Optimal. Leaf size=43 \[ -\frac {\log (x) (A b-a B)}{a^2}+\frac {(A b-a B) \log (a+b x)}{a^2}-\frac {A}{a x} \]
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Rubi [A] time = 0.03, antiderivative size = 43, 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} \[ -\frac {\log (x) (A b-a B)}{a^2}+\frac {(A b-a B) \log (a+b x)}{a^2}-\frac {A}{a x} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 (a+b x)} \, dx &=\int \left (\frac {A}{a x^2}+\frac {-A b+a B}{a^2 x}-\frac {b (-A b+a B)}{a^2 (a+b x)}\right ) \, dx\\ &=-\frac {A}{a x}-\frac {(A b-a B) \log (x)}{a^2}+\frac {(A b-a B) \log (a+b x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.98 \[ \frac {\log (x) (a B-A b)}{a^2}+\frac {(A b-a B) \log (a+b x)}{a^2}-\frac {A}{a x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 41, normalized size = 0.95 \[ -\frac {{\left (B a - A b\right )} x \log \left (b x + a\right ) - {\left (B a - A b\right )} x \log \relax (x) + A a}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 51, normalized size = 1.19 \[ \frac {{\left (B a - A b\right )} \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {A}{a x} - \frac {{\left (B a b - A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 1.19 \[ -\frac {A b \ln \relax (x )}{a^{2}}+\frac {A b \ln \left (b x +a \right )}{a^{2}}+\frac {B \ln \relax (x )}{a}-\frac {B \ln \left (b x +a \right )}{a}-\frac {A}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 43, normalized size = 1.00 \[ -\frac {{\left (B a - A b\right )} \log \left (b x + a\right )}{a^{2}} + \frac {{\left (B a - A b\right )} \log \relax (x)}{a^{2}} - \frac {A}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 33, normalized size = 0.77 \[ \frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )\,\left (A\,b-B\,a\right )}{a^2}-\frac {A}{a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.40, size = 95, normalized size = 2.21 \[ - \frac {A}{a x} + \frac {\left (- A b + B a\right ) \log {\left (x + \frac {- A a b + B a^{2} - a \left (- A b + B a\right )}{- 2 A b^{2} + 2 B a b} \right )}}{a^{2}} - \frac {\left (- A b + B a\right ) \log {\left (x + \frac {- A a b + B a^{2} + a \left (- A b + B a\right )}{- 2 A b^{2} + 2 B a b} \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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