Optimal. Leaf size=40 \[ a^2 A \log (x)+2 a A b x+\frac{B (a+b x)^3}{3 b}+\frac{1}{2} A b^2 x^2 \]
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Rubi [A] time = 0.0111689, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 43} \[ a^2 A \log (x)+2 a A b x+\frac{B (a+b x)^3}{3 b}+\frac{1}{2} A b^2 x^2 \]
Antiderivative was successfully verified.
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Rule 80
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 (A+B x)}{x} \, dx &=\frac{B (a+b x)^3}{3 b}+A \int \frac{(a+b x)^2}{x} \, dx\\ &=\frac{B (a+b x)^3}{3 b}+A \int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx\\ &=2 a A b x+\frac{1}{2} A b^2 x^2+\frac{B (a+b x)^3}{3 b}+a^2 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.0139103, size = 43, normalized size = 1.08 \[ a^2 A \log (x)+a^2 B x+a b x (2 A+B x)+\frac{1}{6} b^2 x^2 (3 A+2 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 46, normalized size = 1.2 \begin{align*}{\frac{B{b}^{2}{x}^{3}}{3}}+{\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 = 1.05566, size = 62, normalized size = 1.55 \begin{align*} \frac{1}{3} \, B b^{2} x^{3} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B a b + A b^{2}\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 = 1.93599, size = 108, normalized size = 2.7 \begin{align*} \frac{1}{3} \, B b^{2} x^{3} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B a b + A b^{2}\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 = 0.320919, size = 46, normalized size = 1.15 \begin{align*} A a^{2} \log{\left (x \right )} + \frac{B b^{2} x^{3}}{3} + x^{2} \left (\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 = 1.16038, size = 62, normalized size = 1.55 \begin{align*} \frac{1}{3} \, B b^{2} x^{3} + B a b x^{2} + \frac{1}{2} \, A b^{2} 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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