Optimal. Leaf size=28 \[ -\frac {1}{a x}-\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, 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*} -\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2}-\frac {1}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)} \, dx &=\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx\\ &=-\frac {1}{a x}-\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -\frac {1}{a x}-\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 29, normalized size = 1.04
method | result | size |
default | \(-\frac {1}{a x}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b x +a \right )}{a^{2}}\) | \(29\) |
norman | \(-\frac {1}{a x}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b x +a \right )}{a^{2}}\) | \(29\) |
risch | \(-\frac {1}{a x}+\frac {b \ln \left (-b x -a \right )}{a^{2}}-\frac {b \ln \left (x \right )}{a^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.36, size = 28, normalized size = 1.00 \begin {gather*} \frac {b \log \left (b x + a\right )}{a^{2}} - \frac {b \log \left (x\right )}{a^{2}} - \frac {1}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.05, size = 26, normalized size = 0.93 \begin {gather*} \frac {b x \log \left (b x + a\right ) - b x \log \left (x\right ) - a}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 19, normalized size = 0.68 \begin {gather*} - \frac {1}{a x} + \frac {b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 30, normalized size = 1.07 \begin {gather*} \frac {b \log \left ({\left | b x + a \right |}\right )}{a^{2}} - \frac {b \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {1}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 25, normalized size = 0.89 \begin {gather*} \frac {2\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^2}-\frac {1}{a\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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