Integrand size = 24, antiderivative size = 79 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {65219}{32 \sqrt {1-2 x}}+\frac {144837}{32} \sqrt {1-2 x}-\frac {21439}{16} (1-2 x)^{3/2}+\frac {5711}{16} (1-2 x)^{5/2}-\frac {12675}{224} (1-2 x)^{7/2}+\frac {125}{32} (1-2 x)^{9/2} \]
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Time = 0.01 (sec) , antiderivative size = 79, 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} \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {125}{32} (1-2 x)^{9/2}-\frac {12675}{224} (1-2 x)^{7/2}+\frac {5711}{16} (1-2 x)^{5/2}-\frac {21439}{16} (1-2 x)^{3/2}+\frac {144837}{32} \sqrt {1-2 x}+\frac {65219}{32 \sqrt {1-2 x}} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {65219}{32 (1-2 x)^{3/2}}-\frac {144837}{32 \sqrt {1-2 x}}+\frac {64317}{16} \sqrt {1-2 x}-\frac {28555}{16} (1-2 x)^{3/2}+\frac {12675}{32} (1-2 x)^{5/2}-\frac {1125}{32} (1-2 x)^{7/2}\right ) \, dx \\ & = \frac {65219}{32 \sqrt {1-2 x}}+\frac {144837}{32} \sqrt {1-2 x}-\frac {21439}{16} (1-2 x)^{3/2}+\frac {5711}{16} (1-2 x)^{5/2}-\frac {12675}{224} (1-2 x)^{7/2}+\frac {125}{32} (1-2 x)^{9/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {38700-37944 x-15948 x^2-9501 x^3-4150 x^4-875 x^5}{7 \sqrt {1-2 x}} \]
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Time = 1.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {875 x^{5}+4150 x^{4}+9501 x^{3}+15948 x^{2}+37944 x -38700}{7 \sqrt {1-2 x}}\) | \(35\) |
risch | \(-\frac {875 x^{5}+4150 x^{4}+9501 x^{3}+15948 x^{2}+37944 x -38700}{7 \sqrt {1-2 x}}\) | \(35\) |
pseudoelliptic | \(\frac {-875 x^{5}-4150 x^{4}-9501 x^{3}-15948 x^{2}-37944 x +38700}{7 \sqrt {1-2 x}}\) | \(35\) |
trager | \(\frac {\left (875 x^{5}+4150 x^{4}+9501 x^{3}+15948 x^{2}+37944 x -38700\right ) \sqrt {1-2 x}}{-7+14 x}\) | \(42\) |
derivativedivides | \(-\frac {21439 \left (1-2 x \right )^{\frac {3}{2}}}{16}+\frac {5711 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {12675 \left (1-2 x \right )^{\frac {7}{2}}}{224}+\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{32}+\frac {65219}{32 \sqrt {1-2 x}}+\frac {144837 \sqrt {1-2 x}}{32}\) | \(56\) |
default | \(-\frac {21439 \left (1-2 x \right )^{\frac {3}{2}}}{16}+\frac {5711 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {12675 \left (1-2 x \right )^{\frac {7}{2}}}{224}+\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{32}+\frac {65219}{32 \sqrt {1-2 x}}+\frac {144837 \sqrt {1-2 x}}{32}\) | \(56\) |
meijerg | \(-\frac {108 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-864 \sqrt {\pi }+\frac {108 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {2763 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{4 \sqrt {\pi }}+\frac {-1766 \sqrt {\pi }+\frac {883 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{64 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {3525 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}+\frac {-\frac {1000 \sqrt {\pi }}{7}+\frac {125 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{896 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(213\) |
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Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {{\left (875 \, x^{5} + 4150 \, x^{4} + 9501 \, x^{3} + 15948 \, x^{2} + 37944 \, x - 38700\right )} \sqrt {-2 \, x + 1}}{7 \, {\left (2 \, x - 1\right )}} \]
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Time = 0.91 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {125 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} - \frac {12675 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} + \frac {5711 \left (1 - 2 x\right )^{\frac {5}{2}}}{16} - \frac {21439 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} + \frac {144837 \sqrt {1 - 2 x}}{32} + \frac {65219}{32 \sqrt {1 - 2 x}} \]
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Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {125}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {12675}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {5711}{16} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {21439}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {144837}{32} \, \sqrt {-2 \, x + 1} + \frac {65219}{32 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {125}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {12675}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {5711}{16} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {21439}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {144837}{32} \, \sqrt {-2 \, x + 1} + \frac {65219}{32 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {65219}{32\,\sqrt {1-2\,x}}+\frac {144837\,\sqrt {1-2\,x}}{32}-\frac {21439\,{\left (1-2\,x\right )}^{3/2}}{16}+\frac {5711\,{\left (1-2\,x\right )}^{5/2}}{16}-\frac {12675\,{\left (1-2\,x\right )}^{7/2}}{224}+\frac {125\,{\left (1-2\,x\right )}^{9/2}}{32} \]
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