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 = 0.0387887, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {72} \[ \frac{(A b-a B) \log (a+b x)}{b (b d-a e)}+\frac{(B d-A e) \log (d+e x)}{e (b d-a e)} \]
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*}
Mathematica [A] time = 0.0272645, size = 50, normalized size = 0.88 \[ \frac{e (A b-a B) \log (a+b x)+b (B d-A e) \log (d+e x)}{b e (b d-a e)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 84, normalized size = 1.5 \begin{align*}{\frac{\ln \left ( ex+d \right ) A}{ae-bd}}-{\frac{\ln \left ( ex+d \right ) Bd}{e \left ( ae-bd \right ) }}-{\frac{\ln \left ( bx+a \right ) A}{ae-bd}}+{\frac{\ln \left ( bx+a \right ) Ba}{b \left ( ae-bd \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15106, size = 78, normalized size = 1.37 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8208, size = 111, normalized size = 1.95 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.30093, size = 226, normalized size = 3.96 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 4.07546, size = 165, normalized size = 2.89 \begin{align*} \frac{B e^{\left (-1\right )} \log \left ({\left | b x^{2} e + b d x + a x e + a d \right |}\right )}{2 \, b} - \frac{{\left (B b d + B a e - 2 \, A b e\right )} e^{\left (-1\right )} \log \left (\frac{{\left | 2 \, b x e + b d + a e -{\left | b d - a e \right |} \right |}}{{\left | 2 \, b x e + b d + a e +{\left | b d - a e \right |} \right |}}\right )}{2 \, b{\left | b d - a e \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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