Optimal. Leaf size=28 \[ -\frac{(d+e x)^2}{2 (a+b x)^2 (b d-a e)} \]
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Rubi [A] time = 0.0043435, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 37} \[ -\frac{(d+e x)^2}{2 (a+b x)^2 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{d+e x}{(a+b x)^3} \, dx\\ &=-\frac{(d+e x)^2}{2 (b d-a e) (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0092757, size = 26, normalized size = 0.93 \[ -\frac{a e+b (d+2 e x)}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 35, normalized size = 1.3 \begin{align*} -{\frac{e}{{b}^{2} \left ( bx+a \right ) }}-{\frac{-ae+bd}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973212, size = 51, normalized size = 1.82 \begin{align*} -\frac{2 \, b e x + b d + a e}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45195, size = 81, normalized size = 2.89 \begin{align*} -\frac{2 \, b e x + b d + a e}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.427896, size = 39, normalized size = 1.39 \begin{align*} - \frac{a e + b d + 2 b e x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10995, size = 35, normalized size = 1.25 \begin{align*} -\frac{2 \, b x e + b d + a e}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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