Optimal. Leaf size=38 \[ \frac{2 (b c-a d)}{d^2 \sqrt{c+d x}}+\frac{2 b \sqrt{c+d x}}{d^2} \]
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Rubi [A] time = 0.0142636, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{2 (b c-a d)}{d^2 \sqrt{c+d x}}+\frac{2 b \sqrt{c+d x}}{d^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{(c+d x)^{3/2}} \, dx &=\int \left (\frac{-b c+a d}{d (c+d x)^{3/2}}+\frac{b}{d \sqrt{c+d x}}\right ) \, dx\\ &=\frac{2 (b c-a d)}{d^2 \sqrt{c+d x}}+\frac{2 b \sqrt{c+d x}}{d^2}\\ \end{align*}
Mathematica [A] time = 0.0193362, size = 27, normalized size = 0.71 \[ \frac{2 (-a d+2 b c+b d x)}{d^2 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 0.7 \begin{align*} -2\,{\frac{-bdx+ad-2\,bc}{\sqrt{dx+c}{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946742, size = 50, normalized size = 1.32 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{d x + c} b}{d} + \frac{b c - a d}{\sqrt{d x + c} d}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97996, size = 74, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (b d x + 2 \, b c - a d\right )} \sqrt{d x + c}}{d^{3} x + c d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.577191, size = 60, normalized size = 1.58 \begin{align*} \begin{cases} - \frac{2 a}{d \sqrt{c + d x}} + \frac{4 b c}{d^{2} \sqrt{c + d x}} + \frac{2 b x}{d \sqrt{c + d x}} & \text{for}\: d \neq 0 \\\frac{a x + \frac{b x^{2}}{2}}{c^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06666, size = 46, normalized size = 1.21 \begin{align*} \frac{2 \, \sqrt{d x + c} b}{d^{2}} + \frac{2 \,{\left (b c - a d\right )}}{\sqrt{d x + c} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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