Optimal. Leaf size=18 \[ \frac {x}{a}-\frac {b \log (a x+b)}{a^2} \]
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Rubi [A] time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {193, 43} \[ \frac {x}{a}-\frac {b \log (a x+b)}{a^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 193
Rubi steps
\begin {align*} \int \frac {1}{a+\frac {b}{x}} \, dx &=\int \frac {x}{b+a x} \, dx\\ &=\int \left (\frac {1}{a}-\frac {b}{a (b+a x)}\right ) \, dx\\ &=\frac {x}{a}-\frac {b \log (b+a x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \[ \frac {x}{a}-\frac {b \log (a x+b)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 17, normalized size = 0.94 \[ \frac {a x - b \log \left (a x + b\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 19, normalized size = 1.06 \[ \frac {x}{a} - \frac {b \log \left ({\left | a x + b \right |}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 1.06 \[ \frac {x}{a}-\frac {b \ln \left (a x +b \right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 18, normalized size = 1.00 \[ \frac {x}{a} - \frac {b \log \left (a x + b\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 18, normalized size = 1.00 \[ -\frac {b\,\ln \left (b+a\,x\right )-a\,x}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 0.78 \[ \frac {x}{a} - \frac {b \log {\left (a x + b \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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