Optimal. Leaf size=30 \[ \frac {A \log (x)}{a}-\frac {(A b-a B) \log (a+b x)}{a b} \]
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Rubi [A]
time = 0.01, antiderivative size = 30, 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 (x)}{a}-\frac {(A b-a B) \log (a+b x)}{a b} \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)} \, dx &=\int \left (\frac {A}{a x}+\frac {-A b+a B}{a (a+b x)}\right ) \, dx\\ &=\frac {A \log (x)}{a}-\frac {(A b-a B) \log (a+b x)}{a b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 0.97 \begin {gather*} \frac {A \log (x)}{a}+\frac {(-A b+a B) \log (a+b x)}{a b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 30, normalized size = 1.00
method | result | size |
default | \(\frac {\left (-A b +B a \right ) \ln \left (b x +a \right )}{a b}+\frac {A \ln \left (x \right )}{a}\) | \(30\) |
norman | \(\frac {A \ln \left (x \right )}{a}-\frac {\left (A b -B a \right ) \ln \left (b x +a \right )}{a b}\) | \(31\) |
risch | \(-\frac {\ln \left (b x +a \right ) A}{a}+\frac {\ln \left (b x +a \right ) B}{b}+\frac {A \ln \left (-x \right )}{a}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 29, normalized size = 0.97 \begin {gather*} \frac {A \log \left (x\right )}{a} + \frac {{\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.32, size = 28, normalized size = 0.93 \begin {gather*} \frac {A b \log \left (x\right ) + {\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 41, normalized size = 1.37 \begin {gather*} \frac {A \log {\left (x \right )}}{a} + \frac {\left (- A b + B a\right ) \log {\left (x + \frac {- A a + \frac {a \left (- A b + B a\right )}{b}}{- 2 A b + B a} \right )}}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.00, size = 31, normalized size = 1.03 \begin {gather*} \frac {A \log \left ({\left | x \right |}\right )}{a} + \frac {{\left (B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 28, normalized size = 0.93 \begin {gather*} \frac {A\,\ln \left (x\right )}{a}-\ln \left (a+b\,x\right )\,\left (\frac {A}{a}-\frac {B}{b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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