Optimal. Leaf size=53 \[ \frac{125}{72} (1-2 x)^{9/2}-\frac{825}{56} (1-2 x)^{7/2}+\frac{363}{8} (1-2 x)^{5/2}-\frac{1331}{24} (1-2 x)^{3/2} \]
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Rubi [A] time = 0.009292, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ \frac{125}{72} (1-2 x)^{9/2}-\frac{825}{56} (1-2 x)^{7/2}+\frac{363}{8} (1-2 x)^{5/2}-\frac{1331}{24} (1-2 x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \sqrt{1-2 x} (3+5 x)^3 \, dx &=\int \left (\frac{1331}{8} \sqrt{1-2 x}-\frac{1815}{8} (1-2 x)^{3/2}+\frac{825}{8} (1-2 x)^{5/2}-\frac{125}{8} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac{1331}{24} (1-2 x)^{3/2}+\frac{363}{8} (1-2 x)^{5/2}-\frac{825}{56} (1-2 x)^{7/2}+\frac{125}{72} (1-2 x)^{9/2}\\ \end{align*}
Mathematica [A] time = 0.0114776, size = 28, normalized size = 0.53 \[ -\frac{1}{63} (1-2 x)^{3/2} \left (875 x^3+2400 x^2+2661 x+1454\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{875\,{x}^{3}+2400\,{x}^{2}+2661\,x+1454}{63} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02515, size = 50, normalized size = 0.94 \begin{align*} \frac{125}{72} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{825}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{363}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1331}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55671, size = 93, normalized size = 1.75 \begin{align*} \frac{1}{63} \,{\left (1750 \, x^{4} + 3925 \, x^{3} + 2922 \, x^{2} + 247 \, x - 1454\right )} \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.767, size = 236, normalized size = 4.45 \begin{align*} \begin{cases} \frac{50 \sqrt{5} i \left (x + \frac{3}{5}\right )^{4} \sqrt{10 x - 5}}{9} - \frac{55 \sqrt{5} i \left (x + \frac{3}{5}\right )^{3} \sqrt{10 x - 5}}{63} - \frac{121 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{105} - \frac{2662 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{1575} - \frac{29282 \sqrt{5} i \sqrt{10 x - 5}}{7875} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{50 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{4}}{9} - \frac{55 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{3}}{63} - \frac{121 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2}}{105} - \frac{2662 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{1575} - \frac{29282 \sqrt{5} \sqrt{5 - 10 x}}{7875} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.14161, size = 78, normalized size = 1.47 \begin{align*} \frac{125}{72} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{825}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{363}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{1331}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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