Integrand size = 26, antiderivative size = 186 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {118054167 \sqrt {1-2 x} \sqrt {3+5 x}}{320000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {3+5 x}}{32000000}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {1298595837 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320000000 \sqrt {10}} \]
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Time = 0.05 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 158, 152, 52, 56, 222} \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=\frac {1298595837 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320000000 \sqrt {10}}-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}+\frac {13}{50} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3+\frac {111 (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^2}{5000}-\frac {(1-2 x)^{5/2} \sqrt {5 x+3} (1990620 x+2725981)}{8000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {5 x+3}}{32000000}+\frac {118054167 \sqrt {1-2 x} \sqrt {5 x+3}}{320000000} \]
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Rule 52
Rule 56
Rule 99
Rule 152
Rule 158
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {(2-39 x) (1-2 x)^{3/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {1}{150} \int \frac {(1-2 x)^{3/2} (2+3 x)^2 \left (-162+\frac {333 x}{2}\right )}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}+\frac {\int \frac {(1-2 x)^{3/2} (2+3 x) \left (\frac {34731}{2}+\frac {99531 x}{4}\right )}{\sqrt {3+5 x}} \, dx}{7500} \\ & = -\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {3577399 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{3200000} \\ & = -\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {3577399 (1-2 x)^{3/2} \sqrt {3+5 x}}{32000000}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {118054167 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{64000000} \\ & = -\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {118054167 \sqrt {1-2 x} \sqrt {3+5 x}}{320000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {3+5 x}}{32000000}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {1298595837 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{640000000} \\ & = -\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {118054167 \sqrt {1-2 x} \sqrt {3+5 x}}{320000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {3+5 x}}{32000000}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {1298595837 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{320000000 \sqrt {5}} \\ & = -\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {118054167 \sqrt {1-2 x} \sqrt {3+5 x}}{320000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {3+5 x}}{32000000}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {1298595837 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320000000 \sqrt {10}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=\frac {10 \sqrt {1-2 x} \left (168414751+1366129125 x+938891620 x^2-3673002400 x^3-2530224000 x^4+4043520000 x^5+3456000000 x^6\right )-1298595837 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{3200000000 \sqrt {3+5 x}} \]
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Time = 1.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\left (69120000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+80870400000 x^{5} \sqrt {-10 x^{2}-x +3}-50604480000 x^{4} \sqrt {-10 x^{2}-x +3}-73460048000 x^{3} \sqrt {-10 x^{2}-x +3}+6492979185 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +18777832400 x^{2} \sqrt {-10 x^{2}-x +3}+3895787511 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+27322582500 x \sqrt {-10 x^{2}-x +3}+3368295020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{6400000000 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(167\) |
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Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.54 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=-\frac {1298595837 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (3456000000 \, x^{6} + 4043520000 \, x^{5} - 2530224000 \, x^{4} - 3673002400 \, x^{3} + 938891620 \, x^{2} + 1366129125 \, x + 168414751\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6400000000 \, {\left (5 \, x + 3\right )}} \]
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\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{4}}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=-\frac {108 \, x^{7}}{5 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1809 \, x^{6}}{125 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {284499 \, x^{5}}{10000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3009863 \, x^{4}}{200000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {138769641 \, x^{3}}{8000000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {179336663 \, x^{2}}{32000000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1298595837}{6400000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1029299623 \, x}{320000000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {168414751}{320000000 \, \sqrt {-10 \, x^{2} - x + 3}} \]
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Time = 0.45 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=\frac {1}{8000000000} \, {\left (4 \, {\left (8 \, {\left (108 \, {\left (16 \, {\left (20 \, \sqrt {5} {\left (5 \, x + 3\right )} - 243 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9263 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 2532859 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 3473645 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 533500275 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {1298595837}{3200000000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {121 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{781250 \, \sqrt {5 \, x + 3}} + \frac {242 \, \sqrt {10} \sqrt {5 \, x + 3}}{390625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4}{{\left (5\,x+3\right )}^{3/2}} \,d x \]
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