Optimal. Leaf size=53 \[ \frac {25}{8} (1-2 x)^{5/2}-\frac {275}{8} (1-2 x)^{3/2}+\frac {1815}{8} \sqrt {1-2 x}+\frac {1331}{8 \sqrt {1-2 x}} \]
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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} \[ \frac {25}{8} (1-2 x)^{5/2}-\frac {275}{8} (1-2 x)^{3/2}+\frac {1815}{8} \sqrt {1-2 x}+\frac {1331}{8 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {1331}{8 (1-2 x)^{3/2}}-\frac {1815}{8 \sqrt {1-2 x}}+\frac {825}{8} \sqrt {1-2 x}-\frac {125}{8} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac {1331}{8 \sqrt {1-2 x}}+\frac {1815}{8} \sqrt {1-2 x}-\frac {275}{8} (1-2 x)^{3/2}+\frac {25}{8} (1-2 x)^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.47 \[ \frac {-25 x^3-100 x^2-335 x+362}{\sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 30, normalized size = 0.57 \[ \frac {{\left (25 \, x^{3} + 100 \, x^{2} + 335 \, x - 362\right )} \sqrt {-2 \, x + 1}}{2 \, x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 44, normalized size = 0.83 \[ \frac {25}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {275}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1815}{8} \, \sqrt {-2 \, x + 1} + \frac {1331}{8 \, \sqrt {-2 \, x + 1}} \]
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 \[ -\frac {25 x^{3}+100 x^{2}+335 x -362}{\sqrt {-2 x +1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 37, normalized size = 0.70 \[ \frac {25}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {275}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1815}{8} \, \sqrt {-2 \, x + 1} + \frac {1331}{8 \, \sqrt {-2 \, x + 1}} \]
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 \[ \frac {1331}{8\,\sqrt {1-2\,x}}+\frac {1815\,\sqrt {1-2\,x}}{8}-\frac {275\,{\left (1-2\,x\right )}^{3/2}}{8}+\frac {25\,{\left (1-2\,x\right )}^{5/2}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.15, size = 435, normalized size = 8.21 \[ \begin {cases} \frac {125 \sqrt {55} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {275 \sqrt {55} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {1210 \sqrt {55} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {26620 \sqrt {5} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {2662 \sqrt {55} i \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {29282 \sqrt {5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {125 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {275 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {1210 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {2662 \sqrt {55} \sqrt {5 - 10 x}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {26620 \sqrt {5} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {29282 \sqrt {5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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