Integrand size = 8, antiderivative size = 28 \[ \int (b+a x) \log (x) \, dx=-b x-\frac {a x^2}{4}+b x \log (x)+\frac {1}{2} a x^2 \log (x) \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2350} \[ \int (b+a x) \log (x) \, dx=-\frac {a x^2}{4}+\frac {1}{2} a x^2 \log (x)-b x+b x \log (x) \]
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Rule 2350
Rubi steps \begin{align*} \text {integral}& = b x \log (x)+\frac {1}{2} a x^2 \log (x)-\int \left (b+\frac {a x}{2}\right ) \, dx \\ & = -b x-\frac {a x^2}{4}+b x \log (x)+\frac {1}{2} a x^2 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int (b+a x) \log (x) \, dx=-b x-\frac {a x^2}{4}+b x \log (x)+\frac {1}{2} a x^2 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
norman | \(-b x -\frac {a \,x^{2}}{4}+b x \ln \left (x \right )+\frac {a \,x^{2} \ln \left (x \right )}{2}\) | \(25\) |
risch | \(\left (\frac {1}{2} a \,x^{2}+b x \right ) \ln \left (x \right )-\frac {a \,x^{2}}{4}-b x\) | \(25\) |
parallelrisch | \(-b x -\frac {a \,x^{2}}{4}+b x \ln \left (x \right )+\frac {a \,x^{2} \ln \left (x \right )}{2}\) | \(25\) |
parts | \(-b x -\frac {a \,x^{2}}{4}+b x \ln \left (x \right )+\frac {a \,x^{2} \ln \left (x \right )}{2}\) | \(25\) |
default | \(a \left (-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x \right )}{2}\right )+b \left (-x +x \ln \left (x \right )\right )\) | \(27\) |
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Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int (b+a x) \log (x) \, dx=-\frac {1}{4} \, a x^{2} - b x + \frac {1}{2} \, {\left (a x^{2} + 2 \, b x\right )} \log \left (x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int (b+a x) \log (x) \, dx=- \frac {a x^{2}}{4} - b x + \left (\frac {a x^{2}}{2} + b x\right ) \log {\left (x \right )} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int (b+a x) \log (x) \, dx=-\frac {1}{4} \, a x^{2} - b x + \frac {1}{2} \, {\left (a x^{2} + 2 \, b x\right )} \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int (b+a x) \log (x) \, dx=\frac {1}{2} \, a x^{2} \log \left (x\right ) - \frac {1}{4} \, a x^{2} + b x \log \left (x\right ) - b x \]
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Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int (b+a x) \log (x) \, dx=-\frac {x\,\left (4\,b+a\,x-4\,b\,\ln \left (x\right )-2\,a\,x\,\ln \left (x\right )\right )}{4} \]
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