Optimal. Leaf size=35 \[ \frac{(a+b x)^3 \log (a+b x)}{3 b}-\frac{(a+b x)^3}{9 b} \]
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Rubi [A] time = 0.0246622, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2390, 2304} \[ \frac{(a+b x)^3 \log (a+b x)}{3 b}-\frac{(a+b x)^3}{9 b} \]
Antiderivative was successfully verified.
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Rule 2390
Rule 2304
Rubi steps
\begin{align*} \int (a+b x)^2 \log (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^2 \log (x) \, dx,x,a+b x\right )}{b}\\ &=-\frac{(a+b x)^3}{9 b}+\frac{(a+b x)^3 \log (a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0167406, size = 44, normalized size = 1.26 \[ \frac{(a+b x)^3 \log (a+b x)}{3 b}-\frac{1}{9} x \left (3 a^2+3 a b x+b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 82, normalized size = 2.3 \begin{align*}{\frac{{b}^{2}\ln \left ( bx+a \right ){x}^{3}}{3}}+b\ln \left ( bx+a \right ){x}^{2}a+\ln \left ( bx+a \right ) x{a}^{2}+{\frac{\ln \left ( bx+a \right ){a}^{3}}{3\,b}}-{\frac{{b}^{2}{x}^{3}}{9}}-{\frac{b{x}^{2}a}{3}}-{\frac{x{a}^{2}}{3}}-{\frac{{a}^{3}}{9\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06081, size = 100, normalized size = 2.86 \begin{align*} \frac{1}{9} \,{\left (\frac{3 \, a^{3} \log \left (b x + a\right )}{b^{2}} - \frac{b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{b}\right )} b + \frac{1}{3} \,{\left (b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x\right )} \log \left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7631, size = 139, normalized size = 3.97 \begin{align*} -\frac{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x - 3 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{9 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.365034, size = 63, normalized size = 1.8 \begin{align*} \frac{a^{3} \log{\left (a + b x \right )}}{3 b} - \frac{a^{2} x}{3} - \frac{a b x^{2}}{3} - \frac{b^{2} x^{3}}{9} + \left (a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}\right ) \log{\left (a + b x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32667, size = 42, normalized size = 1.2 \begin{align*} \frac{{\left (b x + a\right )}^{3} \log \left (b x + a\right )}{3 \, b} - \frac{{\left (b x + a\right )}^{3}}{9 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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