Optimal. Leaf size=35 \[ \frac {(a+b x)^2 \log (a+b x)}{2 b}-\frac {(a+b x)^2}{4 b} \]
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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 = {2390, 2304} \[ \frac {(a+b x)^2 \log (a+b x)}{2 b}-\frac {(a+b x)^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2390
Rubi steps
\begin {align*} \int (a+b x) \log (a+b x) \, dx &=\frac {\operatorname {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 \[ \frac {(a+b x)^2 \log (a+b x)}{2 b}-\frac {1}{4} x (2 a+b x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 42, normalized size = 1.20 \[ -\frac {b^{2} x^{2} + 2 \, a b x - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 31, normalized size = 0.89 \[ \frac {{\left (b x + a\right )}^{2} \log \left (b x + a\right )}{2 \, b} - \frac {{\left (b x + a\right )}^{2}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 55, normalized size = 1.57 \[ \frac {b \,x^{2} \ln \left (b x +a \right )}{2}+a x \ln \left (b x +a \right )-\frac {b \,x^{2}}{4}+\frac {a^{2} \ln \left (b x +a \right )}{2 b}-\frac {a x}{2}-\frac {a^{2}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 52, normalized size = 1.49 \[ \frac {1}{4} \, b {\left (\frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {b x^{2} + 2 \, a x}{b}\right )} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \log \left (b x + a\right ) \]
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 \[ \frac {a^2\,\ln \left (a+b\,x\right )}{2\,b}-\frac {b\,x^2}{4}-\frac {a\,x}{2}+a\,x\,\ln \left (a+b\,x\right )+\frac {b\,x^2\,\ln \left (a+b\,x\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 41, normalized size = 1.17 \[ \frac {a^{2} \log {\left (a + b x \right )}}{2 b} - \frac {a x}{2} - \frac {b x^{2}}{4} + \left (a x + \frac {b x^{2}}{2}\right ) \log {\left (a + b x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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