Optimal. Leaf size=37 \[ \frac{(a+b x) \log ^2(a+b x)}{b}-\frac{2 (a+b x) \log (a+b x)}{b}+2 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0143306, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2389, 2296, 2295} \[ \frac{(a+b x) \log ^2(a+b x)}{b}-\frac{2 (a+b x) \log (a+b x)}{b}+2 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2389
Rule 2296
Rule 2295
Rubi steps
\begin{align*} \int \log ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^2(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \log ^2(a+b x)}{b}-\frac{2 \operatorname{Subst}(\int \log (x) \, dx,x,a+b x)}{b}\\ &=2 x-\frac{2 (a+b x) \log (a+b x)}{b}+\frac{(a+b x) \log ^2(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0043415, size = 36, normalized size = 0.97 \[ \frac{(a+b x) \log ^2(a+b x)-2 (a+b x) \log (a+b x)+2 b x}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.059, size = 55, normalized size = 1.5 \begin{align*} \left ( \ln \left ( bx+a \right ) \right ) ^{2}x+{\frac{ \left ( \ln \left ( bx+a \right ) \right ) ^{2}a}{b}}-2\,\ln \left ( bx+a \right ) x-2\,{\frac{\ln \left ( bx+a \right ) a}{b}}+2\,x+2\,{\frac{a}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.2412, size = 36, normalized size = 0.97 \begin{align*} \frac{{\left (b x + a\right )}{\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) + 2\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.76001, size = 88, normalized size = 2.38 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a\right )^{2} + 2 \, b x - 2 \,{\left (b x + a\right )} \log \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.365685, size = 42, normalized size = 1.14 \begin{align*} 2 b \left (- \frac{a \log{\left (a + b x \right )}}{b^{2}} + \frac{x}{b}\right ) - 2 x \log{\left (a + b x \right )} + \frac{\left (a + b x\right ) \log{\left (a + b x \right )}^{2}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.31965, size = 59, normalized size = 1.59 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a\right )^{2}}{b} - \frac{2 \,{\left (b x + a\right )} \log \left (b x + a\right )}{b} + \frac{2 \,{\left (b x + a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]