Integrand size = 22, antiderivative size = 66 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx=\frac {3773}{48 (1-2 x)^{3/2}}-\frac {3283}{8 \sqrt {1-2 x}}-\frac {1071}{4} \sqrt {1-2 x}+\frac {207}{8} (1-2 x)^{3/2}-\frac {27}{16} (1-2 x)^{5/2} \]
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Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx=-\frac {27}{16} (1-2 x)^{5/2}+\frac {207}{8} (1-2 x)^{3/2}-\frac {1071}{4} \sqrt {1-2 x}-\frac {3283}{8 \sqrt {1-2 x}}+\frac {3773}{48 (1-2 x)^{3/2}} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3773}{16 (1-2 x)^{5/2}}-\frac {3283}{8 (1-2 x)^{3/2}}+\frac {1071}{4 \sqrt {1-2 x}}-\frac {621}{8} \sqrt {1-2 x}+\frac {135}{16} (1-2 x)^{3/2}\right ) \, dx \\ & = \frac {3773}{48 (1-2 x)^{3/2}}-\frac {3283}{8 \sqrt {1-2 x}}-\frac {1071}{4} \sqrt {1-2 x}+\frac {207}{8} (1-2 x)^{3/2}-\frac {27}{16} (1-2 x)^{5/2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.50 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx=-\frac {1726-5250 x+2403 x^2+459 x^3+81 x^4}{3 (1-2 x)^{3/2}} \]
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Time = 1.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45
method | result | size |
gosper | \(-\frac {81 x^{4}+459 x^{3}+2403 x^{2}-5250 x +1726}{3 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(30\) |
pseudoelliptic | \(\frac {-81 x^{4}-459 x^{3}-2403 x^{2}+5250 x -1726}{3 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(30\) |
trager | \(-\frac {\left (81 x^{4}+459 x^{3}+2403 x^{2}-5250 x +1726\right ) \sqrt {1-2 x}}{3 \left (-1+2 x \right )^{2}}\) | \(37\) |
risch | \(\frac {81 x^{4}+459 x^{3}+2403 x^{2}-5250 x +1726}{3 \left (-1+2 x \right ) \sqrt {1-2 x}}\) | \(37\) |
derivativedivides | \(\frac {3773}{48 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {207 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {3283}{8 \sqrt {1-2 x}}-\frac {1071 \sqrt {1-2 x}}{4}\) | \(47\) |
default | \(\frac {3773}{48 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {207 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {3283}{8 \sqrt {1-2 x}}-\frac {1071 \sqrt {1-2 x}}{4}\) | \(47\) |
meijerg | \(-\frac {16 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {\frac {148 \sqrt {\pi }}{3}-\frac {37 \sqrt {\pi }\, \left (-24 x +8\right )}{6 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {57 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {234 \sqrt {\pi }-\frac {117 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{64 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {45 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}\) | \(165\) |
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Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx=-\frac {{\left (81 \, x^{4} + 459 \, x^{3} + 2403 \, x^{2} - 5250 \, x + 1726\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
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Time = 0.88 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx=- \frac {27 \left (1 - 2 x\right )^{\frac {5}{2}}}{16} + \frac {207 \left (1 - 2 x\right )^{\frac {3}{2}}}{8} - \frac {1071 \sqrt {1 - 2 x}}{4} - \frac {3283}{8 \sqrt {1 - 2 x}} + \frac {3773}{48 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
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Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx=-\frac {27}{16} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {207}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1071}{4} \, \sqrt {-2 \, x + 1} + \frac {49 \, {\left (804 \, x - 325\right )}}{48 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx=-\frac {27}{16} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {207}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1071}{4} \, \sqrt {-2 \, x + 1} - \frac {49 \, {\left (804 \, x - 325\right )}}{48 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]
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Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx=\frac {\frac {3283\,x}{4}-\frac {15925}{48}}{{\left (1-2\,x\right )}^{3/2}}-\frac {1071\,\sqrt {1-2\,x}}{4}+\frac {207\,{\left (1-2\,x\right )}^{3/2}}{8}-\frac {27\,{\left (1-2\,x\right )}^{5/2}}{16} \]
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