Optimal. Leaf size=21 \[ \frac{2 \sqrt{d (a+b x)+c}}{b d} \]
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Rubi [A] time = 0.0103886, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {33, 32} \[ \frac{2 \sqrt{d (a+b x)+c}}{b d} \]
Antiderivative was successfully verified.
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Rule 33
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c+d (a+b x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x}} \, dx,x,a+b x\right )}{b}\\ &=\frac{2 \sqrt{c+d (a+b x)}}{b d}\\ \end{align*}
Mathematica [A] time = 0.0127891, size = 21, normalized size = 1. \[ \frac{2 \sqrt{d (a+b x)+c}}{b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{bdx+ad+c}}{bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03585, size = 26, normalized size = 1.24 \begin{align*} \frac{2 \, \sqrt{{\left (b x + a\right )} d + c}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49795, size = 42, normalized size = 2. \begin{align*} \frac{2 \, \sqrt{b d x + a d + c}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.19525, size = 31, normalized size = 1.48 \begin{align*} \begin{cases} \frac{x}{\sqrt{a d + c}} & \text{for}\: b = 0 \\\frac{x}{\sqrt{c}} & \text{for}\: d = 0 \\\frac{2 \sqrt{c + d \left (a + b x\right )}}{b d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13198, size = 26, normalized size = 1.24 \begin{align*} \frac{2 \, \sqrt{b d x + a d + c}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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