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 = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
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*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.53 \begin {gather*} -\frac {1}{7} \sqrt {1-2 x} \left (125 x^3+390 x^2+575 x+764\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 40, normalized size = 0.75 \begin {gather*} \frac {1}{56} \left (125 (1-2 x)^3-1155 (1-2 x)^2+4235 (1-2 x)-9317\right ) \sqrt {1-2 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.38, size = 24, normalized size = 0.45 \begin {gather*} -\frac {1}{7} \, {\left (125 \, x^{3} + 390 \, x^{2} + 575 \, x + 764\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 51, normalized size = 0.96 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.47 \begin {gather*} -\frac {\left (125 x^{3}+390 x^{2}+575 x +764\right ) \sqrt {-2 x +1}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 37, normalized size = 0.70 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 37, normalized size = 0.70 \begin {gather*} \frac {605\,{\left (1-2\,x\right )}^{3/2}}{8}-\frac {1331\,\sqrt {1-2\,x}}{8}-\frac {165\,{\left (1-2\,x\right )}^{5/2}}{8}+\frac {125\,{\left (1-2\,x\right )}^{7/2}}{56} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.05, size = 190, normalized size = 3.58 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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