Integrand size = 26, antiderivative size = 113 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {37 \sqrt {1-2 x} (2+3 x)^2}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (173063+72060 x)}{96800}-\frac {35451 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{800 \sqrt {10}} \]
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Time = 0.02 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 155, 152, 56, 222} \[ \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {35451 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}}+\frac {7 (3 x+2)^3}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^2}{605 \sqrt {5 x+3}}+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (72060 x+173063)}{96800} \]
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Rule 56
Rule 100
Rule 152
Rule 155
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x)^2 \left (152+\frac {519 x}{2}\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^2}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{605} \int \frac {(2+3 x) \left (\frac {5487}{2}+\frac {18015 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^2}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (173063+72060 x)}{96800}-\frac {35451 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1600} \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^2}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (173063+72060 x)}{96800}-\frac {35451 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{800 \sqrt {5}} \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^2}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (173063+72060 x)}{96800}-\frac {35451 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{800 \sqrt {10}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.73 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {10 \left (2026687+2323271 x-1992870 x^2-392040 x^3\right )+4289571 \sqrt {10-20 x} \sqrt {3+5 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{968000 \sqrt {1-2 x} \sqrt {3+5 x}} \]
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Time = 1.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.21
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \left (42895710 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-7840800 x^{3} \sqrt {-10 x^{2}-x +3}+4289571 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -39857400 x^{2} \sqrt {-10 x^{2}-x +3}-12868713 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+46465420 x \sqrt {-10 x^{2}-x +3}+40533740 \sqrt {-10 x^{2}-x +3}\right )}{1936000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(137\) |
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Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {4289571 \, \sqrt {10} {\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (392040 \, x^{3} + 1992870 \, x^{2} - 2323271 \, x - 2026687\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1936000 \, {\left (10 \, x^{2} + x - 3\right )}} \]
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\[ \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {\left (3 x + 2\right )^{4}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {81 \, x^{3}}{20 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1647 \, x^{2}}{80 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {35451}{16000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {2323271 \, x}{96800 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {2026687}{96800 \, \sqrt {-10 \, x^{2} - x + 3}} \]
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Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.16 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {35451}{8000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (6534 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 197 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 21456431 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{12100000 \, {\left (2 \, x - 1\right )}} - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{151250 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{75625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
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Timed out. \[ \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^4}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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