Optimal. Leaf size=19 \[ -b \log (x)+b \log (b x+1)-\frac {1}{x} \]
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Rubi [A] time = 0.01, antiderivative size = 19, 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} \[ -b \log (x)+b \log (b x+1)-\frac {1}{x} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x^2 (1+b x)} \, dx &=\int \left (\frac {1}{x^2}-\frac {b}{x}+\frac {b^2}{1+b x}\right ) \, dx\\ &=-\frac {1}{x}-b \log (x)+b \log (1+b x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \[ -b \log (x)+b \log (b x+1)-\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 21, normalized size = 1.11 \[ \frac {b x \log \left (b x + 1\right ) - b x \log \relax (x) - 1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 21, normalized size = 1.11 \[ b \log \left ({\left | b x + 1 \right |}\right ) - b \log \left ({\left | x \right |}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 1.05 \[ -b \ln \relax (x )+b \ln \left (b x +1\right )-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 19, normalized size = 1.00 \[ b \log \left (b x + 1\right ) - b \log \relax (x) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 16, normalized size = 0.84 \[ 2\,b\,\mathrm {atanh}\left (2\,b\,x+1\right )-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 14, normalized size = 0.74 \[ b \left (- \log {\relax (x )} + \log {\left (x + \frac {1}{b} \right )}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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