Optimal. Leaf size=25 \[ \frac{(b d-a e) \log (a+b x)}{b^2}+\frac{e x}{b} \]
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Rubi [A] time = 0.0205974, antiderivative size = 25, 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, 43} \[ \frac{(b d-a e) \log (a+b x)}{b^2}+\frac{e x}{b} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
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{d+e x}{a+b x} \, dx\\ &=\int \left (\frac{e}{b}+\frac{b d-a e}{b (a+b x)}\right ) \, dx\\ &=\frac{e x}{b}+\frac{(b d-a e) \log (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0071223, size = 25, normalized size = 1. \[ \frac{(b d-a e) \log (a+b x)}{b^2}+\frac{e x}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 1.3 \begin{align*}{\frac{ex}{b}}-{\frac{\ln \left ( bx+a \right ) ae}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) d}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957427, size = 34, normalized size = 1.36 \begin{align*} \frac{e x}{b} + \frac{{\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46514, size = 54, normalized size = 2.16 \begin{align*} \frac{b e x +{\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.308245, size = 20, normalized size = 0.8 \begin{align*} \frac{e x}{b} - \frac{\left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09236, size = 38, normalized size = 1.52 \begin{align*} \frac{x e}{b} + \frac{{\left (b d - a e\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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