Optimal. Leaf size=29 \[ -\frac {\log (a+b x)}{a^2}+\frac {\log (x)}{a^2}+\frac {1}{a (a+b x)} \]
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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 = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {44} \begin {gather*} -\frac {\log (a+b x)}{a^2}+\frac {\log (x)}{a^2}+\frac {1}{a (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)^2} \, dx &=\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx\\ &=\frac {1}{a (a+b x)}+\frac {\log (x)}{a^2}-\frac {\log (a+b x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.83 \begin {gather*} \frac {\frac {a}{a+b x}-\log (a+b x)+\log (x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x (a+b x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.28, size = 39, normalized size = 1.34 \begin {gather*} -\frac {{\left (b x + a\right )} \log \left (b x + a\right ) - {\left (b x + a\right )} \log \relax (x) - a}{a^{2} b x + a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 38, normalized size = 1.31 \begin {gather*} b {\left (\frac {\log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{2} b} + \frac {1}{{\left (b x + a\right )} a b}\right )} \end {gather*}
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 \begin {gather*} \frac {1}{\left (b x +a \right ) a}+\frac {\ln \relax (x )}{a^{2}}-\frac {\ln \left (b x +a \right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 28, normalized size = 0.97 \begin {gather*} \frac {1}{a b x + a^{2}} - \frac {\log \left (b x + a\right )}{a^{2}} + \frac {\log \relax (x)}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 26, normalized size = 0.90 \begin {gather*} \frac {1}{a^2+b\,x\,a}-\frac {\ln \left (\frac {a+b\,x}{x}\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 22, normalized size = 0.76 \begin {gather*} \frac {1}{a^{2} + a b x} + \frac {\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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