Optimal. Leaf size=53 \[ \frac{125}{56} (1-2 x)^{7/2}-\frac{165}{8} (1-2 x)^{5/2}+\frac{605}{8} (1-2 x)^{3/2}-\frac{1331}{8} \sqrt{1-2 x} \]
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Rubi [A] time = 0.0087757, 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} \[ \frac{125}{56} (1-2 x)^{7/2}-\frac{165}{8} (1-2 x)^{5/2}+\frac{605}{8} (1-2 x)^{3/2}-\frac{1331}{8} \sqrt{1-2 x} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(3+5 x)^3}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{1331}{8 \sqrt{1-2 x}}-\frac{1815}{8} \sqrt{1-2 x}+\frac{825}{8} (1-2 x)^{3/2}-\frac{125}{8} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac{1331}{8} \sqrt{1-2 x}+\frac{605}{8} (1-2 x)^{3/2}-\frac{165}{8} (1-2 x)^{5/2}+\frac{125}{56} (1-2 x)^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0095531, size = 28, normalized size = 0.53 \[ -\frac{1}{7} \sqrt{1-2 x} \left (125 x^3+390 x^2+575 x+764\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{125\,{x}^{3}+390\,{x}^{2}+575\,x+764}{7}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14607, size = 50, normalized size = 0.94 \begin{align*} \frac{125}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{165}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{605}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1331}{8} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6277, size = 74, normalized size = 1.4 \begin{align*} -\frac{1}{7} \,{\left (125 \, x^{3} + 390 \, x^{2} + 575 \, x + 764\right )} \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.7952, size = 190, normalized size = 3.58 \begin{align*} \begin{cases} - \frac{25 \sqrt{5} i \left (x + \frac{3}{5}\right )^{3} \sqrt{10 x - 5}}{7} - \frac{33 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{7} - \frac{242 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{35} - \frac{2662 \sqrt{5} i \sqrt{10 x - 5}}{175} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{25 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{3}}{7} - \frac{33 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2}}{7} - \frac{242 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{35} - \frac{2662 \sqrt{5} \sqrt{5 - 10 x}}{175} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.94007, size = 69, normalized size = 1.3 \begin{align*} -\frac{125}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{165}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{605}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1331}{8} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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