1.39 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{2} \, B b c x^{4} + \frac{1}{4} \, A c^{2} x^{4} + \frac{1}{3} \, B b^{2} x^{3} + \frac{2}{3} \, B a c x^{3} + \frac{2}{3} \, A b c x^{3} + B a b x^{2} + \frac{1}{2} \, A b^{2} x^{2} + A a c x^{2} + B a^{2} x + 2 \, A a b x + A a^{2} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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