Optimal. Leaf size=28 \[ \frac {2 b \log (b+a x)}{a}+x \log \left (\frac {c (b+a x)^2}{x^2}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2536, 31}
\begin {gather*} x \log \left (\frac {c (a x+b)^2}{x^2}\right )+\frac {2 b \log (a x+b)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2536
Rubi steps
\begin {align*} \int \log \left (\frac {c (b+a x)^2}{x^2}\right ) \, dx &=x \log \left (\frac {c (b+a x)^2}{x^2}\right )+(2 b) \int \frac {1}{b+a x} \, dx\\ &=\frac {2 b \log (b+a x)}{a}+x \log \left (\frac {c (b+a x)^2}{x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {2 b \log (b+a x)}{a}+x \log \left (\frac {c (b+a x)^2}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 41, normalized size = 1.46
method | result | size |
risch | \(\frac {2 b \ln \left (a x +b \right )}{a}+x \ln \left (\frac {c \left (a x +b \right )^{2}}{x^{2}}\right )\) | \(29\) |
derivativedivides | \(x \ln \left (c \left (a +\frac {b}{x}\right )^{2}\right )-2 b \left (-\frac {\ln \left (a +\frac {b}{x}\right )}{a}+\frac {\ln \left (\frac {1}{x}\right )}{a}\right )\) | \(41\) |
default | \(x \ln \left (c \left (a +\frac {b}{x}\right )^{2}\right )-2 b \left (-\frac {\ln \left (a +\frac {b}{x}\right )}{a}+\frac {\ln \left (\frac {1}{x}\right )}{a}\right )\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 28, normalized size = 1.00 \begin {gather*} x \log \left (\frac {{\left (a x + b\right )}^{2} c}{x^{2}}\right ) + \frac {2 \, b \log \left (a x + b\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 42, normalized size = 1.50 \begin {gather*} \frac {a x \log \left (\frac {a^{2} c x^{2} + 2 \, a b c x + b^{2} c}{x^{2}}\right ) + 2 \, b \log \left (a x + b\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 26, normalized size = 0.93 \begin {gather*} x \log {\left (\frac {c \left (a x + b\right )^{2}}{x^{2}} \right )} + \frac {2 b \log {\left (a x + b \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.96, size = 29, normalized size = 1.04 \begin {gather*} x \log \left (\frac {{\left (a x + b\right )}^{2} c}{x^{2}}\right ) + \frac {2 \, b \log \left ({\left | a x + b \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 28, normalized size = 1.00 \begin {gather*} x\,\ln \left (\frac {c\,{\left (b+a\,x\right )}^2}{x^2}\right )+\frac {2\,b\,\ln \left (b+a\,x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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