Optimal. Leaf size=12 \[ -\frac{1}{d (c+d x)} \]
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Rubi [A] time = 0.0081565, antiderivative size = 12, 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 = {626, 32} \[ -\frac{1}{d (c+d x)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 32
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx &=\int \frac{1}{(c+d x)^2} \, dx\\ &=-\frac{1}{d (c+d x)}\\ \end{align*}
Mathematica [A] time = 0.0029981, size = 12, normalized size = 1. \[ -\frac{1}{d (c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 13, normalized size = 1.1 \begin{align*} -{\frac{1}{d \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02133, size = 18, normalized size = 1.5 \begin{align*} -\frac{1}{d^{2} x + c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51142, size = 24, normalized size = 2. \begin{align*} -\frac{1}{d^{2} x + c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.325532, size = 10, normalized size = 0.83 \begin{align*} - \frac{1}{c d + d^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17501, size = 16, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (d x + c\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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