Optimal. Leaf size=42 \[ \frac {A b-a B}{a b (a+b x)}+\frac {A \log (x)}{a^2}-\frac {A \log (a+b x)}{a^2} \]
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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 = {78}
\begin {gather*} -\frac {A \log (a+b x)}{a^2}+\frac {A \log (x)}{a^2}+\frac {A b-a B}{a b (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
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 = 0.02, size = 38, normalized size = 0.90 \begin {gather*} \frac {\frac {a (A b-a B)}{b (a+b x)}+A \log (x)-A \log (a+b x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 44, normalized size = 1.05
method | result | size |
norman | \(-\frac {\left (A b -B a \right ) x}{a^{2} \left (b x +a \right )}+\frac {A \ln \left (x \right )}{a^{2}}-\frac {A \ln \left (b x +a \right )}{a^{2}}\) | \(42\) |
default | \(-\frac {-A b +B a}{a b \left (b x +a \right )}-\frac {A \ln \left (b x +a \right )}{a^{2}}+\frac {A \ln \left (x \right )}{a^{2}}\) | \(44\) |
risch | \(\frac {A}{a \left (b x +a \right )}-\frac {B}{b \left (b x +a \right )}+\frac {A \ln \left (-x \right )}{a^{2}}-\frac {A \ln \left (b x +a \right )}{a^{2}}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 44, normalized size = 1.05 \begin {gather*} -\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 \left (x\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.27, size = 62, normalized size = 1.48 \begin {gather*} -\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 \left (x\right )}{a^{2} b^{2} x + a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 32, normalized size = 0.76 \begin {gather*} \frac {A \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{2}} + \frac {A b - B a}{a^{2} b + a b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.73, size = 55, normalized size = 1.31 \begin {gather*} 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 )} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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