Integrand size = 17, antiderivative size = 53 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {1331}{8 \sqrt {1-2 x}}+\frac {1815}{8} \sqrt {1-2 x}-\frac {275}{8} (1-2 x)^{3/2}+\frac {25}{8} (1-2 x)^{5/2} \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {25}{8} (1-2 x)^{5/2}-\frac {275}{8} (1-2 x)^{3/2}+\frac {1815}{8} \sqrt {1-2 x}+\frac {1331}{8 \sqrt {1-2 x}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.47 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {362-335 x-100 x^2-25 x^3}{\sqrt {1-2 x}} \]
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Time = 1.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45
method | result | size |
pseudoelliptic | \(\frac {-25 x^{3}-100 x^{2}-335 x +362}{\sqrt {1-2 x}}\) | \(24\) |
gosper | \(-\frac {25 x^{3}+100 x^{2}+335 x -362}{\sqrt {1-2 x}}\) | \(25\) |
risch | \(-\frac {25 x^{3}+100 x^{2}+335 x -362}{\sqrt {1-2 x}}\) | \(25\) |
trager | \(\frac {\left (25 x^{3}+100 x^{2}+335 x -362\right ) \sqrt {1-2 x}}{-1+2 x}\) | \(31\) |
derivativedivides | \(-\frac {275 \left (1-2 x \right )^{\frac {3}{2}}}{8}+\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {1331}{8 \sqrt {1-2 x}}+\frac {1815 \sqrt {1-2 x}}{8}\) | \(38\) |
default | \(-\frac {275 \left (1-2 x \right )^{\frac {3}{2}}}{8}+\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {1331}{8 \sqrt {1-2 x}}+\frac {1815 \sqrt {1-2 x}}{8}\) | \(38\) |
meijerg | \(-\frac {27 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-135 \sqrt {\pi }+\frac {135 \sqrt {\pi }\, \left (-8 x +8\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {225 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{4 \sqrt {\pi }}+\frac {-50 \sqrt {\pi }+\frac {25 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{64 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(122\) |
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Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.57 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {{\left (25 \, x^{3} + 100 \, x^{2} + 335 \, x - 362\right )} \sqrt {-2 \, x + 1}}{2 \, x - 1} \]
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Result contains complex when optimal does not.
Time = 1.15 (sec) , antiderivative size = 434, normalized size of antiderivative = 8.19 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\begin {cases} \frac {125 \sqrt {55} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {275 \sqrt {55} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {1210 \sqrt {55} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {26620 \sqrt {5} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {2662 \sqrt {55} i \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {29282 \sqrt {5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {125 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {275 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {1210 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {2662 \sqrt {55} \sqrt {5 - 10 x}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {26620 \sqrt {5} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {29282 \sqrt {5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {25}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {275}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1815}{8} \, \sqrt {-2 \, x + 1} + \frac {1331}{8 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {25}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {275}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1815}{8} \, \sqrt {-2 \, x + 1} + \frac {1331}{8 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {1331}{8\,\sqrt {1-2\,x}}+\frac {1815\,\sqrt {1-2\,x}}{8}-\frac {275\,{\left (1-2\,x\right )}^{3/2}}{8}+\frac {25\,{\left (1-2\,x\right )}^{5/2}}{8} \]
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