Optimal. Leaf size=18 \[ \frac {\log (x)}{b}-\frac {\log (b+c x)}{b} \]
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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {629}
\begin {gather*} \frac {\log (x)}{b}-\frac {\log (b+c x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin {align*} \int \frac {1}{b x+c x^2} \, dx &=\frac {\log (x)}{b}-\frac {\log (b+c x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{b}-\frac {\log (b+c x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 19, normalized size = 1.06
method | result | size |
default | \(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (c x +b \right )}{b}\) | \(19\) |
norman | \(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (c x +b \right )}{b}\) | \(19\) |
risch | \(\frac {\ln \left (-x \right )}{b}-\frac {\ln \left (c x +b \right )}{b}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\log \left (c x + b\right )}{b} + \frac {\log \left (x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.91, size = 16, normalized size = 0.89 \begin {gather*} -\frac {\log \left (c x + b\right ) - \log \left (x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 10, normalized size = 0.56 \begin {gather*} \frac {\log {\left (x \right )} - \log {\left (\frac {b}{c} + x \right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.32, size = 20, normalized size = 1.11 \begin {gather*} -\frac {\log \left ({\left | c x + b \right |}\right )}{b} + \frac {\log \left ({\left | x \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 15, normalized size = 0.83 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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