Optimal. Leaf size=29 \[ \frac {x \log (x)}{b (a x+b)}-\frac {\log (a x+b)}{a b} \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2314, 31} \[ \frac {x \log (x)}{b (a x+b)}-\frac {\log (a x+b)}{a b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2314
Rubi steps
\begin {align*} \int \frac {\log (x)}{(b+a x)^2} \, dx &=\frac {x \log (x)}{b (b+a x)}-\frac {\int \frac {1}{b+a x} \, dx}{b}\\ &=\frac {x \log (x)}{b (b+a x)}-\frac {\log (b+a x)}{a b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.93 \[ \frac {\frac {x \log (x)}{a x+b}-\frac {\log (a x+b)}{a}}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 34, normalized size = 1.17 \[ \frac {a x \log \relax (x) - {\left (a x + b\right )} \log \left (a x + b\right )}{a^{2} b x + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.19, size = 138, normalized size = 4.76 \[ a^{2} {\left (\frac {\log \left (\frac {{\left (a x + b\right )}^{2} {\left | a \right |} {\left | \frac {b}{a x + b} - 1 \right |}}{a^{2} {\left | a x + b \right |}}\right )}{a^{3} b} + \frac {\log \left (-\frac {b + \frac {{\left (a x + b\right )} a {\left (\frac {b}{a x + b} - 1\right )} - a b}{a}}{a}\right )}{{\left ({\left (a x + b\right )} {\left (\frac {b}{a x + b} - 1\right )} - b\right )} a^{3}} - \frac {\log \left ({\left | -{\left (a x + b\right )} {\left (\frac {b}{a x + b} - 1\right )} + b \right |}\right )}{a^{3} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 1.03 \[ \frac {x \ln \relax (x )}{\left (a x +b \right ) b}-\frac {\ln \left (a x +b \right )}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 38, normalized size = 1.31 \[ -\frac {\frac {\log \left (a x + b\right )}{b} - \frac {\log \relax (x)}{b}}{a} - \frac {\log \relax (x)}{{\left (a x + b\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 35, normalized size = 1.21 \[ \frac {x^2\,\ln \relax (x)}{b\,\left (a\,x^2+b\,x\right )}-\frac {\ln \left (b+a\,x\right )}{a\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 24, normalized size = 0.83 \[ - \frac {\log {\relax (x )}}{a^{2} x + a b} + \frac {\log {\relax (x )} - \log {\left (x + \frac {b}{a} \right )}}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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