Optimal. Leaf size=14 \[ -\frac{1}{2 b (a+b x)^2} \]
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Rubi [A] time = 0.0024815, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 32} \[ -\frac{1}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 32
Rubi steps
\begin{align*} \int \frac{a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^3} \, dx\\ &=-\frac{1}{2 b (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0026421, size = 14, normalized size = 1. \[ -\frac{1}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,b \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968606, size = 31, normalized size = 2.21 \begin{align*} -\frac{1}{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49037, size = 49, normalized size = 3.5 \begin{align*} -\frac{1}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.326833, size = 26, normalized size = 1.86 \begin{align*} - \frac{1}{2 a^{2} b + 4 a b^{2} x + 2 b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14184, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2 \,{\left (b x + a\right )}^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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