Integrand size = 24, antiderivative size = 105 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {2033647}{128 \sqrt {1-2 x}}+\frac {6206585}{128} \sqrt {1-2 x}-\frac {8117095}{384} (1-2 x)^{3/2}+\frac {1179381}{128} (1-2 x)^{5/2}-\frac {367155}{128} (1-2 x)^{7/2}+\frac {74667}{128} (1-2 x)^{9/2}-\frac {97605 (1-2 x)^{11/2}}{1408}+\frac {6075 (1-2 x)^{13/2}}{1664} \]
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Time = 0.02 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {6075 (1-2 x)^{13/2}}{1664}-\frac {97605 (1-2 x)^{11/2}}{1408}+\frac {74667}{128} (1-2 x)^{9/2}-\frac {367155}{128} (1-2 x)^{7/2}+\frac {1179381}{128} (1-2 x)^{5/2}-\frac {8117095}{384} (1-2 x)^{3/2}+\frac {6206585}{128} \sqrt {1-2 x}+\frac {2033647}{128 \sqrt {1-2 x}} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2033647}{128 (1-2 x)^{3/2}}-\frac {6206585}{128 \sqrt {1-2 x}}+\frac {8117095}{128} \sqrt {1-2 x}-\frac {5896905}{128} (1-2 x)^{3/2}+\frac {2570085}{128} (1-2 x)^{5/2}-\frac {672003}{128} (1-2 x)^{7/2}+\frac {97605}{128} (1-2 x)^{9/2}-\frac {6075}{128} (1-2 x)^{11/2}\right ) \, dx \\ & = \frac {2033647}{128 \sqrt {1-2 x}}+\frac {6206585}{128} \sqrt {1-2 x}-\frac {8117095}{384} (1-2 x)^{3/2}+\frac {1179381}{128} (1-2 x)^{5/2}-\frac {367155}{128} (1-2 x)^{7/2}+\frac {74667}{128} (1-2 x)^{9/2}-\frac {97605 (1-2 x)^{11/2}}{1408}+\frac {6075 (1-2 x)^{13/2}}{1664} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {21493640-21370088 x-10015804 x^2-7945164 x^3-5928885 x^4-3350637 x^5-1201635 x^6-200475 x^7}{429 \sqrt {1-2 x}} \]
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Time = 1.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {200475 x^{7}+1201635 x^{6}+3350637 x^{5}+5928885 x^{4}+7945164 x^{3}+10015804 x^{2}+21370088 x -21493640}{429 \sqrt {1-2 x}}\) | \(45\) |
risch | \(-\frac {200475 x^{7}+1201635 x^{6}+3350637 x^{5}+5928885 x^{4}+7945164 x^{3}+10015804 x^{2}+21370088 x -21493640}{429 \sqrt {1-2 x}}\) | \(45\) |
pseudoelliptic | \(\frac {-200475 x^{7}-1201635 x^{6}-3350637 x^{5}-5928885 x^{4}-7945164 x^{3}-10015804 x^{2}-21370088 x +21493640}{429 \sqrt {1-2 x}}\) | \(45\) |
trager | \(\frac {\left (200475 x^{7}+1201635 x^{6}+3350637 x^{5}+5928885 x^{4}+7945164 x^{3}+10015804 x^{2}+21370088 x -21493640\right ) \sqrt {1-2 x}}{-429+858 x}\) | \(52\) |
derivativedivides | \(-\frac {8117095 \left (1-2 x \right )^{\frac {3}{2}}}{384}+\frac {1179381 \left (1-2 x \right )^{\frac {5}{2}}}{128}-\frac {367155 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {74667 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {97605 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {6075 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {2033647}{128 \sqrt {1-2 x}}+\frac {6206585 \sqrt {1-2 x}}{128}\) | \(74\) |
default | \(-\frac {8117095 \left (1-2 x \right )^{\frac {3}{2}}}{384}+\frac {1179381 \left (1-2 x \right )^{\frac {5}{2}}}{128}-\frac {367155 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {74667 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {97605 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {6075 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {2033647}{128 \sqrt {1-2 x}}+\frac {6206585 \sqrt {1-2 x}}{128}\) | \(74\) |
meijerg | \(-\frac {288 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-3120 \sqrt {\pi }+\frac {390 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {3620 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-14928 \sqrt {\pi }+\frac {933 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {28845 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{8 \sqrt {\pi }}+\frac {-6792 \sqrt {\pi }+\frac {849 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{128 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {6885 \left (\frac {1024 \sqrt {\pi }}{231}-\frac {\sqrt {\pi }\, \left (-2688 x^{6}-1792 x^{5}-1280 x^{4}-1024 x^{3}-1024 x^{2}-2048 x +2048\right )}{462 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}+\frac {-\frac {32400 \sqrt {\pi }}{143}+\frac {2025 \sqrt {\pi }\, \left (-67584 x^{7}-43008 x^{6}-28672 x^{5}-20480 x^{4}-16384 x^{3}-16384 x^{2}-32768 x +32768\right )}{292864 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(324\) |
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Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.49 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {{\left (200475 \, x^{7} + 1201635 \, x^{6} + 3350637 \, x^{5} + 5928885 \, x^{4} + 7945164 \, x^{3} + 10015804 \, x^{2} + 21370088 \, x - 21493640\right )} \sqrt {-2 \, x + 1}}{429 \, {\left (2 \, x - 1\right )}} \]
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Time = 1.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {6075 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {97605 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} + \frac {74667 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} - \frac {367155 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} + \frac {1179381 \left (1 - 2 x\right )^{\frac {5}{2}}}{128} - \frac {8117095 \left (1 - 2 x\right )^{\frac {3}{2}}}{384} + \frac {6206585 \sqrt {1 - 2 x}}{128} + \frac {2033647}{128 \sqrt {1 - 2 x}} \]
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Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {6075}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {97605}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {74667}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {367155}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {1179381}{128} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {8117095}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6206585}{128} \, \sqrt {-2 \, x + 1} + \frac {2033647}{128 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.03 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {6075}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {97605}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {74667}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {367155}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {1179381}{128} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {8117095}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6206585}{128} \, \sqrt {-2 \, x + 1} + \frac {2033647}{128 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {2033647}{128\,\sqrt {1-2\,x}}+\frac {6206585\,\sqrt {1-2\,x}}{128}-\frac {8117095\,{\left (1-2\,x\right )}^{3/2}}{384}+\frac {1179381\,{\left (1-2\,x\right )}^{5/2}}{128}-\frac {367155\,{\left (1-2\,x\right )}^{7/2}}{128}+\frac {74667\,{\left (1-2\,x\right )}^{9/2}}{128}-\frac {97605\,{\left (1-2\,x\right )}^{11/2}}{1408}+\frac {6075\,{\left (1-2\,x\right )}^{13/2}}{1664} \]
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