Optimal. Leaf size=32 \[ \frac{e \log (a+b x)}{b^2}-\frac{b d-a e}{b^2 (a+b x)} \]
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Rubi [A] time = 0.0216175, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 43} \[ \frac{e \log (a+b x)}{b^2}-\frac{b d-a e}{b^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{d+e x}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{d+e x}{(a+b x)^2} \, dx\\ &=\int \left (\frac{b d-a e}{b (a+b x)^2}+\frac{e}{b (a+b x)}\right ) \, dx\\ &=-\frac{b d-a e}{b^2 (a+b x)}+\frac{e \log (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0110032, size = 31, normalized size = 0.97 \[ \frac{a e-b d}{b^2 (a+b x)}+\frac{e \log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 39, normalized size = 1.2 \begin{align*}{\frac{e\ln \left ( bx+a \right ) }{{b}^{2}}}+{\frac{ae}{{b}^{2} \left ( bx+a \right ) }}-{\frac{d}{b \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15544, size = 47, normalized size = 1.47 \begin{align*} -\frac{b d - a e}{b^{3} x + a b^{2}} + \frac{e \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 = 1.69713, size = 80, normalized size = 2.5 \begin{align*} -\frac{b d - a e -{\left (b e x + a e\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.391281, size = 27, normalized size = 0.84 \begin{align*} \frac{a e - b d}{a b^{2} + b^{3} x} + \frac{e \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 = 1.17031, size = 47, normalized size = 1.47 \begin{align*} \frac{e \log \left ({\left | b x + a \right |}\right )}{b^{2}} - \frac{b d - a e}{{\left (b x + a\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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