Optimal. Leaf size=45 \[ \frac {20 \sqrt {5 x+3}}{363 \sqrt {1-2 x}}+\frac {2 \sqrt {5 x+3}}{33 (1-2 x)^{3/2}} \]
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Rubi [A] time = 0.00, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac {20 \sqrt {5 x+3}}{363 \sqrt {1-2 x}}+\frac {2 \sqrt {5 x+3}}{33 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {10}{33} \int \frac {1}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {20 \sqrt {3+5 x}}{363 \sqrt {1-2 x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.60 \[ -\frac {2 \sqrt {5 x+3} (20 x-21)}{363 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 33, normalized size = 0.73 \[ -\frac {2 \, {\left (20 \, x - 21\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{363 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 39, normalized size = 0.87 \[ -\frac {2 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} - 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1815 \, {\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.49 \[ -\frac {2 \sqrt {5 x +3}\, \left (20 x -21\right )}{363 \left (-2 x +1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 48, normalized size = 1.07 \[ \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{33 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {20 \, \sqrt {-10 \, x^{2} - x + 3}}{363 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 34, normalized size = 0.76 \[ \frac {\sqrt {5\,x+3}\,\left (\frac {20\,x}{363}-\frac {7}{121}\right )}{x\,\sqrt {1-2\,x}-\frac {\sqrt {1-2\,x}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.86, size = 177, normalized size = 3.93 \[ \begin {cases} \frac {100 \sqrt {10} \left (x + \frac {3}{5}\right )}{3630 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right ) - 3993 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}} - \frac {165 \sqrt {10}}{3630 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right ) - 3993 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\- \frac {100 \sqrt {10} i \left (x + \frac {3}{5}\right )}{3630 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right ) - 3993 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}} + \frac {165 \sqrt {10} i}{3630 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right ) - 3993 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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