Optimal. Leaf size=18 \[ \frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1598, 36, 29,
31} \begin {gather*} \frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 1598
Rubi steps
\begin {align*} \int \frac {x}{a x^2+b x^3} \, dx &=\int \frac {1}{x (a+b x)} \, dx\\ &=\frac {\int \frac {1}{x} \, dx}{a}-\frac {b \int \frac {1}{a+b x} \, dx}{a}\\ &=\frac {\log (x)}{a}-\frac {\log (a+b x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 19, normalized size = 1.06
method | result | size |
default | \(\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b x +a \right )}{a}\) | \(19\) |
norman | \(\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b x +a \right )}{a}\) | \(19\) |
risch | \(-\frac {\ln \left (b x +a \right )}{a}+\frac {\ln \left (-x \right )}{a}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b x + a\right )}{a} + \frac {\log \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.49, size = 16, normalized size = 0.89 \begin {gather*} -\frac {\log \left (b x + a\right ) - \log \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 10, normalized size = 0.56 \begin {gather*} \frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.90, size = 20, normalized size = 1.11 \begin {gather*} -\frac {\log \left ({\left | b x + a \right |}\right )}{a} + \frac {\log \left ({\left | x \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.12, size = 15, normalized size = 0.83 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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