Optimal. Leaf size=35 \[ -\frac {(a+b x)^2}{4 b}+\frac {(a+b x)^2 \log (a+b x)}{2 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2437, 2341}
\begin {gather*} \frac {(a+b x)^2 \log (a+b x)}{2 b}-\frac {(a+b x)^2}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rule 2437
Rubi steps
\begin {align*} \int (a+b x) \log (a+b x) \, dx &=\frac {\text {Subst}(\int x \log (x) \, dx,x,a+b x)}{b}\\ &=-\frac {(a+b x)^2}{4 b}+\frac {(a+b x)^2 \log (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.94 \begin {gather*} -\frac {1}{4} x (2 a+b x)+\frac {(a+b x)^2 \log (a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 30, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {\left (b x +a \right )^{2} \ln \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}}{b}\) | \(30\) |
default | \(\frac {\frac {\left (b x +a \right )^{2} \ln \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}}{b}\) | \(30\) |
risch | \(\left (\frac {1}{2} b \,x^{2}+a x \right ) \ln \left (b x +a \right )-\frac {b \,x^{2}}{4}-\frac {a x}{2}+\frac {a^{2} \ln \left (b x +a \right )}{2 b}\) | \(43\) |
norman | \(a x \ln \left (b x +a \right )-\frac {a x}{2}-\frac {b \,x^{2}}{4}+\frac {a^{2} \ln \left (b x +a \right )}{2 b}+\frac {b \,x^{2} \ln \left (b x +a \right )}{2}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 52, normalized size = 1.49 \begin {gather*} \frac {1}{4} \, b {\left (\frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {b x^{2} + 2 \, a x}{b}\right )} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \log \left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 42, normalized size = 1.20 \begin {gather*} -\frac {b^{2} x^{2} + 2 \, a b x - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 41, normalized size = 1.17 \begin {gather*} \frac {a^{2} \log {\left (a + b x \right )}}{2 b} - \frac {a x}{2} - \frac {b x^{2}}{4} + \left (a x + \frac {b x^{2}}{2}\right ) \log {\left (a + b x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.53, size = 31, normalized size = 0.89 \begin {gather*} \frac {{\left (b x + a\right )}^{2} \log \left (b x + a\right )}{2 \, b} - \frac {{\left (b x + a\right )}^{2}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 46, normalized size = 1.31 \begin {gather*} \frac {a^2\,\ln \left (a+b\,x\right )}{2\,b}-\frac {b\,x^2}{4}-\frac {a\,x}{2}+a\,x\,\ln \left (a+b\,x\right )+\frac {b\,x^2\,\ln \left (a+b\,x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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