Optimal. Leaf size=28 \[ \frac{1}{2} x^2 (A c+b B)+A b x+\frac{1}{3} B c x^3 \]
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Rubi [A] time = 0.0182864, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {765} \[ \frac{1}{2} x^2 (A c+b B)+A b x+\frac{1}{3} B c x^3 \]
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 )}{x} \, dx &=\int \left (A b+(b B+A c) x+B c x^2\right ) \, dx\\ &=A b x+\frac{1}{2} (b B+A c) x^2+\frac{1}{3} B c x^3\\ \end{align*}
Mathematica [A] time = 0.0044914, size = 28, normalized size = 1. \[ \frac{1}{2} x^2 (A c+b B)+A b x+\frac{1}{3} B c x^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 25, normalized size = 0.9 \begin{align*} Abx+{\frac{ \left ( Ac+bB \right ){x}^{2}}{2}}+{\frac{Bc{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96317, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{3} \, B c x^{3} + A b x + \frac{1}{2} \,{\left (B b + A c\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65916, size = 58, normalized size = 2.07 \begin{align*} \frac{1}{3} \, B c x^{3} + A b x + \frac{1}{2} \,{\left (B b + A c\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.062886, size = 26, normalized size = 0.93 \begin{align*} A b x + \frac{B c x^{3}}{3} + x^{2} \left (\frac{A c}{2} + \frac{B b}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13863, size = 35, normalized size = 1.25 \begin{align*} \frac{1}{3} \, B c x^{3} + \frac{1}{2} \, B b x^{2} + \frac{1}{2} \, A c x^{2} + A b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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