Optimal. Leaf size=37 \[ 2 x-\frac {2 (a+b x) \log (a+b x)}{b}+\frac {(a+b x) \log ^2(a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 = {2436, 2333,
2332} \begin {gather*} \frac {(a+b x) \log ^2(a+b x)}{b}-\frac {2 (a+b x) \log (a+b x)}{b}+2 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2332
Rule 2333
Rule 2436
Rubi steps
\begin {align*} \int \log ^2(a+b x) \, dx &=\frac {\text {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 \text {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 = 0.00, size = 36, normalized size = 0.97 \begin {gather*} \frac {2 b x-2 (a+b x) \log (a+b x)+(a+b x) \log ^2(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 40, normalized size = 1.08
method | result | size |
derivativedivides | \(\frac {\ln \left (b x +a \right )^{2} \left (b x +a \right )-2 \left (b x +a \right ) \ln \left (b x +a \right )+2 b x +2 a}{b}\) | \(40\) |
default | \(\frac {\ln \left (b x +a \right )^{2} \left (b x +a \right )-2 \left (b x +a \right ) \ln \left (b x +a \right )+2 b x +2 a}{b}\) | \(40\) |
risch | \(\frac {\left (b x +a \right ) \ln \left (b x +a \right )^{2}}{b}-2 x \ln \left (b x +a \right )+2 x -\frac {2 a \ln \left (b x +a \right )}{b}\) | \(43\) |
norman | \(x \ln \left (b x +a \right )^{2}+\frac {a \ln \left (b x +a \right )^{2}}{b}+2 x -2 x \ln \left (b x +a \right )-\frac {2 a \ln \left (b x +a \right )}{b}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 27, normalized size = 0.73 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 36, normalized size = 0.97 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.07, size = 42, normalized size = 1.14 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.82, size = 44, normalized size = 1.19 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.29, size = 48, normalized size = 1.30 \begin {gather*} 2\,x-2\,x\,\ln \left (a+b\,x\right )+x\,{\ln \left (a+b\,x\right )}^2+\frac {a\,{\ln \left (a+b\,x\right )}^2}{b}-\frac {2\,a\,\ln \left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________