Optimal. Leaf size=21 \[ -\frac{2}{b d \sqrt{d (a+b x)+c}} \]
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Rubi [A] time = 0.0105183, 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} \[ -\frac{2}{b d \sqrt{d (a+b x)+c}} \]
Antiderivative was successfully verified.
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Rule 33
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{(c+d (a+b x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(c+d x)^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{2}{b d \sqrt{c+d (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0130418, size = 21, normalized size = 1. \[ -\frac{2}{b d \sqrt{d (a+b x)+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 20, normalized size = 1. \begin{align*} -2\,{\frac{1}{\sqrt{bdx+ad+c}bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09262, 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 = 1.48526, size = 76, normalized size = 3.62 \begin{align*} -\frac{2 \, \sqrt{b d x + a d + c}}{b^{2} d^{2} x + a b d^{2} + b c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.39265, size = 58, normalized size = 2.76 \begin{align*} \begin{cases} \frac{x}{c^{\frac{3}{2}}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x}{\left (a d + c\right )^{\frac{3}{2}}} & \text{for}\: b = 0 \\\frac{x}{c^{\frac{3}{2}}} & \text{for}\: d = 0 \\- \frac{2 \sqrt{a d + b d x + c}}{a b d^{2} + b^{2} d^{2} x + b c d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16669, 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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