Integrand size = 22, antiderivative size = 66 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx=\frac {3773}{16 \sqrt {1-2 x}}+\frac {3283}{8} \sqrt {1-2 x}-\frac {357}{4} (1-2 x)^{3/2}+\frac {621}{40} (1-2 x)^{5/2}-\frac {135}{112} (1-2 x)^{7/2} \]
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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} \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx=-\frac {135}{112} (1-2 x)^{7/2}+\frac {621}{40} (1-2 x)^{5/2}-\frac {357}{4} (1-2 x)^{3/2}+\frac {3283}{8} \sqrt {1-2 x}+\frac {3773}{16 \sqrt {1-2 x}} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3773}{16 (1-2 x)^{3/2}}-\frac {3283}{8 \sqrt {1-2 x}}+\frac {1071}{4} \sqrt {1-2 x}-\frac {621}{8} (1-2 x)^{3/2}+\frac {135}{16} (1-2 x)^{5/2}\right ) \, dx \\ & = \frac {3773}{16 \sqrt {1-2 x}}+\frac {3283}{8} \sqrt {1-2 x}-\frac {357}{4} (1-2 x)^{3/2}+\frac {621}{40} (1-2 x)^{5/2}-\frac {135}{112} (1-2 x)^{7/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.50 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx=\frac {19994-19154 x-6987 x^2-2997 x^3-675 x^4}{35 \sqrt {1-2 x}} \]
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Time = 1.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45
method | result | size |
gosper | \(-\frac {675 x^{4}+2997 x^{3}+6987 x^{2}+19154 x -19994}{35 \sqrt {1-2 x}}\) | \(30\) |
risch | \(-\frac {675 x^{4}+2997 x^{3}+6987 x^{2}+19154 x -19994}{35 \sqrt {1-2 x}}\) | \(30\) |
pseudoelliptic | \(\frac {-675 x^{4}-2997 x^{3}-6987 x^{2}-19154 x +19994}{35 \sqrt {1-2 x}}\) | \(30\) |
trager | \(\frac {\left (675 x^{4}+2997 x^{3}+6987 x^{2}+19154 x -19994\right ) \sqrt {1-2 x}}{-35+70 x}\) | \(37\) |
derivativedivides | \(-\frac {357 \left (1-2 x \right )^{\frac {3}{2}}}{4}+\frac {621 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {135 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {3773}{16 \sqrt {1-2 x}}+\frac {3283 \sqrt {1-2 x}}{8}\) | \(47\) |
default | \(-\frac {357 \left (1-2 x \right )^{\frac {3}{2}}}{4}+\frac {621 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {135 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {3773}{16 \sqrt {1-2 x}}+\frac {3283 \sqrt {1-2 x}}{8}\) | \(47\) |
meijerg | \(-\frac {24 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-148 \sqrt {\pi }+\frac {37 \sqrt {\pi }\, \left (-8 x +8\right )}{2 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {171 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{2 \sqrt {\pi }}+\frac {-\frac {702 \sqrt {\pi }}{5}+\frac {351 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{320 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {135 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}\) | \(165\) |
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Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.55 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx=\frac {{\left (675 \, x^{4} + 2997 \, x^{3} + 6987 \, x^{2} + 19154 \, x - 19994\right )} \sqrt {-2 \, x + 1}}{35 \, {\left (2 \, x - 1\right )}} \]
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Time = 0.85 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx=- \frac {135 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} + \frac {621 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} - \frac {357 \left (1 - 2 x\right )^{\frac {3}{2}}}{4} + \frac {3283 \sqrt {1 - 2 x}}{8} + \frac {3773}{16 \sqrt {1 - 2 x}} \]
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Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx=-\frac {135}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {621}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {357}{4} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {3283}{8} \, \sqrt {-2 \, x + 1} + \frac {3773}{16 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx=\frac {135}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {621}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {357}{4} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {3283}{8} \, \sqrt {-2 \, x + 1} + \frac {3773}{16 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx=\frac {3773}{16\,\sqrt {1-2\,x}}+\frac {3283\,\sqrt {1-2\,x}}{8}-\frac {357\,{\left (1-2\,x\right )}^{3/2}}{4}+\frac {621\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {135\,{\left (1-2\,x\right )}^{7/2}}{112} \]
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