Optimal. Leaf size=14 \[ -\frac{2}{d \sqrt{c+d x}} \]
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Rubi [A] time = 0.0015836, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {32} \[ -\frac{2}{d \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 32
Rubi steps
\begin{align*} \int \frac{1}{(c+d x)^{3/2}} \, dx &=-\frac{2}{d \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0037861, size = 14, normalized size = 1. \[ -\frac{2}{d \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 13, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{d\sqrt{dx+c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931832, size = 16, normalized size = 1.14 \begin{align*} -\frac{2}{\sqrt{d x + c} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10794, size = 43, normalized size = 3.07 \begin{align*} -\frac{2 \, \sqrt{d x + c}}{d^{2} x + c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.056988, size = 12, normalized size = 0.86 \begin{align*} - \frac{2}{d \sqrt{c + d x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06052, size = 16, normalized size = 1.14 \begin{align*} -\frac{2}{\sqrt{d x + c} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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