Optimal. Leaf size=46 \[ A b^2 \log (x)+\frac{1}{2} c x^2 (A c+2 b B)+b x (2 A c+b B)+\frac{1}{3} B c^2 x^3 \]
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Rubi [A] time = 0.0271827, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ A b^2 \log (x)+\frac{1}{2} c x^2 (A c+2 b B)+b x (2 A c+b B)+\frac{1}{3} B c^2 x^3 \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{x^3} \, dx &=\int \left (b (b B+2 A c)+\frac{A b^2}{x}+c (2 b B+A c) x+B c^2 x^2\right ) \, dx\\ &=b (b B+2 A c) x+\frac{1}{2} c (2 b B+A c) x^2+\frac{1}{3} B c^2 x^3+A b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0142341, size = 43, normalized size = 0.93 \[ A b^2 \log (x)+b c x (2 A+B x)+\frac{1}{6} c^2 x^2 (3 A+2 B x)+b^2 B x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 46, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{3}}{3}}+{\frac{A{c}^{2}{x}^{2}}{2}}+B{x}^{2}bc+2\,Abcx+{b}^{2}Bx+A{b}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14898, size = 62, normalized size = 1.35 \begin{align*} \frac{1}{3} \, B c^{2} x^{3} + A b^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B b c + A c^{2}\right )} x^{2} +{\left (B b^{2} + 2 \, A b c\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61684, size = 108, normalized size = 2.35 \begin{align*} \frac{1}{3} \, B c^{2} x^{3} + A b^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B b c + A c^{2}\right )} x^{2} +{\left (B b^{2} + 2 \, A b c\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.297472, size = 46, normalized size = 1. \begin{align*} A b^{2} \log{\left (x \right )} + \frac{B c^{2} x^{3}}{3} + x^{2} \left (\frac{A c^{2}}{2} + B b c\right ) + x \left (2 A b c + B b^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11434, size = 62, normalized size = 1.35 \begin{align*} \frac{1}{3} \, B c^{2} x^{3} + B b c x^{2} + \frac{1}{2} \, A c^{2} x^{2} + B b^{2} x + 2 \, A b c x + A b^{2} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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