Optimal. Leaf size=42 \[ -\frac {A \log (a+b x)}{a^2}+\frac {A \log (x)}{a^2}+\frac {A b-a B}{a b (a+b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ -\frac {A \log (a+b x)}{a^2}+\frac {A \log (x)}{a^2}+\frac {A b-a B}{a b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x (a+b x)^2} \, dx &=\int \left (\frac {A}{a^2 x}+\frac {-A b+a B}{a (a+b x)^2}-\frac {A b}{a^2 (a+b x)}\right ) \, dx\\ &=\frac {A b-a B}{a b (a+b x)}+\frac {A \log (x)}{a^2}-\frac {A \log (a+b x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 0.90 \[ \frac {\frac {a (A b-a B)}{b (a+b x)}-A \log (a+b x)+A \log (x)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 62, normalized size = 1.48 \[ -\frac {B a^{2} - A a b + {\left (A b^{2} x + A a b\right )} \log \left (b x + a\right ) - {\left (A b^{2} x + A a b\right )} \log \relax (x)}{a^{2} b^{2} x + a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 55, normalized size = 1.31 \[ b {\left (\frac {A \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{2} b} - \frac {\frac {B a}{b x + a} - \frac {A b}{b x + a}}{a b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 1.10 \[ \frac {A}{\left (b x +a \right ) a}+\frac {A \ln \relax (x )}{a^{2}}-\frac {A \ln \left (b x +a \right )}{a^{2}}-\frac {B}{\left (b x +a \right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 44, normalized size = 1.05 \[ -\frac {B a - A b}{a b^{2} x + a^{2} b} - \frac {A \log \left (b x + a\right )}{a^{2}} + \frac {A \log \relax (x)}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 39, normalized size = 0.93 \[ \frac {A\,b-B\,a}{a\,b\,\left (a+b\,x\right )}-\frac {2\,A\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 32, normalized size = 0.76 \[ \frac {A \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{2}} + \frac {A b - B a}{a^{2} b + a b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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