Optimal. Leaf size=25 \[ x \log \left (a c+\frac {b c}{x}\right )+\frac {b \log (a x+b)}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2453, 2448, 263, 31} \[ x \log \left (a c+\frac {b c}{x}\right )+\frac {b \log (a x+b)}{a} \]
Antiderivative was successfully verified.
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Rule 31
Rule 263
Rule 2448
Rule 2453
Rubi steps
\begin {align*} \int \log \left (\frac {c (b+a x)}{x}\right ) \, dx &=\int \log \left (a c+\frac {b c}{x}\right ) \, dx\\ &=x \log \left (a c+\frac {b c}{x}\right )+(b c) \int \frac {1}{\left (a c+\frac {b c}{x}\right ) x} \, dx\\ &=x \log \left (a c+\frac {b c}{x}\right )+(b c) \int \frac {1}{b c+a c x} \, dx\\ &=x \log \left (a c+\frac {b c}{x}\right )+\frac {b \log (b+a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 28, normalized size = 1.12 \[ \frac {(a x+b) \log \left (\frac {c (a x+b)}{x}\right )}{a}+\frac {b \log (x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 29, normalized size = 1.16 \[ \frac {a x \log \left (\frac {a c x + b c}{x}\right ) + b \log \left (a x + b\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 153, normalized size = 6.12 \[ \frac {b^{2} c^{2} {\left (\frac {\log \left (\frac {{\left | a c x + b c \right |}}{{\left | x \right |}}\right )}{a c} - \frac {\log \left ({\left | -a c + \frac {a c x + b c}{x} \right |}\right )}{a c}\right )} - \frac {b^{2} c^{2} \log \left (-{\left (b - \frac {a}{\frac {a}{b} - \frac {a c x + b c}{b c x}}\right )} c {\left (\frac {a}{b} - \frac {a c x + b c}{b c x}\right )}\right )}{a c - \frac {a c x + b c}{x}}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 44, normalized size = 1.76 \[ x \ln \left (a c +\frac {b c}{x}\right )-\frac {b \ln \left (\frac {b c}{x}\right )}{a}+\frac {b \ln \left (a c +\frac {b c}{x}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 25, normalized size = 1.00 \[ x \log \left (\frac {{\left (a x + b\right )} c}{x}\right ) + \frac {b \log \left (a x + b\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 25, normalized size = 1.00 \[ x\,\ln \left (\frac {c\,\left (b+a\,x\right )}{x}\right )+\frac {b\,\ln \left (b+a\,x\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 20, normalized size = 0.80 \[ x \log {\left (\frac {c \left (a x + b\right )}{x} \right )} + \frac {b \log {\left (a x + b \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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