Optimal. Leaf size=60 \[ -\frac{(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac{\log (a+b x) (-2 a B e+A b e+b B d)}{b^3}+\frac{B e x}{b^2} \]
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Rubi [A] time = 0.0560364, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 77} \[ -\frac{(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac{\log (a+b x) (-2 a B e+A b e+b B d)}{b^3}+\frac{B e x}{b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(A+B x) (d+e x)}{(a+b x)^2} \, dx\\ &=\int \left (\frac{B e}{b^2}+\frac{(A b-a B) (b d-a e)}{b^2 (a+b x)^2}+\frac{b B d+A b e-2 a B e}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{B e x}{b^2}-\frac{(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac{(b B d+A b e-2 a B e) \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0474741, size = 56, normalized size = 0.93 \[ \frac{-\frac{(A b-a B) (b d-a e)}{a+b x}+\log (a+b x) (-2 a B e+A b e+b B d)+b B e x}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 106, normalized size = 1.8 \begin{align*}{\frac{Bex}{{b}^{2}}}+{\frac{aAe}{{b}^{2} \left ( bx+a \right ) }}-{\frac{Ad}{b \left ( bx+a \right ) }}-{\frac{Be{a}^{2}}{{b}^{3} \left ( bx+a \right ) }}+{\frac{aBd}{{b}^{2} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) Ae}{{b}^{2}}}-2\,{\frac{\ln \left ( bx+a \right ) aBe}{{b}^{3}}}+{\frac{\ln \left ( bx+a \right ) Bd}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05196, size = 104, normalized size = 1.73 \begin{align*} \frac{B e x}{b^{2}} + \frac{{\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e}{b^{4} x + a b^{3}} + \frac{{\left (B b d -{\left (2 \, B a - A b\right )} e\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47672, size = 224, normalized size = 3.73 \begin{align*} \frac{B b^{2} e x^{2} + B a b e x +{\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e +{\left (B a b d -{\left (2 \, B a^{2} - A a b\right )} e +{\left (B b^{2} d -{\left (2 \, B a b - A b^{2}\right )} e\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.665877, size = 71, normalized size = 1.18 \begin{align*} \frac{B e x}{b^{2}} - \frac{- A a b e + A b^{2} d + B a^{2} e - B a b d}{a b^{3} + b^{4} x} - \frac{\left (- A b e + 2 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16444, size = 100, normalized size = 1.67 \begin{align*} \frac{B x e}{b^{2}} + \frac{{\left (B b d - 2 \, B a e + A b e\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{B a b d - A b^{2} d - B a^{2} e + A a b e}{{\left (b x + a\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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