Optimal. Leaf size=22 \[ -\frac{2 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]
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Rubi [A] time = 0.0018495, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ -\frac{2 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x}}{11 \sqrt{3+5 x}}\\ \end{align*}
Mathematica [A] time = 0.0034236, size = 22, normalized size = 1. \[ -\frac{2 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 17, normalized size = 0.8 \begin{align*} -{\frac{2}{11}\sqrt{1-2\,x}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.45186, size = 28, normalized size = 1.27 \begin{align*} -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{11 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66171, size = 49, normalized size = 2.23 \begin{align*} -\frac{2 \, \sqrt{-2 \, x + 1}}{11 \, \sqrt{5 \, x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.06604, size = 53, normalized size = 2.41 \begin{align*} \begin{cases} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{55} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{55} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.84967, size = 82, normalized size = 3.73 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{110 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{55 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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