Optimal. Leaf size=57 \[ \frac {(A b-a B) \log (a+b x)}{b (b d-a e)}+\frac {(B d-A e) \log (d+e x)}{e (b d-a e)} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {72} \begin {gather*} \frac {(A b-a B) \log (a+b x)}{b (b d-a e)}+\frac {(B d-A e) \log (d+e x)}{e (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x) (d+e x)} \, dx &=\int \left (\frac {A b-a B}{(b d-a e) (a+b x)}+\frac {B d-A e}{(b d-a e) (d+e x)}\right ) \, dx\\ &=\frac {(A b-a B) \log (a+b x)}{b (b d-a e)}+\frac {(B d-A e) \log (d+e x)}{e (b d-a e)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.88 \begin {gather*} \frac {e (A b-a B) \log (a+b x)+b (B d-A e) \log (d+e x)}{b e (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{(a+b x) (d+e x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.10, size = 53, normalized size = 0.93 \begin {gather*} -\frac {{\left (B a - A b\right )} e \log \left (b x + a\right ) - {\left (B b d - A b e\right )} \log \left (e x + d\right )}{b^{2} d e - a b e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 63, normalized size = 1.11 \begin {gather*} -\frac {{\left (B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2} d - a b e} + \frac {{\left (B d - A e\right )} \log \left ({\left | x e + d \right |}\right )}{b d e - a e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 1.47 \begin {gather*} -\frac {A \ln \left (b x +a \right )}{a e -b d}+\frac {A \ln \left (e x +d \right )}{a e -b d}+\frac {B a \ln \left (b x +a \right )}{\left (a e -b d \right ) b}-\frac {B d \ln \left (e x +d \right )}{\left (a e -b d \right ) e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 58, normalized size = 1.02 \begin {gather*} -\frac {{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2} d - a b e} + \frac {{\left (B d - A e\right )} \log \left (e x + d\right )}{b d e - a e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 57, normalized size = 1.00 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,e-B\,d\right )}{a\,e^2-b\,d\,e}+\frac {\ln \left (a+b\,x\right )\,\left (A\,b-B\,a\right )}{b^2\,d-a\,b\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.56, size = 226, normalized size = 3.96 \begin {gather*} - \frac {\left (- A e + B d\right ) \log {\left (x + \frac {- A a e - A b d + 2 B a d - \frac {a^{2} e \left (- A e + B d\right )}{a e - b d} + \frac {2 a b d \left (- A e + B d\right )}{a e - b d} - \frac {b^{2} d^{2} \left (- A e + B d\right )}{e \left (a e - b d\right )}}{- 2 A b e + B a e + B b d} \right )}}{e \left (a e - b d\right )} + \frac {\left (- A b + B a\right ) \log {\left (x + \frac {- A a e - A b d + 2 B a d + \frac {a^{2} e^{2} \left (- A b + B a\right )}{b \left (a e - b d\right )} - \frac {2 a d e \left (- A b + B a\right )}{a e - b d} + \frac {b d^{2} \left (- A b + B a\right )}{a e - b d}}{- 2 A b e + B a e + B b d} \right )}}{b \left (a e - b d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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