Optimal. Leaf size=82 \[ -\frac{B d-A e}{e (d+e x) (b d-a e)}+\frac{(A b-a B) \log (a+b x)}{(b d-a e)^2}-\frac{(A b-a B) \log (d+e x)}{(b d-a e)^2} \]
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Rubi [A] time = 0.0601968, antiderivative size = 82, 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 = {77} \[ -\frac{B d-A e}{e (d+e x) (b d-a e)}+\frac{(A b-a B) \log (a+b x)}{(b d-a e)^2}-\frac{(A b-a B) \log (d+e x)}{(b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{(a+b x) (d+e x)^2} \, dx &=\int \left (\frac{b (A b-a B)}{(b d-a e)^2 (a+b x)}+\frac{B d-A e}{(b d-a e) (d+e x)^2}+\frac{(-A b+a B) e}{(b d-a e)^2 (d+e x)}\right ) \, dx\\ &=-\frac{B d-A e}{e (b d-a e) (d+e x)}+\frac{(A b-a B) \log (a+b x)}{(b d-a e)^2}-\frac{(A b-a B) \log (d+e x)}{(b d-a e)^2}\\ \end{align*}
Mathematica [A] time = 0.0797107, size = 80, normalized size = 0.98 \[ \frac{B d-A e}{e (d+e x) (a e-b d)}+\frac{(A b-a B) \log (a+b x)}{(b d-a e)^2}+\frac{(a B-A b) \log (d+e x)}{(b d-a e)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 123, normalized size = 1.5 \begin{align*} -{\frac{A}{ \left ( ae-bd \right ) \left ( ex+d \right ) }}+{\frac{Bd}{e \left ( ae-bd \right ) \left ( ex+d \right ) }}-{\frac{\ln \left ( ex+d \right ) Ab}{ \left ( ae-bd \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ) Ba}{ \left ( ae-bd \right ) ^{2}}}+{\frac{\ln \left ( bx+a \right ) Ab}{ \left ( ae-bd \right ) ^{2}}}-{\frac{\ln \left ( bx+a \right ) Ba}{ \left ( ae-bd \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19441, size = 161, normalized size = 1.96 \begin{align*} -\frac{{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac{{\left (B a - A b\right )} \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac{B d - A e}{b d^{2} e - a d e^{2} +{\left (b d e^{2} - a e^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87502, size = 308, normalized size = 3.76 \begin{align*} -\frac{B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e +{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \log \left (b x + a\right ) -{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \log \left (e x + d\right )}{b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.29751, size = 355, normalized size = 4.33 \begin{align*} \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a b e - A b^{2} d + B a^{2} e + B a b d - \frac{a^{3} e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} + \frac{3 a^{2} b d e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} - \frac{3 a b^{2} d^{2} e \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} + \frac{b^{3} d^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}}}{- 2 A b^{2} e + 2 B a b e} \right )}}{\left (a e - b d\right )^{2}} - \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a b e - A b^{2} d + B a^{2} e + B a b d + \frac{a^{3} e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} - \frac{3 a^{2} b d e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} + \frac{3 a b^{2} d^{2} e \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} - \frac{b^{3} d^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}}}{- 2 A b^{2} e + 2 B a b e} \right )}}{\left (a e - b d\right )^{2}} + \frac{- A e + B d}{a d e^{2} - b d^{2} e + x \left (a e^{3} - b d e^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.92496, size = 149, normalized size = 1.82 \begin{align*} -\frac{{\left (B a e - A b e\right )} \log \left ({\left | -b + \frac{b d}{x e + d} - \frac{a e}{x e + d} \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} - \frac{\frac{B d}{x e + d} - \frac{A e}{x e + d}}{b d e - a e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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