Optimal. Leaf size=22 \[ -\frac {2 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}} \]
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Rubi [A] time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{3/2}}{33 (3+5 x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ -\frac {2 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.08, size = 33, normalized size = 1.50 \[ \frac {2 \, \sqrt {5 \, x + 3} {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}}{33 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.24, size = 125, normalized size = 5.68 \[ -\frac {1}{13200} \, \sqrt {5} {\left (\sqrt {2} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} - \frac {12 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {16 \, \sqrt {2} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {3 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.77 \[ -\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{33 \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.18, size = 48, normalized size = 2.18 \[ -\frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{15 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {4 \, \sqrt {-10 \, x^{2} - x + 3}}{165 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.06, size = 38, normalized size = 1.73 \[ \frac {\sqrt {5\,x+3}\,\left (\frac {4\,x\,\sqrt {1-2\,x}}{825}-\frac {2\,\sqrt {1-2\,x}}{825}\right )}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.68, size = 99, normalized size = 4.50 \[ \begin {cases} \frac {4 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{825} - \frac {2 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{375 \left (x + \frac {3}{5}\right )} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\\frac {4 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{825} - \frac {2 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{375 \left (x + \frac {3}{5}\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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