Optimal. Leaf size=29 \[ \frac {x \log (x)}{a (a+b x)}-\frac {\log (a+b x)}{a b} \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2314, 31} \[ \frac {x \log (x)}{a (a+b x)}-\frac {\log (a+b x)}{a b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2314
Rubi steps
\begin {align*} \int \frac {\log (x)}{(a+b x)^2} \, dx &=\frac {x \log (x)}{a (a+b x)}-\frac {\int \frac {1}{a+b x} \, dx}{a}\\ &=\frac {x \log (x)}{a (a+b x)}-\frac {\log (a+b x)}{a b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 0.93 \[ \frac {\frac {x \log (x)}{a+b x}-\frac {\log (a+b x)}{b}}{a} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log (x)}{(a+b x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.87, size = 34, normalized size = 1.17 \[ \frac {b x \log \relax (x) - {\left (b x + a\right )} \log \left (b x + a\right )}{a b^{2} x + a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.61, size = 138, normalized size = 4.76 \[ b^{2} {\left (\frac {\log \left (\frac {{\left (b x + a\right )}^{2} {\left | b \right |} {\left | \frac {a}{b x + a} - 1 \right |}}{b^{2} {\left | b x + a \right |}}\right )}{a b^{3}} + \frac {\log \left (-\frac {a + \frac {{\left (b x + a\right )} b {\left (\frac {a}{b x + a} - 1\right )} - a b}{b}}{b}\right )}{{\left ({\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a\right )} b^{3}} - \frac {\log \left ({\left | -{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} + a \right |}\right )}{a b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 30, normalized size = 1.03
method | result | size |
default | \(\frac {x \ln \relax (x )}{a \left (b x +a \right )}-\frac {\ln \left (b x +a \right )}{a b}\) | \(30\) |
norman | \(\frac {x \ln \relax (x )}{a \left (b x +a \right )}-\frac {\ln \left (b x +a \right )}{a b}\) | \(30\) |
risch | \(-\frac {\ln \relax (x )}{b \left (b x +a \right )}+\frac {\ln \left (-x \right )}{a b}-\frac {\ln \left (b x +a \right )}{a b}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 38, normalized size = 1.31 \[ -\frac {\frac {\log \left (b x + a\right )}{a} - \frac {\log \relax (x)}{a}}{b} - \frac {\log \relax (x)}{{\left (b x + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 35, normalized size = 1.21 \[ \frac {x^2\,\ln \relax (x)}{a\,\left (b\,x^2+a\,x\right )}-\frac {\ln \left (a+b\,x\right )}{a\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 24, normalized size = 0.83 \[ - \frac {\log {\relax (x )}}{a b + b^{2} x} + \frac {\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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