Optimal. Leaf size=55 \[ \frac{1}{2} A b^2 x^2+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{5} B c^2 x^5 \]
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Rubi [A] time = 0.0358139, antiderivative size = 55, 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} \[ \frac{1}{2} A b^2 x^2+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{5} B c^2 x^5 \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{x} \, dx &=\int \left (A b^2 x+b (b B+2 A c) x^2+c (2 b B+A c) x^3+B c^2 x^4\right ) \, dx\\ &=\frac{1}{2} A b^2 x^2+\frac{1}{3} b (b B+2 A c) x^3+\frac{1}{4} c (2 b B+A c) x^4+\frac{1}{5} B c^2 x^5\\ \end{align*}
Mathematica [A] time = 0.0102534, size = 49, normalized size = 0.89 \[ \frac{1}{60} x^2 \left (30 A b^2+15 c x^2 (A c+2 b B)+20 b x (2 A c+b B)+12 B c^2 x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{5}}{5}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,Abc+{b}^{2}B \right ){x}^{3}}{3}}+{\frac{A{b}^{2}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13337, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{2} \, A b^{2} x^{2} + \frac{1}{4} \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73533, size = 117, normalized size = 2.13 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{2} \, A b^{2} x^{2} + \frac{1}{4} \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.071358, size = 54, normalized size = 0.98 \begin{align*} \frac{A b^{2} x^{2}}{2} + \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{B b^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1302, size = 72, normalized size = 1.31 \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} \, A b c x^{3} + \frac{1}{2} \, A b^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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