Optimal. Leaf size=19 \[ -\frac {1}{x}-b \log (x)+b \log (1+b x) \]
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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 = {46}
\begin {gather*} -b \log (x)+b \log (b x+1)-\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
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 \begin {gather*} -\frac {1}{x}-b \log (x)+b \log (1+b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 20, normalized size = 1.05
method | result | size |
default | \(-\frac {1}{x}-b \ln \left (x \right )+b \ln \left (b x +1\right )\) | \(20\) |
norman | \(-\frac {1}{x}-b \ln \left (x \right )+b \ln \left (b x +1\right )\) | \(20\) |
risch | \(-\frac {1}{x}+b \ln \left (-b x -1\right )-b \ln \left (x \right )\) | \(21\) |
meijerg | \(b \left (-\frac {1}{x b}-\ln \left (x \right )-\ln \left (b \right )+\ln \left (b x +1\right )\right )\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 19, normalized size = 1.00 \begin {gather*} b \log \left (b x + 1\right ) - b \log \left (x\right ) - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 21, normalized size = 1.11 \begin {gather*} \frac {b x \log \left (b x + 1\right ) - b x \log \left (x\right ) - 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 14, normalized size = 0.74 \begin {gather*} b \left (- \log {\left (x \right )} + \log {\left (x + \frac {1}{b} \right )}\right ) - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 21, normalized size = 1.11 \begin {gather*} b \log \left ({\left | b x + 1 \right |}\right ) - b \log \left ({\left | x \right |}\right ) - \frac {1}{x} \end {gather*}
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 \begin {gather*} 2\,b\,\mathrm {atanh}\left (2\,b\,x+1\right )-\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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