Optimal. Leaf size=44 \[ \frac{(a+b x)^{n+1} \log (a+b x)}{b (n+1)}-\frac{(a+b x)^{n+1}}{b (n+1)^2} \]
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Rubi [A] time = 0.0302658, antiderivative size = 44, 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} \[ \frac{(a+b x)^{n+1} \log (a+b x)}{b (n+1)}-\frac{(a+b x)^{n+1}}{b (n+1)^2} \]
Antiderivative was successfully verified.
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Rule 2390
Rule 2304
Rubi steps
\begin{align*} \int (a+b x)^n \log (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^n \log (x) \, dx,x,a+b x\right )}{b}\\ &=-\frac{(a+b x)^{1+n}}{b (1+n)^2}+\frac{(a+b x)^{1+n} \log (a+b x)}{b (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0142109, size = 30, normalized size = 0.68 \[ \frac{(a+b x)^{n+1} ((n+1) \log (a+b x)-1)}{b (n+1)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 96, normalized size = 2.2 \begin{align*}{\frac{\ln \left ( bx+a \right ) x{{\rm e}^{\ln \left ( bx+a \right ) n}}}{1+n}}+{\frac{a\ln \left ( bx+a \right ){{\rm e}^{\ln \left ( bx+a \right ) n}}}{b \left ( 1+n \right ) }}-{\frac{x{{\rm e}^{\ln \left ( bx+a \right ) n}}}{{n}^{2}+2\,n+1}}-{\frac{a{{\rm e}^{\ln \left ( bx+a \right ) n}}}{b \left ({n}^{2}+2\,n+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83709, size = 112, normalized size = 2.55 \begin{align*} -\frac{{\left (b x -{\left (a n +{\left (b n + b\right )} x + a\right )} \log \left (b x + a\right ) + a\right )}{\left (b x + a\right )}^{n}}{b n^{2} + 2 \, b n + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n} \log \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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