Optimal. Leaf size=45 \[ -\frac{8 \sqrt{1-2 x}}{363 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{33 (5 x+3)^{3/2}} \]
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Rubi [A] time = 0.0050273, antiderivative size = 45, normalized size of antiderivative = 1., 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} \[ -\frac{8 \sqrt{1-2 x}}{363 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{33 (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x}}{33 (3+5 x)^{3/2}}+\frac{4}{33} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x}}{33 (3+5 x)^{3/2}}-\frac{8 \sqrt{1-2 x}}{363 \sqrt{3+5 x}}\\ \end{align*}
Mathematica [A] time = 0.0059106, size = 27, normalized size = 0.6 \[ -\frac{2 \sqrt{1-2 x} (20 x+23)}{363 (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 22, normalized size = 0.5 \begin{align*} -{\frac{46+40\,x}{363}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.8694, size = 65, normalized size = 1.44 \begin{align*} -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{33 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{8 \, \sqrt{-10 \, x^{2} - x + 3}}{363 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71167, size = 93, normalized size = 2.07 \begin{align*} -\frac{2 \,{\left (20 \, x + 23\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{363 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.09515, size = 102, normalized size = 2.27 \begin{align*} \begin{cases} - \frac{8 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1815} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{825 \left (x + \frac{3}{5}\right )} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{8 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1815} - \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{825 \left (x + \frac{3}{5}\right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.17013, size = 170, normalized size = 3.78 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{29040 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{3 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2420 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{9 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1815 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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